q. yaWiaSvili avtomatizebuli marTvis modelebi. statistikuri modelebi

q. yaWiaSvili 

avtomatizebuli marTvis modelebi. 

statistikuri modelebi 

“teqnikuri universiteti”

saqarTvelos saqarTvelos teqnikuri teqnikuri universiteti 

universiteti 

q. q. yaWiaSvili 

yaWiaSvili 

avtomatizebuli avtomatizebuli marTvis marTvis modelebi. 

modelebi. 

statistikuri statistikuri modelebi 

modelebi 

Tbilisi 

Tbilisi 

2004 

2004 

damtkicebulia damtkicebulia stu stu – s 

s 

saswavlo saswavlo – meToduri meToduri 

sabWos sabWos mier 

mier

uak 519.2, 681.3 

saxelmZRvanelo gankuTvnilia 2202 – informaciis damuSavebisa da mar- 

Tvis avtomatizebuli sistemebis specialobis studentebisaTvis. 

masSi warmodgenilia gamoyenebiTi statistikis ZiriTadi cnebebi, albaTobebis 

ganawilebis mniSvnelovani kanonebi, statistikuri hipoTezebis Semowmebisa 

da SefasebaTa Teriis safuZvlebi, erTi da ori normaluri amonarCevis 

analizi, dispersiuli analizi, regresiuli analizi. masala warmodgenilia 

ara mkacri formaluri aparatis gamoyenebiT, rac aadvilebs mis Seswavlas 

teqnikuri dargis studentebisaTvis. 

saxelmZRvaneloSi warmodgenili masalis ukeT aTvisebisaTvis praqtikul 

mecadineobebze da damoukideblad garCevis mizniT, mocemulia agreTve sakmaod 

didi raodenobis mravalferovani praqtikuli amocanebi. saxelmZRvanelos 

boloSi moyvanilia yvela is statistikuri cxrili, romlebic saWiroa 

masSi warmodgenili meTodebis praqtikuli gamoyenebisaTvis; kerZod, 

saxelmZRvaneloSi mocemuli praqtikuli amocanebis gadawyvetisaTvis. 

saxelmZRvanelo gaTvaliswinebulia marTvis avtomatizebuli sistemebis 

studentebisaTvis, Tumca is agreTve sasargeblo iqneba sxva specialobis 

studentebisa da aspirantebisaTvis, romlebic dainteresebuli arian gamoyenebiTi 

statistikis Tanamedrove meTodebis SeiswavliT. 

recenzentebi: prof. n. jiblaZe 

doc. i. qarTveliSvili 

redaqtorebi: prof. g. gogiCaiSvili 

v. oTaraSvili 

gamomcemloba “teqnikuri universiteti”, 2004 

ISBN 99940-35-27-4

s a r C e v i 

Sesavali…………………………………………………………………………………6 

Tavi 1. gamoyenebiTi statistikis ZiriTadi cnebebi…………………………...9 G 

1.1 SemTxveviTi cvalebadoba………………………………………………...9 

1.2 xdomileba da misi albaToba……………………………………………10 

1.3 albaTobebis gazomva………………………………………………………12 

1.4 SemTxveviTi sidideebi. ganawilebis funqcia………………………....13 

1.5 albaTobebis ganawilebebis ricxviTi maxasiaTeblebi…………….15 

1.6 damoukidebeli da damokidebuli SemTxveviTi sidideebi………...18 

1.7 SemTxveviTi amonarCevi...............................................................................19 

1.8 amonarCevebi da maTi aRwera.....................................................................20 

1.9 rangi da ranJireba………………………………………………………….22 

1.10 aRweriTi statistikis meTodebi…………………………………………23 

Tavi 2. albaTobebis ganawilebis mniSvnelovani kanonebi……………………26 

2.1 binomialuri ganawileba……………………………………………………26 

2.2 puasonis ganawileba…………………………………………………………28 

2.3 maCvenebliani anu eqsponencialuri ganawileba………………………29 

2.4 normaluri ganawileba……………………………………………………...30 

2.5 organzomilebiani normaluri ganawileba……………………………..32 

2.6 normalur kanonTan dakavSirebuli ganawilebebi……………………33 

2.6.1 

2 

χ ganawileba…………………………………………………………..33 

2.6.2 stiudentis ganawileba………………………………………………..34 

2.6.3 fiSeris ganawileba……………………………………………………35 

2.7 Tanabari ganawilebis kanoni………………………………………………36 

Tavi 3. statistikuri hipoTezebis Semowmebis safuZvlebi…………………….38 

3.1 statistikuri modelebi……………………………………………………38 

3.2 statistikuri hipoTezebis Semowmeba……………………………………39 

3.3 statistikuri modelebisa da hipoTezebis magaliTebi……………...42 

3.4 statistikuri hipoTezebis Semowmeba (gamoyenebiTi amocanebi)…...45 

3.4.1. bernulis gamocdebis sqema.........................................................................45 

3.4.2. niSnebis kriteriumi erTi amonarCevisaTvis.......................................47 

3.5 hipoTezebis Semowmeba or amonarCevian amocanebSi………………...48 

3.5.1. mani uitnis kriteriumi……………………………………………………49 

3.5.2. uilkoksonis kriteriumi…………………………………………………51 

3.6 Sewyvilebuli dakvirvebebi……………………………………………….53 

3.6.1. niSnebis kriteriumi Sewyvilebuli amonarCevis analizisaTvis...53 

3.6.2. ganmeorebadi Sewyvilebuli dakvirvebebis analizi niSnebis 

rangebis mixedviT (uilkoksonis niSnebis rangebis jamebis 

kriteriumi)………………………………………………………………………54 

Tavi 4. SefasebaTa Teoriis safuZvlebi.................................................................56 

4.1 Sesavali…………………………………………………………………….…..56 

4

4.2 did ricxvTa kanoni………………………………………………………….58 

4.3 statistikuri parametrebi………………………………………………….59 

4.4 ganawilebis parametrebis Sefaseba amonarCeviT…………………….60 

4.5 Sefasebebis Tvisebebi. intervaluri Sefasebebi……………………...62 

4.6 maqsimaluri (udidesi) SesaZleblobebis meTodi................................63 

Tavi 5. erTi da ori normaluri amonarCevis analizi………………………65 

5.1 normaluri amonarCevis gamokvlevia…………………………………..65 

5.2 normalurobis Semowmebis grafikuli meTodi……………………….66 

5.3 normaluri ganawilebis parametrebis Sefaseba da maTi 

Tvisebebi……………………………………………………………………….67 

5.4 normaluri ganawilebis parametrebTan dakavSirebuli 

hipoTezebis Semowmeba………………………………………………………70 

5.4.1. erTi amonarCevi……………………………………………………………..70 

5.4.2. ori amonarCevi………………………………………………………………71 

5.4.3. Sewyvilebuli monacemebi………………………………………………….73 

Tavi 6. dispersiuli analizi………………………………………………………74 

6.1 amocanis dasma………………………………………………………………..74 

6.2 erTfaqtoruli dispersiuli analizi………………………………….75 

6.3 orfaqtoruli dispersiuli analizi.......................................................79 

Tavi 7. regresiuli analizi………………………………………………………84 

7.1 Sesavali………………………………………………………………………84 

7.2 miaxloebiTi regresiis gamoTvla da analizi………………………85 

7.3 wrfivi regresia……………………………………………………………88 

7.4 arawrfivi regresia……………………………………………………….90 

amocanebi praqtikuli mecadineobisaTvis……………………………………….…91 

literatura……………………………………………………………………………...102 

danarTi 1. binomialuri ganawilebis zeda boloebis albaTobebi …………105 

danarTi 2. puasonis ganawileba……………………………………………………..110 

danarTi 3. standartuli normaluri ganawilebis zeda bolos 

albaTobebi………………………………………………………………...118 

2 

danarTi 4. pirsonis ganawilebis kvantilebi χ ………………………..……119 

danarTi 5. stiudentis ganawilebis kvantilebi 

5 

1− p 

1 

2 

p t 

− 

………………….….…….121 

danarTi 6. fiSeris ganawilebis kvantilebi F1 − p ……………………..………..122 

danarTi 7. mani – uitnis kriteriumis U statistikis zeda kritikuli 

mniSvnelobebi …………….……………………………………………...126 

danarTi 8. uilkoksonis kriteriumis W statistikis qveda kritikuli 

mniSvnelobebi ……………………………………………………………131 

danarTi 9. amonarCeviT gamoTvlili korelaciis koeficientis 

ganwilebis kvantilebi 

1 

2 

p r 

− 

……………………………………………135 

danarTi danarTi danarTi 10. uilkoksonis niSnebis rangebis jamebis statistikis zeda da 

qveda procentuli wertilebi…………………………..……………………………136

Sesavali 

adamians yoveldRiur cxovrebaSi aqvs Sexeba problemebTan, romelic 

dakavSirebulia mosalodneli Sedegis zusti gamocnobis SesaZleblobis ar 

arsebobasTan. magaliTad, nebismieri CvenTaganisaTvis SeuZlebelia winaswar 

zustad ganvsazRvroT saWiro drois xangrZlioba daniSnulebis adgilamde, 

saswavleblamde, samuSaomde an sxva adgilamde, misasvlelad. aseve, 

maRaziis gamyidvelisaTvis SeuZlebelia winaswar gansazRvros Tu ramdeni 

myidveli Seva im dRes maRaziaSi. aseT martiv SemTxvevebSi adamiani wina gamocdilebis 

safuZvelze akeTebs miaxloebiT Sefasebas da, magaliTad, mgzavrobisaTvis 

saWiro droze aTi wuTiT adre gamodis saxlidan, rom droze 

mivides daniSnulebis adgilze. magram rodesac saqme exeba seriozul sakiTxebs, 

magaliTad, kompaniis mier axali saqmis wamowyebasTan dakavSirebuli 

sakiTxebis gadawyvetas, bankebis mier garkveuli finansuri operaciebis Catarebas 

da a.S. resursebis aseTi darezerveba, anu gamouyenebeli maragebis 

Seqmna, konkurenciis Tanamedrove pirobebSi, ararentabeluri anu wamgebiania, 

radgan mas biznesSi gaaswreben is konkurentebi, romlebic ukeTesad 

iTvlian da Rebuloben ufro zust gadawyvetilebebs. 

aseTi SemTxvevebisaTvis, rodesac Sedegis zustad gansazRvra winaswar 

SeuZlebelia da atarebs SemTxveviT xasiaTs, gamoiyeneba maTematikuri statistikis 

meTodebi, romelTa safuZvlebsac Cven SeviswavliT winamdebare 

kursSi. 

maTematikuri statistikis meTodebis farTod gamoyenebas cxovrebaSi xeli 

Seuwyo gasuli saukunis 60 – 70 - ian wlebSi gamoTvliTi teqnikis far- 

Tod ganviTarebam da danergvam cxovrebaSi. gansakuTrebulad intensiurad 

daiwyo statistikuri meTodebis gamoyeneba gasuli saukunis 80-ian wlebSi, 

rac ganapiroba personaluri kompiuteris Seqmnam da farTod gavrcelebam. 

garda personaluri kompiuterebis farTo gavrceldebisa, statistikuri 

meTodebis farTod gamoyenebas didad Seuwyo xeli maTTvis monacemebis 

statistikuri damuSavebis specializebuli programuli paketebis Seqmnam. 

aseTma paketebma saSualeba misca maTematikuri statistikis ara specialistebs, 

anu sxva dargis specialistebs, romlebic Rrmad ar floben statistikur 

meTodebs, aramed ician am meTodebis daniSnuleba da maTi SesaZleblobebi, 

kvalificiurad gamoiyenon es meTodebi Tavisi dargis amocanebis gadasawyvetad. 

daniSnulebisa da Sesabamisad, maTSi realizebuli amocanebis 

mixedviT, arseboben universaluri da specializirebuli statistikuri paketebi. 

universaluri statistikuri paketebidan dReisaTvis yvelaze farTod 

gavrcalebuli da gamoyenebuli paketebia: SPSS, STATISTICA, STATGRAPHICS, 

STADIA da sxva. specializirebuli paketebidan aRsaniSnavia: Эвриста, 

Мезозавр (droiTi mwkrivebis dasamuSaveblad), КЛАСС-МАСТЕР (raodenobrivi, 

Tvisobrivi da logikuri monacemebis analizisaTvis) da sxva. calke 

gamovyofT winamdebare naSromis avtoris xelmZRvanelobiT saqarTveloSi 

Seqmnil monacemTa statistikuri analizis universalur pakets SDpro – s, 

romelic Seqmnilia analogiur produqciaze arsebuli saerTaSoriso 

standartebis Sesabamisad. ukanaskneli zemoT dasaxelebuli paketebisagan 

gansxvavdeba imiT, rom is orientirebulia ara profesional momxmareblze, 

ris gamoc misi Seswavla da gamoyeneba statistikur meTodebSi cotad Tu 

bevrad garkveuli momxmareblisaTvis ar warmoadgens sirTules. garda 

6

amisa, sxvebisagan gansxvavebiT, paketi mravalenovania. Sesabamisi ofciis 

SerCeviT,DmuSaobis nebismier etapze, SesaZlebelia masSi realizebul nebismier 

enaze gadasvla. dReisaTvis paketSi realizebulia qarTuli, rusuli 

da inglisuri enebi. paketis Help – Si mocemuli instruqciis gamoyenebiT paketis 

momxmarebels SeuZlia nebismieri enis damateba paketSi gamoyenebuli 

winadadebebis am eneze Targmnis da, instruqciis Tanaxmad, paketSi CarTvis 

gziT. 

moviyvanoT sxvadasxva praqtikuli amocanebis gadasawyvetad statistikuri 

meTodebis gamoyenebis ramodenime magaliTi. 

1. mewarme, romelsac Seaqvs Tavis sawarmoSi Sromis anazRaurebis axali 

sistema an nergavs axal teqnologiur process, dainteresebulia rac SeiZleba 

swrafad darwmundes imaSi, rom warmoebis gaumjobeseba dakavSirebulia 

am siaxlesTan da ar atarebs SemTxveviT xasaTs da ramodenime xnis Semdeg 

aseve SemTxveviT ar gauaresdeba situacia. 

am amocanis gadawyveta SesaZlebelia maTematikuri statistikis meTodebis 

gamoyenebiT, romlebic saSualebas iZlevian erTmaneTTan Sedarebuli 

iqnas siaxlis Semotanamde arsebuli da Semotanis Semdeg miRebuli Sedegebi 

da garkveuli garantiiT iyos miRebuli gadawyvetileba maTi erTgvarovnebis 

an gansxvavebis Sesaxeb. 

2. vTqvaT saWiroa raime procesis ganviTarebis winaswar ganWvreta anu 

procesis ganviTarebis prognozi. magaliTad, birJaze valutis kursis cvalebadobis, 

garemos temperaturis cvalebadobis da a.S. prognozi. procesis 

ganviTarebis winaswar ganWvretis saSualebas, wina periodSi miRebuli dakvirvebis 

Sedegebis safuZvelze, iZlevian maTematikuri statistikis meTodebi, 

romlebic gaerTianebuli arian regresiuli analizis, an SemTxveviTi 

procesebis analizisa da prognozis saxelwodebiT. 

3. yoveli mewarme dainteresebulia, rom mis mier gamoSvebuli produqcia 

iyos erTgvarovani da rac SeiZleba maRali xarisxis. amis miRweva SesaZlebelia 

teqnologiuri procesis mkacri dacviT, risTvisac saWiroa samuSaos 

yoveli etapis mudmivi, obieqturi kontroli. aseTi kontrolis gansaxorcieleblad 

damuSavebulia produqciis xarisxis kontrolis statistikuri 

meTodebi, romelTa gamoyeneba sawarmoSi saSualebas iZleva warmoebis 

yovel etapze gavakontroloT Sesrulebuli samuSaos xarisxi, raTa 

problemis warmoSobisTanave, rac SeiZleba swrafad, davafiqsiroT da aRmovfxvraT 

is. kontrolis aseTi meTodebis sayovelTao danergva da gamoyeneba 

gaxda mTavari mizezi iaponiis ekonomikis arnaxuli ganviTarebisa meore 

msoflio omis Semdeg. 

4. bankebis kreditis gamcemi ganyofilebebi yoveldRiurad dgebian 

problemis winaSe, miscen Tu ara krediti ama Tu im mTxovnels. SeZlebs Tu 

ara valis amRebi fulis dabrunebas droulad. am problemis gadasawyvetadac 

gamoiyeneba maTematikuri statistikis meTodebi, romelTac klaster 

analizis meTodebs uwodeben. meTodis arsi mdgomareobs SemdegSi: yoveli 

firma xasiaTdeba garkveuli parametrebis simravliT. magaliTad, ZiriTadi 

saSualebebis moculoba, sabrunavi kapitalis moculoba, gamoSvebuli produqciis 

saxeoba da raodenoba, moxmarebuli nedleuli da a.S. am parametrebiT 

xdeba firmebis, romlebmac ukve miiRes krediti am bankidan, dajgufeba 

karg gadamxdelebad, cud gadamxdelebad da firmebad, romlebmac ver daabrunes 

krediti. klaster analizis meTodebis gamoyenebiT axal firmas mi- 

7

akuTvneben erT-erT zemoTdasaxelebul jgufs sakontrolo parametrebis 

mniSvnelobebis mixedviT da Rebuloben Sesabamis gadawyvetilebas. 

5. sahaero Tavdacvis amocanebis gadawyvetisas radiolokaciuri gazomili 

informaciis pirveladi damuSavebiT mravalganzomilebian sivrceSi gamoiyofa 

wertilebis simravle, romelTa garkveul qvesimravleSic SesaZlebelia 

mowinaaRmdegis moZravi obieqtebis arseboba. saWiroa mocemuli garantiiT 

gadawyvetilebis miReba Tu romel qvesimravleSi imyofebian moZravi 

obieqtebi. am amocanis gadasawyvetad gamoiyeneba statistikuri hipo- 

Tezebis Semowmebis kriteriumebi, radgan radiolokaciuri gazomvis Sedegebi 

xasiaTdebian SemTxveviTi damaxinjebebiT. 

6. garemos obieqtebis mdgomareoba xasiaTdeba garkveuli parametrebis 

simravliT, romelTa mniSvnelobebis gazomvac xdeba drois sxvadasxva 

momentSi sivrcis sxvadasxva wertilSi. saWiroa gadawyvetilebis miReba garemos 

sakontrolo obieqtSi parametrebis cvalebadoba droSi da sivrceSi 

ganpirobebuli garkveuli gareSe zemoqmedebiT Tu parametrebis SemTxvevi- 

Ti cvalebadobiT. am amocanis gadawyveta SesaZlebelia faqtoruli analizis 

meTodebiT da a.S. 

maTematikuri statistikis meTodebi arian universaluri meTodebi im 

TvalsazrisiT, rom ara aqvs mniSvneloba Tu codnis romel dargs miekuTvnebian 

is monacemebi, romelTa damuSavebac xdeba am meTodebiT; mTavaria 

amocanis arsi, romlis gadawyvetac gvinda am meTodebiT. Zalze iSviaT Sem- 

TxvevebSi gvxvdeba iseTi amocanebi, romelTa gadaWrac moiTxovs specialuri 

meTodebis damuSavebas. am SemTxvevaSi saWiroa am amocanis maTematikuri 

formalizeba misi specifikis gaTvaliswinebiT, amoxsnis algoriTmis damu- 

Saveba da realizeba kompiuterze programis saxiT. winamdebare kursSi Cven 

SeviswavliT maTematikuri statistikis universalur meTodebs, romlebic 

erTnairad gamodgebian nebismieri dargis monacemebis dasamuSaveblad Sesabamisi 

amocanebis gadawyvetisaTvis. 

monacemebis damuSavebisas pirvel etaps warmoadgens maTi vizualizacia 

anu monacemebis TvalsaCino warmodgena grafikebis saxiT. es saSualebas iZleva 

TvalsaCinod warmovidginoT procesis mimdinareobis xasiaTi da SevirCioT 

Sesabamisi meTodebi saWiro amocanis gadasawyvetad. amitom zemoT 

naxseneb monacemTa damuSavebis programul paketebSi farTod aris warmodgenili 

monacemTa erT, or, samganzomilebian grafikebad warmodgenis sa- 

Sualebebi da SesaZlebelia maTi gamoyeneba rogorc monacemTa damuSavebis 

sawyis etapze, aseve miRebuli Sedegebis TvalsaCino warmodgenisaTvis. 

monacemTa pirveladi damuSavebisas saWiroa monacemebidan gamoiricxos 

“uxeSi Secdomebi”. “uxeSi Secdomebi” es iseTi monacemebia, romlebic ar Seesabamebian 

monacemTa ZiriTad simravles da maTi warmoSoba gamowveulia 

garkveuli subieqturi an obieqturi mizezebiT. magaliTad, xelsawyos Cvenebis 

aRebisas daSvebuli Secdoma gamowveuli Zabvis myisieri cvlilebiT, an 

anaTvalis amRebi pirovnebis subieqturi Secdoma daSvebuli Canaweris gake- 

Tebisas da a.S. cnobilia [56], rom saSualod eqsperimentalur monacemebSi 

aT procentamde “uxeSi Secdomebia”, romlebic iwveven damuSavebis Sedegebis 

mkveTr gauaresebas. amitom saWiroa damuSavebis sawyis etapze maTi 

gamovlena da ugulebelyofa. 

8

Tavi 1. gamoyenebiTi statistikis ZiriTadi cnebebi 

winamdebare TavSi SeviswavliT ZiriTad cnebebsa da gansazRvrebebs, 

romlebic dagvWirdeba mocemuli kursis Semdeg TavebSi Sesaswavli gamoyenebiTi 

statistikis meTodebisa da algoriTmebis gasagebad. aseTi cnebebia: 

SemTxveviTi movlena, SemTxveviTi sidide, SemTxveviTi movlenisa da sididis 

albaToba, albaTobebis ganawilebis kanonebi, SemTxveviTi sididis ricxviTi 

maxasiaTeblebi, amonarCevi da misi warmodgenis meTodebi. 

1.1. SemTxveviTi cvalebadoba 

bunebaSi Zalian xSirad gvxvdebian movlenebi, romlTa Sedegebis winaswar 

ganWvreta SeuZlebelia, radgan isini Rebuloben SemTxveviT mniSvnelobebs 

mniSvnelobaTa garkveuli areebidan. aseT movlenebs SemTxveviTi movlenebi 

hqviaT. Tumca ki, meores mxriv, cnobilia bunebis kanonzomierebebi, 

romlebic mkacrad emroCilebian garkveul damokidebulebebs da maTi mniSvnelobebis 

gansazRvra zustad aris SesaZlebeli masTan dakavSirebuli meore 

sididis mniSvnelobis mixedviT. magaliTad, fizikidan cnobilia siTxe- 

Si CaSvebul sxeulze moqmedi wnevis CaSvebis siRrmeze damokidebuleba da 

yovelgvari eqsperimentis gareSe am damokidebulebiT SegviZlia zustad 

ganvsazRvroT sxeulze moqmedi wneva siRrmis mixedviT. zustad aseve Segvi- 

Zlia ganvsazRvroT sxeulis Tavisufali vardnisas mis mier ganvlili 

manZili vardnis drois mixedviT. aseT procesebs determinirebuli procesebi 

hqvia. 

rogorc ukve vTqviT, determinirebuli procesebisagan gansxvavebiT, SeuZlebelia 

SemTxveviTi procesebis mniSvnelobebis winaswar zusti gansaz- 

Rvra. magaliTad, SeuZlebelia zustad ganvsazRvroT Tu rogori mosavali 

iqneba miRebuli wlis bolos ama Tu im regionSi; haeris zustad rogori 

temperatura iqneba garkveuli periodis Semdeg da a.S. Tumca ki SesaZlebelia 

garkveuli saimedobiT mivuTiToT intervali, romlidanac miiRebs 

mniSvnelobebs esa Tu is SemTxveviTi sidide. SemTxveviTi sidide SeiZleba 

warmodgenili iqnas ori mdgenelis jamis saxiT: a) determinirebuli mdgeneli, 

romelic droSi icvleba garkveuli kanonzomierebiT da b) SemTxvevi- 

Ti mdgeneli, romelic emateba determinirebul mdgenels da cvlis mis xasiaTs 

SemTxveviT. magaliTad [1], naxaz 1.1 – ze sqematurad warmodgenilia 1948- 

1989 wlebSi sabWoTa kavSirSi erT heqtarze marcvleuli kulturebis mosavlianobis 

cvalebadoba centnerebSi. miuxedavad imisa, rom mosavlianobis 

mniSvneloba SemTxveviT icvleba wlidan wlamde, mainc aRiniSneba mosavlianobis 

zrdis tendencia rac ganpirobebuli iyo progresuli agromeTodebis 

da sasuqebis gamoyenebiT. mosavlianobis SemTxveviTi cvalebadoba sxvadasxva 

wlebSi ganpirobebulia amindis da kidev mravali sxva faqtorebis 

cvalebadobiT, romlebsac aqvT SemTxveviTi xasiaTi. maTematikuri statistikis 

meTodebi saSualebas iZlevian aseT SemTxvevebSi SevafasoT arsebuli 

kanonzomierebis parametrebi, SevamowmoT esa Tu is hipoTezebi am kanonzomierebis 

Sesaxeb da a.S. aseTi meTodebis Seswavlas garkveul doneze isaxavs 

miznad winamdebare kursi. 

9

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0 

1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 

nax. 1.1. 1948 – 1989 wlebSi sabWoTa kavSirSi erT heqtarze yvela 

marcvleuli kulturebis mosavlianobis cvalebadoba (c/ha) 

1.2. xdomileba da misi albaToba 

SemTxveviTi xdomileba hqvia iseT movlenas, romelsac cdis Sedegad SeiZleba 

hqondes an ar hqondes adgili. Tu cdis Sedegad movlenas yovel- 

Tvis aqvs adgili, WeSmariti xdomileba ewodeba. xolo Tu cdis Sedegad 

movlenas ar SeiZleba hqondes adgili, SeuZlebeli xdomileba hqvia. 

xdomilebis moxdenis SesaZleblobis zomas misi albaToba ewodeba. 

xdomilebas, rogorc wesi, aRniSnaven didi laTinuri asoebiT, magaliTad, 

A , B , C ,…, xolo Sesabamis albaTobas P asoTi. SeuZlebeli xdomilebis 

albaToba nulis tolia, anu P (SeuZl. xdom.)=0, xolo WeSmariti xdomilebis 

albaToba erTis tolia, anu P (WeSm. xdom.)=1. zogadad albaToba Rebulobs 

mniSvnelobebs nolidan erTamde intervalSi, anu raime A xdomilebisaTvis 

0 ≤ P( 

A) 

≤ 1 

adgili aqvs pirobas 

. 

xdomilebebze SesaZlebelia moqmedebis Catareba. magaliTad: 

1) ori A da B xdomilebis gaerTianeba an jami hqvia iseT C xdomilebas, 

romelsac adgili aqvs maSin, rodesac adgili aqvs an A – s, an B – s an 

orives erTad. xdomilebebis jami aRiniSneba Semdegnairad C = A + B an 

C = A∪ B . xdomilebebis gaerTianebis geometriul interpretacias aqvs saxe 

10

A B 

nax. 1.2. 

2) ori A da B xdomilebis gadakveTa an namravli hqvia iseT C xdomilebas, 

romelsac adgili aqvs maSin, rodesac adgili aqvs A – s da B – s 

erTdroulad. xdomilebebis namravli aRiniSneba Semdegnairad C = A⋅ 

B an 

C = A∩ B . xdomilebebis gadakveTis geometriul interpretacias aqvs saxe 

nax. 1.3. 

3) A xdomilebis uaryofa hqvia iseT A xdomilebas, romelsac adgili 

aqvs maSin, rodesac A xdomilebas ara aqvs adgili da piriqiT. geometriul 

interpretacias aqvs saxe 

A 

nax. 1.4. 

Tu xdomilebebi A da B ar SeiZleba moxdnen erTdroulad, maSin aseT 

xdomilebebs ewodebaT SeuTavsebadi xdomilebebi. SeuTavsebadi xdomilebebis 

magaliTebia A da A , xolo xdomileba A + A WeSmariti xdomilebaa. 

moviyvanoT magaliTebi: 

• vTqvaT A aris xdomileba, rom kamaTelis gagorebis Sedegad gamova 

4 – ze naklebi cifri. B aris xdomileba, rom kamaTelis ga- 

11 

A

gorebisas gamova cifri 3 an 6. maSin A + B aris xdomileba, rom 

kamaTelis gagorebisas gamova cifri 1, 2, 3 an 6; xolo A ⋅ B aris 

xdomileba, rom kamaTlis gagorebisas gamova cifri 3. A aris 

xdomileba, rom kamaTlis gagorebisas gamova cifri meti an tol 4 

– is da naklebi an toli 6 – is. 

moviyvanoT albaTobebis Tvisebebi. 

1) nebismieri A xdomilebisaTvis 

0 ≤ P( 

A) 

≤ 1 

; 

2) ori A da B SeuTavsebadi xdomilebebis jamis albaToba tolia am 

xdomilebebis albaTobebis jamisa 

P ( A + B) 

= P( 

A) 

+ P( 

B) 

, xolo zogadad 

P( A + B) 

= P( 

A) 

+ P( 

B) 

− P( 

A⋅ 

B) 

. 

3) WeSmariri xdomilebis albaToba tolia 1 – is, xolo SeuZlebeli 

xdomilebis albaToba tolia 0 – is. 

A da B xdomilebebs ewodebaT damoukidebeli, Tu 

P( A ⋅ B) 

= P( 

A) 

⋅ P( 

B) 

. 

SemovitanoT pirobiTi albaTobis cneba. A xdomilebis albaToba im 

P( A⋅ B) 

pirobiT, rom adgili hqonda B xdomilebas Caiwereba ase P( A | B) 

= . 

P( B) 

aqedan P( A⋅ B) = P( A | B) ⋅ P( B) = P( B | A) ⋅ P( A). 

Tu xdomileba A damokidebulia 

B xdomilebisagan, maSin B xdomilebac damokidebulia A xdomilebisagan 

da piriqiT. Tu xdomileba A ar aris damokidebuli B xdomilebisagan, 

maSin B xdomilebac ar aris damokidebuli A xdomilebisagan. 

1.3. albaTobebis gazomva 

radgan albaToba axasiaTebs ama Tu im movlenis moxdenis SesaZleblobas, 

bunebrivia moviTxovoT misi gazomvis SesaZlebloba. sxva fizikuri sidideebisagan 

gansxvavebiT, magaliTad, wona, denis Zabva, siswrafe, sigrZe da 

a.S., albaTobis gazomva raime xelsawyos gamoyenebiT SeuZlebelia. 

arsebobs albaTobis gazomvis ori gza: logikuri daskvnebis safuZvelze an 

pirdapiri gazomva. albaTobis logikuri gazomva mdgomareobs miRebuli 

daSvebebis pirobebSi logikuri msjelobis safuZvelze albaTobis gamoTvlaSi. 

magaliTad, kamaTelis gagorebis SemTxvevaSi, Tu viciT, rom is simetriuli 

da erTgvarovania, bunebrivia davuSvaT, rom nebismieri gverdis 

gamosvlis albaToba ernairia. radgan yvela SesaZlo Sedegebis raodenoba 

eqvsia, amitom 1 – dan 6 – de nebismieri cifris gamosvlis albaToba tolia 

1 / 6 - is. wyvili cifrebis gamosvlis albaToba tolia 1 / 2 - is. cifrebis 3 – 

is an 5 – is gamosvlis albaToba tolia 1 / 3 – is. 

pirdapiri gazomvis arsi mdgomareobs SemdegSi. atareben rac SeiZleba 

didi raodenobis eqsperimentebs. aRvniSnoT maTi saerTo raodenoba N N 

– iT, xolo N (A) 

aRvniSnoT eqsperimentebis ricxvi, rodesac adgili 

12

hqonda A A xdomilebas. cxadia, rom N( A) 

≤ N . A xdomilebis albaTobis mi- 

N( 

A) 

axloebiTi mniSvneloba gamoiTvleba formuliT P( 

A) 

= . 

N 

eqsperimentebis ricxvis usasrulod gazrdisas, anu rodesac N → ∞ 

albaTobis gamoTvlili mniSvneloba miiswrafvis misi namdvili mniSvnelobebisaken 

(ix. paragrafi 4..2). 

1.4. SemTxveviTi sidideebi. ganawilebis funqcia 

rogorc zemoT aRvniSneT, SemTxveviTi sidide es iseTi sididea, romelic 

Tavisi gansazRvris aredan Rebulobs SemTxveviT mniSvnelobebs. yoveli 

mniSvnelobis miRebas gaaCnia garkveuli SesaZlebloba, romelsac am 

mniSvnelobis Sesabamisi albaToba hqvia. Sesabamisobas SemTxveviTi sididis 

mniSvnelobebsa da am mniSvnelobebis miRebis albaTobebs Soris SemTxvevi- 

Ti sididis ganawilebis kanoni hqvia. 

praqtikuli amocanebis amoxsnis dros ZiriTadad gvxvdeba ori saxis 

SemTxveviTi sidideebi: diskretuli da uwyveti. Tumca arseboben sxva saxis 

SemTxveviTi sidideebi, romelTac winamdebare kursSi ar ganvixilavT. diskretuli 

SemTxveviTi sidide hqvia iseT sidides, romlis SesaZlo mniSvnelobebi 

sasrulo an Tvladia. magaliTad, diskretuli SemTxveviTi sididea 

saTamaSo kamaTelis gagorebis Sedegad gamosuli cifri. am SemTxveviTi 

sididis SesaZlo mniSvnelobebia 1, 2, 3, 4, 5, 6. diskreuli SemTxveviTi 

sididea, agreTve, ori kamaTelis agdebisas gamosuli cifrebis jami, 

romelic Rebulobs mniSvnelobbs 2 – dan 12 - de. 

uwveti SemTxveviTi sidide hqvia iseT SemTxveviT sidides, romlis SesaZlo 

mniSvnelobaTa CamoTvla SeuZlebelia, radgan is Rebulobs yvela 

SesaZlo mniSvnelobebs romelime sasrulo an usasrulo aredan. uwyveti 

SemTxveviTi sididis magaliTad SeiZleba davasaxeloT eleqtro naTuris 

muSaobis xangrZlivoba. am SemTxveviT sidides Teoriulad SeuZlia miiRos 

mniSvneloba nolidan usasrulobamde drois intervalSi. 

zemoT aRvniSneT, rom SemTxveviTi sididis mniSvnelobebsa da am mniSvnelobebis 

miRebis SesaZleblobebs, anu Sesabamis albaTobebs Soris damokidebulebas 

hqvia SemTxveviTi sididis ganawilebis kanoni. diskretuli SemTxveviTi 

sididis magaliTze ganvixiloT ganawilebis kanonis arsi. ganvixiloT 

ori kamaTelis agdebis Sedegad gamosuli ori cifris jami. am Sem- 

TxveviT sidides SeuZlia miiRos mniSvnelobebi 2, 3, 4, …, 12. SemTxveviTi sididis 

yovel SesaZlo mniSvnelobas Seesabameba garkveuli albaToba, 

romelic axasiaTebs am mniSvnelobis miRebis SesaZleblobas. cxril 1.1 – Si 

mocemulia gansaxilveli diskretuli SemTxveviTi sididis SesaZlo 

mniSvnelobebi da Sesabamisi albaTobebi. 

cxrili 1.1. 

A A 2 3 4 5 6 7 8 9 10 11 12 

P P (A) 

1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 

13

cxriliT mocemulia Sesabamisi diskretuli SemTxveviTi sididis ganawilebis 

kanoni, radgan is amyarebs Sesabamisobas SemTxveviTi sididis mniSvnelobebsa 

da am mniSvnelobebis miRebis albaTobebs Soris. 

uwyveti SemTxveviTi sididisaTvis ganawilebis kanonis aseve martivad 

Cawera SeuZlebelia misi SesaZlo mniSvnelobebis usasrulo raodenobis 

gamo. amas garda, uwyveti SemTxveviTi sididis SemTxvevaSi albaToba imisa, 

rom SemTxveviTi sidide miiRebs erT romelime mniSvnelobas misi 

gansazRvris aredan nolis tolia (misi SesaZlo mniSvnelobebis usasrulo 

raodenobis gamo). am SemTxvevaSi SeiZleba vilaparakoT albaTobaze, rom 

SemTxveviTi sidide miiRebs mniSvnelobas misi gansazRvris aris romelime 

qvesimravlidan. kerZod, albaToba imisa, rom uwyveti SemTxveviTi sidide ξ 

miiRebs x cvladze nakleb mniSvnelobas Caiwereba ase F ( x) 

= P( 

ξ < x) 

da mas 

ewodeba SemTxveviTi sididis ganawilebis funqcia. ganawilebis funqcia SemTxveviTi 

sididis ganawilebis kanonis warmodgenis yvelaze universaluri 

formaa. is arsebobs rogorc diskretuli aseve uwyveti SemTxveviTi sidideebisaTvis. 

ganmartebidan Cans, rom ganawilebis funqcia arsebobs rogorc diskretuli, 

aseve uwyveti SemTxveviTi sidideebisaTvis. 

moviyvanoT ganawilebis funqciis Tvisebebi. 

1) F ( x) 

≥ 0; 

2) F ( −∞) 

= 0; 

3) F ( +∞) 

= 1; 

4) 0 ≤ F ( x) 

≤ 1. 

albaTobebis ganawilebis kanonis Cawris Semdegi formaa ganawilebis 

simkvrive. magram ganawilebis funqciisagan gansxvavebiT albaTobebis ganawilebis 

simkrive arsebobs mxolod uwyveti SemTxveviTi sidideebisaTvis. 

ganawilebis simkvrivis cneba SeiZleba SemovitanoT integraluri da diferenciuli 

formiT. 

integraluri integraluri forma. forma. ξ SemTxveviTi sididis ganawilebis simkvrive ewodeba 

p (x) 

funqcias, Tu is akmayofilebs pirobas 

14 

∫ ∞ 

− 

F ( x) 

= p( 

t) 

dt, 

sadac F (x) 

aris gansaxilveli SemTxveviTi sididis ganawilebis funqcia. 

diferencialuri diferencialuri forma. forma. ξ SemTxveviTi sididis ganawilebis simkvrive 

ewodeba p (x) 

funqcias, Tu albaToba imisa, rom ξ SemTxveviTi sidide moxvdeba 

Δ x sigrZis mqone elementarul intervalSi gamoiTvleba formuliT 

P( x ≤ x + Δx) 

= p( 

x) 

⋅ Δx 

+ OΔx, 

sadac OΔ x aris Δ x SedarebiT usasrulod mcire sidide. 

moviyvanoT ganawilebis simkvrivis Tvisebebi: 

1) p ( −∞) = p( 

+∞) 

= 0; 

2) p ( x) 

≥ 0, 

x ∈ ( −∞; 

+∞); 

+∞ 

3) ∫ 

−∞ 

p ( x) 

dx 

= 1. 

naxazebze 1.5 da 1.6 moyvanilia Sesabamisad ganawilebis funqciisa da simkvrivis 

grafikuli saxeebi 

x

nax. 1.5. nax. 1.6. 

albaTobebis ganawilebis funqciis da albaTobebis simkvrivis ganawilebis 

integraluri formis safuZvelze advilad vrwmundebiT, rom ξ SemTxveviTi 

sididis misi gansazRvris aredan ( a , b) 

intervalSi moxvedris albaToba 

ganisazrvreba Semdegnairad 

P (ξ ∈ ( a, 

b)) 

= F( 

b) 

− F( 

a) 

= p( 

x) 

dx, 

sadac F (x) 

aris ξ SemTxveviTi sididis ganawilebis funqcia, xolo p (x) 

aris misi albaTobebis ganawilebis simkvrive. 

1.5. albaTobebis ganawilebis ricxviTi maxasiaTeblebi 

SemTxveviTi sididis ganawilebis funqcia da simkvrive moicaven SemTxveviTi 

sididis Sesaxeb srul informacias, anu SemTxveviTi sididis ganawilebis 

kanonis codna Seesabameba SemTxveviTi sididis bunebis Sesaxeb sruli 

informaciis qonas (misi gansazRvris are, diskretuli SemTxveviTi sididis 

dros SesaZlo mniSvnelobebis simravle da maTi miRebis albaTobebi, 

SemTxveviTi sididis romelime areSi moxvedris albaToba da sxva). Zalze 

xSirad SemTxveviTi sididis ganawilebis kanoni ucnobia da misi dadgena 

xdeba eqsperimentaluri monacemebis safuZvelze, rac sakmaod Sromatevad 

da, xSir SemTxvevaSi, Zvirad Rirebul samuSaos warmoadgens. mravali praqtikuli 

amocanis gadawyvetisas arc aris saWiro ganawilebis kanonis codna 

anu SemTxveviTi sididis mTeli bunebis codna. sakmarisia misi calkeuli maxasiaTeblebis 

codna. SemTxveviTi sididis calkeul Tvisebebs axasiaTeben 

misi ricxviTi maxasiaTeblebi. arsebobs SemTxveviTi sididis mravali ricxviTi 

maxasiaTebeli. maT Soris yvelaze gavrcelebulia momentebi da kvantilebi. 

Cven SeviswavliT ori saxis momentebs. SemTxveviTi sididis sawyis 

momentebs da centralur momentebs. 

diskretuli SemTxveviTi sididis k - uri rigis sawyisi momenti hqvia si- 

α 

dides k , romelic gamoiTvleba formuliT 

15 

b 

∫ 

a

α 

k 

= 

n 

∑ 

i= 

1 

x 

16 

k 

i 

⋅ p , 

sadac n x x x ,..., , 1 2 arian diskretuli SemTxveviTi sididis SesaZlo mniSvnelobebi, 

xolo n p p p ,..., , 1 2 arian Sesabamisi mniSvnelobebis miRebis albaTobebi. 

imisaTvis rom k - uri rigis sawyisi momenti α k arsebobdes mwkrivi unda 

ikribebodes absoluturad. 

analogiurad ganimarteba uwyveti SemTxveviTi sididis k - uri rigis sawyisi 

momenti 

α 

x k 

+∞ 

k = ∫ 

−∞ 

i 

⋅ p( 

x) 

dx, 

sadac p (x) 

Sesabamisi SemTxveviTi sididis ganawilebis simkvrivea. 

sawyisi momentebi damokidebuli arian ricxviT RerZze aTvlis wertilis 

mdebareobaze. momentebs, romelTa mniSvnelobebic am mdebareobaze ar arian 

damikidebuli, ewodebaT centraluri momentebi.diskretuli ξ SemTxvevi- 

Ti sididis k - uri rigis centraluri momenti hqvia sidides μ k , romelic 

ganisazRvreba Semdegnairad 

μ 

k 

= 

n 

∑ 

i= 

1 

( x − Mξ 

) 

i 

k 

⋅ p , 

sadac M ξ aRniSnulia SemTxveviTi sididis pirveli sawyisi momenti, romelsac 

SemTxveviTi sididis maTematikuri molodini hqvia. imis gamo, rom 

maTematikur molodins aqvs didi mniSvneloba aqvs albaTobis TeoriaSi da 

maTematikur statistikaSi, SemoRebulia misi universaluri aRniSvna M ξ an 

M (ξ ) . inglisur literaturaSi xmaroben aRniSvnas E (ξ ) . maTematikuri molodini 

axasiaTebs SemTxveviTi sididis mniSvnelobebis mdebareobas ricxviT 

RerZze. anu es aris iseTi mniSvneloba, romlis axlo–maxloTac ganlagdebian 

SemTxveviTi sididis mniSvnelobaTa didi umravlesoba. 

uwyveti ξ SemTxveviTi sididis k - uri momenti ganisazRvreba analogiurad. 

kerZod, 

k 

+∞ 

∫ 

−∞ 

k 

μ = ( x − Mξ 

) ⋅ p( 

x) 

dx, 

sadac M ξ aris ξ SemTxveviTi sididis pirveli rigis sawyisi momenti. 

moviyvanoT maTematikuri molodinis Tvisebebi. avRniSnoT: a – raime 

ricxvia, xolo ξ - SemTxveviTi ricxvi. maSin 

1) M ( a) 

= a; 

2) M ( ξ + η) 

= M ( ξ ) + M ( η), 

3) M ( a ⋅ ξ ) = a ⋅ M ( ξ ). 

SemTxveviTi sididis ricxviT maxasiaTeblebs Soris mniSvnelovani 

adgili ukavia meore rigis centralur moments, romelsac SemTxveviTi 

sididis dispersia hqvia. diskretuli SemTxveviTi sididisaTvis is gamoiT- 

n 

2 

vleba formuliT Dξ 

= ∑ ( xi 

− Mξ 

) ⋅ pi 

, xolo uwyveti SemTxveviTi sididisai= 

1 

i

Tvis - 

+∞ 

= ∫ 

−∞ 

2 

D ξ ( x − Mξ 

) ⋅ p( 

x) 

dx . inglisur literaturaSi xmaroben aRniSvnas 

V ( ξ ) . 

dispersia axasiaTebs SemTxveviTi sididis mniSvnelobebis gabnevas maTematikuri 

molodinis mimarT. dispersias aqvs SemTxveviTi sididis ganzomilebis 

kvadratis ganzomileba, amitom xSirad moxerxebulia gabneva davaxasiaToT 

sididiT, romelsac aqvs SemTxveviTi sididis ganzomileba. aseT maxasiaTebels 

saSualo kvadratuli gadaxra ewodeba, aRiniSneba σ da gamo- 

iTvleba σ = Dξ 

. 

moviyvanoT dispersiis Tvisebebi. 

1) D ( a) 

= 0; 

2) D ( a + ξ ) = Dξ; 

2 

3) D( a ⋅ξ 

) = a ⋅ Dξ 

. 

iseve rogorc maTematikuri molodini da dispersia sxva sawyisi da 

centraluri momentebic axasiaTeben SemTxveviTi sididis ama Tu im Tvisebebs. 

SemTxveviTi sididis yvela momentis gansazRvris SesaZlebloba niSnavs 

imas, rom SesaZlebelia SemTxveviTi sididis Sesaxeb miviRoT sruli informacia 

anu es tolfasia SemTxveviTi sididis ganawilebis kanonis, ker- 

Zod, albaTobebis ganawilebis funqciis an simkvrivis codnisa. 

moviyvanoT SemTxveviTi sididis ramodenime mniSvnelovani ricxviTi maxasiaTebli. 

sawyisi da centraluri momentebis gansazRvrebidan Cans, rom 

maTi mniSvnelobebi damokidebuli arian SemTxveviTi sididis gazomvis 

erTeulze. zogjer moxerxebulia iseTi momentebis gamoyeneba, romlebic ar 

arian gazomvis erTeulze damokidebuli. aseTi momentebidan yvelaze 

xSirad gamoiyenebian mesame da meoTxe rigis normirebuli centrirebuli 

momentebi, romlebsac asimetriisa da eqscesis koeficientebi hqviaT. asimetriis 

koeficienti gamoiTvleba formuliT 

3 

M ( ξ − Mξ 

) 

Ассим. 

= 

3 / 2 

Dξ 

. 

asimetriis koeficienti axasiaTebs SemTxveviTi sididis ganawilebis 

arasimetriulobas. 

eqscesis koeficienti gamoiTvleba formuliT 

4 

M ( ξ − Mξ 

) 

Эксцесс = 

2 

Dξ 

. 

eqscesis koeficienti axasiaTebs SemTxveviTi sididis ganawilebis simkvrivis 

gverdebis cicaboobas, anu imas, Tu ramdenad xSirad Rebulobs Sem- 

TxveviTi sidide maTematikuri molodinidan daSorebul mniSvnelobebs. 

SemTxveviTi sididis Semdegi mniSvnelovani ricxviTi maxasiaTeblebia 

kvantilebi. 

SemTxveviTi ξ sididis p - uri rigirs kvantili ewodeba iseT p x 

ricxvs, romelic akmayofilebs pirobas F( x p ) = p , sadac F (x) 

aris ξ Sem- 

TxveviTi sididis ganawilebis funqcia. 0,75 da 0,25 doneebis mqone kvantilebs 

kvartilebs uZaxian. 0.1, 0.2, 0.3 da a.S. 0.9 doneebis mqone kvantilebs decilebi 

hqviaT. 

17

1.6. damoukidebeli da damokidebuli SemTxveviTi sidideebi 

damoukidebloba da damokidebuleba SemTxveviTi sidideebisaTvis Zalze 

mniSvnelovani maxasiaTeblebia. zemoT SemovitaneT damokidebulebis cneba 

xdomilebebisaTvis. analogiured SemovitanoT SemTxveviTi sidideebis damokidebulebis 

cneba. 

or SemTxveviT ξ da η sidides ewodeba damoukidebeli SemTxveviTi sidideebi, 

Tu adgili aqvs tolobas P( A ⋅ B) 

= P( 

A) 

⋅ P( 

B), 

sadac A da B arian 

xdomilebebi A = ( a1 

< ξ < a2 

), B = ( b1 

< η < b2 

), xolo a 1 , a2 

, b1, 

b2 

– nebismieri 

ricxvebia. 

damoukidebeli SemTxveviTi sidideebisaTvis adgili aqvs tolobebs 

M ( ξ ⋅ η) 

= Mξ 

⋅ Mη 

, 

D ( ξ + η) 

= Dξ 

+ Dη 

. 

SemTxveviTi sidide ξ damokidebulia meore SemTxveviT η sidideze, 

Tu ξ SemTxveviTi sididis albaTobebis ganawilebis kanoni aris damokidebuli 

imaze, Tu ra mniSvneloba miiRo η SemTxveviTma sididem. SemovitanoT 

SemTxveviTi sidideebis damokidebulebis xarisxis damaxasiaTebeli zoma. 

SemTxveviTi sidideebis damokidebulebis uamravi maxasiaTebeli arsebobs. 

maT Soris yvelaze gavrcelebulia kovariaciisa da korelaciis koeficientebi, 

romlebic axasiaTeben SemTxveviTi sidideebis wrfiv damokidebulebebs. 

ori SemTxveviTi ξ da η sididis kovariaciis koeficienti gansazRvrebiT 

aris Semdegnairad gamoTvlili ricxvi 

cov( ξ, η) 

= M[( ξ − Mξ 

)( η − Mη)] 

= Mξη 

− MξMη 

. 

kovariacia, dispersiis analogiurad, aris meore rigis centraluri 

momenti. kovariaciis koeficientis mniSvneloba damokidebulia ξ da η Sem- 

TxveviTi sidideebis gazomvis erTeulze. erTi erTeulidan meoreze gadasvlisas 

misi mniSvneloba icvleba, Tumca ξ da η Soris damokidebulebis 

xarisxi rCeba ucvleli. amitom ufro mosaxerxebelia damokidebulebis xarisxis 

iseTi maxasiaTeblis arseboba, romelic ar iqneba damokidebuli 

gazomvis erTeulisagan. aseTi maxasiaTebelia korelaciis koeficienti. 

M[( 

ξ − Mξ 

)( η − Mη)] 

cor( 

ξ, 

η) 

= ρ( 

ξ, 

η) 

= 

, 

Dξ 

Dη 

sadac D ξ > 0 , Dη 

> 0 . 

korelaciis koeficientis Tvisebebia: 

1) 

' ' 

ρ ( ξ, 

η) 

= ρ( 

ξ , η , sadac ' 

' 

ξ = a 1 + a2ξ 

, η = b1 

+ b2η, 

ricxvebia; 

2) −1 ≤ ρ ( ξ, 

η) 

≤ + 1; 

18 

a , a , b , b - nebismieri 

3) ρ ( ξ, 

η) 

= 1 maSin da mxolod maSin, roca ξ da η SemTxveviT sidideebs So- 

ris arsebobs wrfivi kavSiri; 

4) ρ ( ξ, 

η) 

= 0 Tu ξ da η damoukidebeli SemTxveviTi sidideebia. zoga- 

dad Sebrunebul mtkicebas adgili ara aqvs, anu SeiZleba korelaciis koeficienti 

iyos nolis toli, magram SemTxveviTi sidideebi ar iyvnen damoukideblebi, 

radgan korelaciis koeficienti asaxavs mxolod wrfiv damokidebulebas. 

amitom SeiZleba korelaciis koeficienti nolis toli iyos, magram 

1 

2 

1 

2

SemTxveviTi sidideebs Soris arsebobdes ara wrfivi kavSiri. damoukidebloba 

da korelaciis koeficientis nolTan toloba sinonimebia praqtikaSi 

Zalzed farTod gavrcelebuli normalurad ganawilebuli SemTxveviTi 

sidideebisaTvis. amis Sesaxeb ufro dawvrilebiT qvemoT iqneba naTqvami, 

organzomilebiani normaluri kanonis ganxilvisas. 

rodesac korelaciis koeficienti erTis tolia anu ρ ( ξ, 

η) 

= + 1, 

maSin 

ξ da η SemTxveviTi sidideebs Soris arsebobs dadebiTi wrfivi kavSiri, 

xolo Tu korelaciis koeeficienti minus erTis tolia, anu ρ ( ξ, 

η) 

= −1, 

ma- 

Sin ξ da η SemTxveviTi sidideebs Soris arsebobs uaryofiTi wrfivi kavSiri. 

1.7. SemTxveviTi amonarCevi 

SemTxveviT movlenebis, maT Soris SemTxveviTi sidideebis Seswavlisas 

Zalze iSviaTadad aris cnobili albaTobebis ganawilebis kanoni. amitom 

Seswavla eyrdnoba SemTxveviTi sididis mniSvnelobebze dakvirvebis Sedegebs, 

anu atareben eqsperimentebis simravles da afiqsireben n x x x ,..., , 1 2 

SemTxveviTi sididis mier miRebul mniSvnelobebs. dakvirvebis am 

mniSvnelobebis gamoyenebiT gamoiTvleba Sesaswavli SemTxveviTi sididis 

esa Tu is maxasiaTeblebi. imis gamo, rom dakvirvebis Sedegebis miReba 

dakavSirebulia garkveul materialur da droiT danaxarjebTan, romlebic 

xSirad sakmaod mniSvnelovania, praqtikaSi maTi raodenoba SezRudulia. 

SemTxveviTi sididis yvela SesaZlo mniSvnelobebis simravles generalur 

amonarCevs uwodeben, xolo SemTxveviTi sididis yvela SesaZlo 

mniSvnelobebidan n x x x ,..., , 1 2 dakvirvebebis sasrulo raodenobas, romelTa 

saSualebiTac Seiswavlian SemTxveviT sidides, amonarCevs eZaxian. Amocana 

mdgomareobs iseTi amonarCevis miRebaSi, romelic maqsimalurad srulad 

asaxavs generaluri amonarCevis yvela Tvisebas. es miiRweva generaluri 

amonarCevidan TiTo – TiTo obieqtis mimdevrobiT da wminda SemTxveviT 

amorCeviT. magaliTad, konkretul SemTxvevaSi, n raodenobis obieqtebidan 

erTis amorCevisas, aucilebelia, rom yoveli obieqtis amorCevis albaToba 

iyos 1 / n - is toli. im SemTxvevaSi, rodesac amonarCevi asaxavs generaluri 

amonarCevis ara yvela Tvisebas, aramed mis romelime mxares, amonarCevs 

hqvia wanacvlebuli amonarCevi. wanacvlebul amonarCevs, rogorc wesi, 

mivyavarT mcdar daskvnamde, radgan is srulad ver asaxavs SemTxveviTi 

sididis Tvisebebs. 

SemTxveviTi amorCevis principis darRvevas zogjer mivyevarT seriozul 

Secdomebamde. warumatebeli amonarCevis magaliTad mogvyavs amerikis 

SeerTebuli Statebis mosaxleobis gamokiTxvis Sedegebi, romelic Catarda 

1936 wels, rodesac qveynis prezidentis postze kenWs iyrida ori kandidati, 

ruzvelti da landoni. avtoritetulma Jurnalma „literaturuli 

mimoxilva“, sazogadoebaSi Tavisi reitingis amaRlebis mizniT, moaxdina 4 

milioni ameriklis gamokiTxva imisaTvis, rom ewinaswarmetyvela amerikis 

momavali prezidentis vinaoba. satelefono wignebidan amoiweres 4 milioni 

adamianis misamarTi da gaugzavnes Sesabamisi SekiTxva. miRebuli pasuxebis 

19

damuSavebis Sedegad Jurnalma gamoaqveyna informacia, rom arCevnebSi didi 

upiratesobiT gaimarjvebda landoni, Tumca ki cxovrebam sapirispiro Sedegi 

aCvena. aseTive gamokiTxva Caatares amerikelma sociologebma gelapma 

da rouperma. maT gamokiTxes mxolod oTxi aTasi amerikeli, amasTan cdilobdnen 

rac SeiZleba srulad moecvaT mosaxleobis yvela fenebi. maTi 

Sedegi aRmoCnda Jurnalis mier miRebuli Sedegis sawinaaRmdego. miuxedavad 

imisa, rom amonarCevi iyo gacilebiT ufro mcire moculobis, Sedegi 

aRmoCnda swori wina Sedegisagan gansxvavebiT. mizezi mdgomareobda imaSi, 

rom redaqciis specialistebma dauSves ramodenime seriozuli Secdoma respondentebis 

amonarCevis formirebisas: a) maT ver gaiTvaliswines, rom telefonis 

wignebSi, gansakuTrebiT im dros, warmodgenili iyo mosaxleobis 

SeZlebuli fena; b) pasuxi daubruna ara yvela gamokiTxulma, aramed im saqmianma 

adamianebma, romlebic miCveulni iyvnen korespondenciaze pasuxis 

gacemas. landons swored es fena uWerda mxars da miRebul SedegebSi aisaxa 

swored maTi ganwyoba. meore SemTxvevaSi amonarCevi Tavisufali iyo am 

Secdomisagan. cnobilia, rom sazogadoebis erTi da igive fenis warmomadgenlebs 

aqvT daaxloebiT erTnairi fsiqologia da damokidebuleba movlenebisadmi. 

amitom mosaxleobis dayofa fenebad da TiToeuli fenidan Tanabari 

raodenobis respodentis SerCeva iZleva obieqtur warmodgenas mTeli 

sazogadoebis ganwyobis Sesaxeb. pirvel SemTxvevaSi amonarCevi wanacvlebuli 

iyo garkveuli mimarTulebiT, amitom mis safuZvelze miRebuli iyo 

araswori Sedegi. rogorc ukve avRniSneT, aseT amonarCevebs wanacvlebuli 

amonarCevebi hqviaT. arseboben praqtikaSi waunacvlebeli amonarCevebis 

miRebis meTodebi. dawvrilebiT es sakiTxi ganxilulia [55] – Si. 

1.8. amonarCevebi da maTi aRwera 

SemTxveviTi sididis yvela SesaZlo mniSvnelobebidan amonarCevi 

ewodeba erTmaneTisagan damoukidebel, erTnairad ganawilebul SemTxveviT 

sidideTa mimdevrobas n x x x ,..., , 1 2 . rogorc wesi, amonarCevs warmoadgenen 

cxrilis saxiT. amonarCevis didi n moculobisas dakvirvebis yvela Sedegis 

mimoxilva SeuZlebeli xdeba, amitom cdiloben amonarCevi warmoadginon 

SesaZleblobis farglebSi kompaqturad da damuSavebisaTvis mosaxerxebeli 

formiT. aseTi warmodgenis erT – erT SesaZlebel formas warmoadgens 

alaTobebis ganawilebis empiriuli (statistikuri) funqcia (x) 

. ξ Sem- 

TxveviTi sididis empiriuli ganawilebis funqcia Fn (x) 

tolia iseTi i x 

mniSvnelobebis wilisa, romelTaTvisac adgili aqvs pirobas xi ≤ x, 

i = 1,..., 

n . 

avRniSnoT x( i ) , i = 1,..., 

n, 

dakvirvebis Sedegebis variaciuli rigi. maSin 

ni 

Fn ( x( i) ) = P( ξ ≤ x( 

i) 

) = , sadac n i aris dakvirvabis Sedegebis raodenoba rom- 

n 

≤ x . naxaz 1.7. – ze mocemulia empiriuli ganawilebis funqciis 

lebic (i) 

grafikuli saxe. 

20 

F n

nax. 1.7. 

ganawilebis empiriul funqcias aqvs diskretuli SemTxveviTi sididis 

ganawilebis funqciis analogiuri saxe. es aixsneba Semdegnairad. uwyveti ξ 

SemTxveviTi sididis amonarCevi n x x x ,..., , 1 2 aris n diskretul 

mniSvnelobaTa erToblioba. amonarCevis ganmartebis Tanaxmad n x x x ,..., , 1 2 

arian damoukidebeli SemTxveviTi sidideebi da TiToeuli maTganis 

Sesabamisi albaToba tolia 1 / n – is. amitom amonarCeviT empiriuli ganawilebis 

funqciis ageba Seesabameba iseTi diskretuli SemTxveviTi sididis 

ganawilebis funqciis agebas, romlis SesaZlo mniSvnelobebia n x x x ,..., , 1 2 da 

TiToeul am mniSvnelobis miRebis albaToba tolia 1 / n – is. iseve, rogorc 

xdomilebis sixSire miiswrafvis Sesabamisi albaTobisken (ix. paragrafi 1.3), 

rodesac dakvirvebis ricxvi n usasrulod izrdeba, empiriuli ganawilebis 

funqcia Fn (x) 

miiswrafis uwyveti SemTxveviTi sididis F (x) 

ganawilebis 

funqciisaken, rodesac n → ∞ . es imas niSnavs, rom n – is usasrulod 

gazrdisas naxazze naCvenebi safexurovani funqcia Fn (x) 

gadadis Sesabamis 

F (x) 

uwyvet funqciaSi. 

rogorc viciT, xSir SemTxvevaSi sakmarisia SemTxveviTi sididis ara 

mTliani bunebis codna (rasac gvaZlevs ganawilebis funqciis codna), aramed 

misi calkeuli mxareebis, anu ricxviTi maxasiaTeblebis codna. 

ganvixiloT dakvirvebis SedegebiT ricxviTi maxasiaTeblebis gansaz- 

Rvris sakiTxi, kerZod, ZiriTadi ricxviTi maxasiaTeblebisa, rogoricaa ma- 

Tematikuri molodini, dispersia, saSualo kvadratuli gadaxra, kovariaciisa 

da korelaciis koeficientebi, kvantilebi. maT Sesabamisad hqviaT amonarCevis 

saSualo, amonarCevis dispersia, amonarCevis kovariacia da korelacia, 

amonarCevis kvantili. 

amonarCevis saSualo aRiniSneba Semdegnairad x . mas dakvirvebis Sedegebis 

saSualo ariTmetikuls eZaxian da gamoiTvleba formuliT 

n 1 

x = ∑ xi 

. dakvirvebis SedegebiT SemTxveviTi sididis dispersiis mniSvne- 

n i= 

1 

2 1 n 

2 

loba gamoiTvleba formuliT S = ∑ ( x − x) 

. xSirad dispersiis mniSvne- 

lobas iTvlian formuliT 

2 

M ( S ) = D( 

ξ ) . 

* 

amonarCevis kvantili ewodeba sidides 

tolebis amonaxsni 

n i= 

1 

i 

n 

2 1 

S * = ∑= ( xi 

n −1 

i 1 

2 

− x) 

, radgan adgili aqvs 

21 

* 

x p , romelic aris Semdegi gan

Fn ( x) 

= p, 

(1.1) 

sadac Fn (x) 

aris albaTobebis empiriuli ganawilebis funqcia, p aris alba- 

Toba 0 

gasagebia, rom gantolebas (1.1) yovelTvis ara aqvs amoxsna. amitom mi- 

Rebulia Semdegi SeTanxmeba. magaliTad, empiriuli medianis ∗ 

Med gansaz- 

Rvrisas saWiroa Fn ( x ) = 0.5 gantolebis amoxsna. dakvirvebebis kenti ricxvis 

∗ 

dros n = 2 k + 1 empiriuli mediana Med = x(k 

) , xolo dakvirvebebis wyvili 

∗ 

ricxvis dros n = 2k 

empiriuli mediana Med = ( x( 

k ) + x( 

k+ 

1) 

) / 2 . 

kovariaciis koeficientis mniSvneloba amonarCeviT gamoiTvleba formuliT 

n 1 

cov( ξ, 

η) 

= ( x − x)( 

y − y), 

∑ n i= 

1 

sadac xi , yi 

, i = 1,..., 

n aris ξ da η SemTxveviT sidideebze dakvirvebis Sedegebi. 

korelaciis koeficientis mniSvneloba amonarCeviT gamoiTvleba formuliT 

ρ = 

n 

∑ 

i= 

1 

n 

∑ 

i= 

1 

( x − x) 

i 

22 

i 

( x − x)( 

y − y) 

i 

2 

n 

i 

∑ 

i= 

1 

i 

( y 

i 

− y) 

danarTi 9 – Si mocemulia amonarCeviT gamoTvlili korelaciis koeficientis 

ganwilebis kvantilebi . 

1 

2 

p r 

− 

1.9. rangi da ranJireba 

xSirad eqsperimentis Sedegebi arian ara ricxvebi, aramed sidideebi, 

romlebic erTmaneTTan mimarTebas axasiaTeben cnebebiT “meti” an “naklebi”. 

magaliTad, codnis Sefasebisas, raime sagnis, movlenis an pirovnebis mimarT 

damokidebulebis gamoxatvisas, an ferTa gamis gansazrvrisas dawerili 

qulaTa raodenoba axasiaTebs ukeTes an uares codnas, met an nakleb simpatias 

da a.S. dawerili balebis mixedviT ar SeiZleba zustad gainsazRvros 

ramdenad metia an naklebia codna. magaliTad, ar SeiZleba Tqma, rom 

2 

“xuTianis” mimRebma studentma sagani zustad 5 / 3 = 1 - jer ukeTesad icis 

3 

sagani, vidre “sami” qulis mimRebma studentma. aseTi monacemebis damuSavebisas 

gamoiyeneba maTematikuri statistikis specialuri meTodebi, romlebic 

operireben eqsperimentis ara konkretul mniSvnelobebze, aramed maT adgilebze 

dakvirvebaTa miRebul mwkrivSi. aseT meTodebs ranguli meTodebi 

hqviaT. dakvirvebis i x Sedegis rangi hqvia mis rigiT nomers x ( 1) 

, x( 

2) 

,..., x( 

n) 

variaciul rigSi. magaliTad, vTqvaT dakvirvebis Sedegebi Sesdgeba ricxvebisagan 

5, 7, 3, 1, 12, maSin dakvirvebaTa am mwkrivSi, maTi Sesabamisi rangebi 

2 

.

iqneba 3, 4, 2, 1, 5. dakvirvebaTa simravlidan maTi rangebis mimdevrobaze gadasvlis 

proceduras ranJirebis procesi hqvia, xolo miRebul Sedegs – 

ranJirebis Sedegi. 

praqtikuli amocanebis gadawyvetisas, dakvirvebis Sedegebis, anu amonarCevis, 

damrgvalebis gamo, gvxvdeba erTnairi Sedegebi. am SemTxvevaSi Sesabamisi 

elementebis rangebis gansazRvra xdeba Semdegnairad. erTnairi 

mniSvnelobis mqone dakvirvebis Sedegebs amonarCevSi Sekvra an kona davarqvaT, 

xolo dakvirvebis raodenobas SekvraSi – misi zoma. SekvraSi moxvedrili 

dakvirvebis Sdegebis rangebi erTmaneTis tolia da utoldeba am dakvirvebis 

Sedegebis rangebis saSualo ariTmetikuls, romlebic eqneboda 

dakvirvebis variaciul rigSi, maTi mniSvnelobebis mixedviT, erTmeneTis 

gverdiT ganTavsebisas. magaliTad, dakvirvebis Sedegebis 3; 7; 3; 1; 12 Sesabamisi 

rangebi iqneba 2,5; 4; 2,5; 1; 5. 

statistikuri meTodebis umravlesoba dafuZnebulia daSvebaze, rom gansaxilvel 

SemTxveviT sidides aqvs albaTobebis garkveuli ganawilebis kanoni. 

am meTodebs parametruli meTodebi hqviaT. umravlesoba am meTodebisa 

uSveben ganawilebis kanonis normalurobas. Tu realuri amocanebis gadawyvetisas 

aRmoCnda, rom es daSveba ara sworia, maSin miRebuli Sedegebi, 

didi albaTobiT, iqnebian ara swori. ranguli meTodebi ar iTxoven aseT 

daSvebebs. amitom maT iyeneben agreTve dakvirvebebis ricxviTi monacemebis 

damuSavebisas, rodesac albaTobebis ganawilebis kanoni ucnobia da dakvirvebebis 

mcire ricxvi ar iZleva maTi identifikaciis saSualebas. 

1.10. aRweriTi statistikis meTodebi 

dakvirvebaTa simravle xSirad sakmaod didi moculobisaa, Sedgeba aTeulobiT, 

aseulobiT, aTaseulobiT dakvirvebebis Sedegebisagan, rac aZnelebs 

dakvirvebaTa Sedegebis uSualo mimoxilvasa da analizs. amitom warmoiSoba 

dakvirveebaTa Sedegebis kompaqturi warmodgenis amocana. idealSi 

aseTi kompaqturi warmodgena iqneboda faqti, rom dakvirvebis Sedegebi 

warmoadgenen amonarCevs, anu damoukidebel dakvirvebis Sedegebs mocemuli 

ganawilebis kanonis mqone SemTxveviTi sididisaTvis. es saSualebas mogvcemda 

yvela saWiro gamoTvlebi Cagvetarebina Teoriulad. magram, saubedurod, 

xSirad, praqtikuli amocanebis amoxsnisas, ganawilebis kanonebi 

ucnobia. amitom, aseT SemTxvevebSi, amonarCevis kompaqturi warmodgenisa- 

Tvis sargebloben aRweriTi statistikis meTodebiT. 

aRweriTi statistikis meTodebs uwodeben sxvadasxva maxasiaTeblebiT 

amonarCevis aRweris da grafikuli warmodgenis meTodebs. amonarCevis ama 

Tu im Tvisebis maxasiaTeblad zemoT ganvixileT amonarCeviT gansazRvruli 

SemTxveviTi sididis ricxviTi maxasiaTeblebi, romlebic SeiZleba davyoT 

ramodenime jgufad. 

1. mdebareobis maxasiaTeblebi. es maxasiaTeblebi ricxviT RerZze miuTiTeben 

mdebareobas, sadac ganlagdebian dakvirvebis Sedegebi. maT miekuTvnebian: 

amonarCevis minimaluri da maqsimaluri elementebi, zeda da qveda kvartilebi, 

dakvirvebis Sedegebis saSualo ariTmetikuli ( x ), amonarCevis mediana 

( med )da sxva analogiuri maxasiaTeblebi. 

23

2. ganbnevis maxasiaTeblebi. isini axasiaTeben dakvirvebis Sedegebis gabnevas 

ricxviT RerZze TaviaanTi centris mimarT. maT miekuTvnebian: amonarCevis 

dispersia 2 

S , standartuli gadaxra, sxvaoba amonarCevis maqsimalur da minimalur 

elementebs Soris, romelsac gaqanebas eZaxian h = xmax 

− xmin 

, sxvaoba 

zeda da qveda kvantilebs Soris, eqscesis koeficienti da sxva. 

3. asimetriis maxasiaTeblebi. isini axasiaTeben monacemebis TaviaanTi centris 

mimarT ganlagebis simetrias. maT miekuTvnebian: asimetriis koeficienti, 

amonarCevis medianis mdebareoba saSualo ariTmetikulis mimarT da 

sxva. 

4. empiriuli ganawilebis kanonebi. esenia empiriuli ganawilebis funqcia, 

wertilovani diagrama, histograma, sixSireebis cxrilebi. wertilovani 

diagrama gamoiyeneba rodesac dakvirvebis SedegebSi erTi da igive dakvirvebis 

Sedegebi gvxvdeba mravaljer. aseT SemTxvevaSi abscisis Sesabamis wertilSi 

ixateba imdeni wertili, ramdenjerac es mniSvneloba gvxvdeba dakvirvebis 

SedegebSi. maxasiaTeblis saerTo saxe naCvenebia nax. 1.8 – ze. 

nax. 1.8. wertilovani diagrama nax. 1.9. histograma 

histograma aris albaTobebis ganawilebis empiruli analogi. is aigeba 

dakvirvebis SedegebiT Semdegnairad. dakvirvebis Sedegebidan x ( 1) 

, x( 

2) 

,..., x( 

n) 

ganisazRvreba x min = x( 

1) 

da x max = x( 

n) 

. amonarCevis warmodgenis intervali 

iyofa k qveintervalad. zogadad intervalebi SeiZleba iyvnen ara Tanaba- 

xmax − xmin 

ri. Tu intervalebi tolebia, maSin maTi sigrZe Δ x = . i – i interva- 

k 

lis sasazRvro wertilebi Semdegnairad ganisazRvrebian = x + ( i −1) 

⋅ Δx, 

x i = x + i ⋅ Δx, 

i = 1,..., 

k −1 

. iTvleba dakvirvebebis ricxvi yovel 

( + 1) 

( 1) 

intervalSi. dakvirvebebis raodenoba yovel intervalSi avRniSnoT n i , 

xolo p i aris am intervalSi dakvirvebis Sedegebis moxvedris sixSire. 

k 

cxadia, rom ∑ ni = n, i= 1 

ni 

pi = , 

n 

k 

∑ pi 

i= 

1 

= 1. 

yovel intervalze igeba sworkuTxedi, 

romlis farTobic p i – s tolia. histogramis saerTo saxe naCvenebia nax. 1.9 

24 

x( i) 

( 1)

– ze. intervalebis raodenoba k airCeva dakvirvebis Sedegebis raodenobisagan 

damokidebulebiT. dakvirvebis didi ricxvis dros intervalebis 

raodenoba izrdeba. 

albaTobebis ganawilebis empiriuli funqciis analogiurad, romelic 

amonarCevis moculobis gazrdisas, anu rodesac n → ∞ asimptoturad miiswrafis 

ganawilebis Teoriuli funqciisaken Fn ( x) → F( x) 

, histograma, rodesac 

n → ∞ , miiswrafis albaTobebis ganawilebis Teoriuli simkvrivisaken 

fn ( x) → f ( x) 

. 

dajgufebuli monacemebiT SemTxveviTi sididis maxasiaTeblebi gamoiTvleba 

Semdegnairad. 

0 

avRniSnoT x i - iT Sua wertili i – ri intervalis romlis sigrZe tolia 

Δ xi = x( i+ 1) − xi , i = 1,..., k . rogorc adre n i aris i intervalSi moxvedrili 

dakvirvebebis raodenoba. dajgufebuli monacemebiT amonarCevis saSualo 

ganisazRvreba Semdegnairad 

k 

0 n 1 k 

i 

0 

x = ∑ xi ⋅ = ∑ xi ⋅ ni 

. 

i= 1 n n i= 

1 

amonarCevis dispersia gamoiTvleba formuliT 

2 1 k 

0 2 

S = ∑ ( x − x) ⋅ n . 

n −1 

i= 

1 

25 

i i 

analogiurad gamoiTvleba sxva ricxviTi maxasiaTeblebis sidideebi 

dajgufebuli monacemebiT.

Tavi 2. albaTobebis ganawilebis mniSvnelovani kanonebi 

rogorc ukve viciT, ganawilebis kanonebi asaxaven damokidebulebas Sem- 

TxveviTi sididis mniSvnelobebsa da am mniSvnelobebis miRebis SesaZleblobebs 

anu albaTobebs Soris. viciT, rom yvelaze gavrcelebuli SemTxveviTi 

sidideebia diskretuli da uwyveti SemTxveviTi sidideebi. uwyveti Sem- 

TxveviTi sidideebis ganawilebis kanonebia: ganawilebis funqcia da ganawilebis 

simkvrive, xolo diskretuli SemTxveviTi sididisaTvis – ganawilebis 

funqcia da ganawilebis mwkrivi. 

winamdebare TavSi SeviswavliT yvelaze ufro farTod gavrcelebul da 

gamoyenebul diskretuli da uwyveti SemTxveviTi sidideebis ganawilebis 

kanonebs da maT Tvisebebs. kerZod, diskretuli SemTxveviTi sidideebisa- 

Tvis – binomaluri da puasonis, uwyveti SemTxveviTi sidideebisTvis – normaluri, 

stiudentis, maCvenebliani, Tanabari, xi – kvadrati da fiSeris ganawilebis 

kanonebi. 

2.1. binomialuri ganawileba 

binomaluri ganawilebis kanoni diskretuli SemTxveviTi sididis 

ganawilebis kanonia. am kanoniT aRiwereba mravali bunebrivi da teqnikuri 

procesebi. es iseTi procesebia, romlebsac eqsperimentis Sedegad SeuZliaT 

hqondeT ori mniSvneloba: warmateba da warumatebloba. magaliTad, sawarmoSi 

produqciis xarisxis kontrolis Sedegi SeiZleba iyos oridan erTi 

gadawyvetileba: nakeToba vargisia an uvargisia. masiuri warmoebis dros 

yoveli nakeTobis xarisxis Semowmeba ekonomiurad ara xelsayrelia. amitom 

kontrols axorcieleben Semdegnairad. drois garkveul monakveTSi gamoSvebuli 

nakeTobebis saerTo raodnobidan airCeven n nakeTobas da 

amowmeben maT xarisxs. nakeTobis xarisxianobas aRniSnaven, magaliTad, noliT, 

xolo uxarisxobas – erTiT. iTvlian erTianebis saerTo raodenobas. 

is tolia uxarisxo nakeTobebis ricxvis. bunebrivia 0 ≤ m ≤ n . igulisxmeba, 

rom calkeuli nakeTobis vargisianoba an uvargisoba damoukidebeli xdomilebebia, 

anu erTi nakeTobis vargisianoba an uvargisoba gavlenas ar axdens 

meore nakeTobis vargisianobaze. avRniSnoT p albaToba imisa, rom 

Sesamowmebeli nakeToba iqneba uxarisxo, maSin (1 − p) 

aris albaToba imisa, 

rom nakeToba iqneba xarisxiani. zemoT aRniSnuli damoukidebloba niSnavs, 

rom p albaTobis mniSneloba yoveli nakeTobisaTvis erTnairia. p - s mi- 

m 

axloebiTi mniSvneloba gamoiTvleba formuliT p ≈ . adgili aqvs 

n 

m 

→ p , 

n 

roca n → ∞ . 

aRvniSnoT X - iT SemTxveviTi diskretuli sidide, romelic Seesabameba 

uxarisxo nakeTobebis raodnobas n Semowmebul nakeTobaSi. X SeuZlia miiRos 

nebismieri mniSvneloba 0 – dan n – de, magram TiToeuli am mniSvnelobis 

miRebis albaToba sxvadasxvaa. zemoT moyvanili pirobebis Sesrulebisas, 

Sesabamisoba X SemTxveviTi sididis mniSvnelobebsa da albaTobebs 

Soris, romlebiTac SemTxveviTi sidide Rebulobs am mnSvnelobebs, aRiwe- 

26

eba ganawilebis kanoniT, romelsac ganawilebis binomialuri kanoni ewodeba 

da Semdegi saxe aqvs 

k k n k 

P( X k) C p (1 p) − 

= = − , (2.1) 

n 

k n! 

sadac Cn 

= da hqvia n - dan k dajgufebaTa ricxvi. 

k !( n − k)! 

im faqtis aRsaniSnavad, rom bernulis ganawileba damokidebulia p da 

n - gan albaTobas P( X = k) 

Caweren ase P( X = k | n, p) 

. 

SemTxveviTi sidide X – is maTematikuri molodini da dispersia Sesaba- 

misad tolia MX = np, DX = np(1 − p) 

. 

albaToba imisa, rom X SemTxveviTi sidide Rebulobs k – ze nakleb an 

tol mniSvnelobebs gamoiTvleba formuliT 

k 

m m n m 

P( X k) C p (1 p) − 

≤ = ∑ − . (2.2) 

m= 

0 

n 

nax. 2.1. – ze mocemulia binomialuri ganawilebis grafikebi p da 

n parametrebis sxvadasxva mniSvnelobebisaTvis. 

p 

= 0, 

2 

nax. 2.1. binomialuri ganawilebis kanonis saxe sxvadasxva p - Tvis, rodesac 

10 

n = . 

binomaluri ganawilebis kanoni mWidro kavSirSia sxva ganawilebis kanonebTan; 

magaliTad, puasonisa da normalur ganawilebi kanonebTan. Tu 

sruldeba piroba 0.1≤ p ≤ 0.9 da np(1 − p) 

> 5 , binomaluri kanoni kargad aproqsimirdeba 

normaluri ganawilebis kanoniT np – s toli maTematikuri 

molodiniTa da dispersiiT np(1 − p) 

. rodesac np(1 − p) 

> 25 es aproqsimacia 

SeiZleba gamoyenebuli iqnas p mniSvnelobisagan damoukideblad. 

sakmaod didi n – is da p < 0.1 dros binomaluri kanonis aproqsimacia 

SesaZlebelia puasonis ganawilebis kanoniT np – s toli maTematikuri molodiniT. 

albaTobebis ganawilebis binomaluri kanonis didi mniSvnelobis gamo 

misi mniSvnelobebi gamoTvlili (2.1) an (2.2) formuliT da mocemulia 

Sesabamis statistikur cxrilebSi sxvadasxva n - sa da p - Tvis [2, 29]. 

winamdebare wignis boloSi, danarT 1 – Si mocemulia es mniSvnelobebi 

zogierTi n da p - Tvis. 

binomaluri ganawilebis kanons agreTve eZaxian bernulis ganawilebis 

kanons. 

27 

p = 0, 

4

2.2. puasonis ganawileba 

puasonis ganawilebis kanonic aris diskretuli SemTxveviTi sididis ganawilebis 

kanoni. am kanons aqvs adgili bunebisa da teqnikis mraval amocanebSi. 

misi saSualebiT aRiwereba im movlenebis albaTobebic, romlebic dakavSirebulia 

SemTxveviTi raodenobis movlenebis warmoqmnasTan drois 

mocemul intervalSi. magaliTad, atomur fizikaSi – radioaqtiuri 

nivTierebis daSla drois mocemul periodSi; astronomiaSi – meteoritebis 

gamoCena drois mocemul intervalSi; radoilokaciaSi – yalbi signalebis 

warmoSoba arekvlili radiosignalebis miRebisas; kavSirgabmulobaSi – 

telefonis sadgurSi darekvebis raodenoba erTeulovani drois 

intervalSi da a.S. puasonis kanons adgili aqvs im SemTxvevebSi, rodesac 

movlenis warmoSobis albaToba proporciulia im Δ x intervalis, 

romelSic es movlena xdeba da tolia a ⋅ Δ x + OΔ x , sadac a > 0 mocemuli 

ricxvia, OΔ x aris usasrulod mcire sidide Δ x - Tan SedarebiT. am SemTxvevaSi, 

albaToba imisa, rom X SemTxveviTi sidide T drois ganmavlobaSi 

Rebulobs k - s tol mniSvnelobas gamoiTvleba formuliT 

k 

λ − 

λ 

P( X = k) = e , k = 0,1, 2,..., 

k! 

sadac λ = a ⋅ T - aris puasonis kanonis intensiuroba. Adgili aqvs pirobebs 

MX = λ , DX = λ . 

puasonis kanoni dakavSirebulia bernulisa da normalur ganawilebis 

kanonebTan. roca λ > 9 puasonis ganawilebis kanoni kargad aproqsimdeba 

albaTobebis normaluri ganawilebiT λ toli maTematikuri molodiniTa 

da dispersiiT. 

damoukidebeli n SemTxveviTi sididis jami, romlebic ganawilebuli 

arian Sesabamisad λ1, λ2,..., λ n parametrebiani puasonis kanonebiT, agreTve ga- 

nawilbulia puasonis kanoniT λ = λ1 + λ2 + ... + λn 

parametriT. 

naxaz 2.2 – ze mocemulia puasonis ganawilebis kanonis sqematuri saxe λ 

- s sxvadasxva mniSvnlobisaTvis. 

λ = 1 

λ 

= 6 

nax. 2.2. puasonis ganawilebis saxe k da λ sxvadasxva mniSvnelobebisa- 

Tvis. 

puasonis ganawilebis kanonis mniSvnelobebi gamoTvlilia da mocemulia 

Sesabamis cxrilebSi λ - s sxvadasxva mniSvnelobisaTvis. zogierT maT- 

28

ganSi mocemulia albaTobebi P( X = k) 

[29], xolo zogierTSi – dagrovili 

k 

m 

λ − 

λ 

albaTobebi P( X ≤ k) = ∑ e . winamdebare wignis boloSi, danarT 2 – Si, 

m= 

0 m! 

moyvanilia albaTobebi P( X = k) 

zogierTi λ - Tvis. 

2.3. maCvenebliani anu eqsponencialuri ganawileba 

eqsponencialuri kanoni aris uwyveti SemTxveviTi sididis ganawilebis 

kanoni. es kanoni xSirad gamoiyeneba e.w. “sicocxlis xangrZliobis” amocanebSi, 

anu amocanebSi, sadac movlenis albaToba damokidebulia im drois 

xangrZliobisagan, romlis ganmavlobaSic mas adgili aqvs. magaliTad, medicinaSi 

– pacientis sicocxlis xangrlivoba; saimedobis TeoriaSi – nakeTobis 

umtyuno muSaobis xangrZlioba; masobrivi momsaxurebis TeoriaSi – momsaxurebaze 

lodinis dro. 

eqsponencialuri ganawilebis simkvrives aqvs Semdegi saxe 

x 

p( , x) e θ − ⋅ 

θ = θ ⋅ , θ > 0 . 

sadac θ – ganawilebis parametria. albaTobebis ganawilebis funqcias aqvs 

saxe 

⎧ −θ 

⋅x 

⎪1 

−θ 

⋅ e , at x ≥ 0, 

F( 

θ , x) 

= ⎨ 

⎪ 

⎩0, 

at x < 0 

eqsponencialuri ganawilebas xSirad eZaxian mexsierebis armqone 

ganawilebasac, radgan adgili aqvs Semdeg pirobas 

P( X ≥ s + t | X ≥ t) = P( x ≥ s) 

nebismieri s, t ≥ 0 . 

vTqvaT X aris raime nakeTobis umtyuno muSaobis xangrZlioba (magali- 

Tad, televizoris). maSin aRniSnuli Tviseba niSnavs, rom mowyobilobisa- 

Tvis, romelmac imsaxura t drois ganmavlobaSi, albaToba rom damatebiT 

imuSavebs kidev s drois ganmavlobaSi, iseTivea rogorc albaToba imisa, 

rom aseve s drois ganmavlobaSi imuSavebs axali mowyobiloba, romelmac 

mxolod axla daiwyo muSaoba. Znelia ar aris imis mixvedra, rom praqtika- 

Si es piroba, rogorc wesi, ar sruldeba. am naklis aRmofxvris mizniT im 

amocanebisaTvis, romlebisTvisac movlenis wina istoria mniSvnelovania, 

eqsponencialuri kanonis nacvlad iyeneben ufro zogad kanonenebs, romlebic 

iTvaliswineben winaistorias. magaliTad, gama ganawileba, veibulis ganawileba, 

an romelime sxva ganawileba, romelTa kerZo SemTxvevasac warmoadgens 

eqsponencialuri ganawileba. 

eqsponencialuri ganawilebis mqone SemTxveviTi sididis maTematikuri 

1 

1 

molodini MX = , xolo dispersia – DX = . nax. 2.3 – ze naCvenebia θ pa- 

2 

θ 

θ 

rametris mqone eqsponencialuri ganawilebis simkvrivis saxe. 

29

nax. 2.3. 

2.4. normaluri ganawileba 

normalur ganawilebas centraluri adgili ukavia albaTobis Teoriasa 

da maTematikur statistikaSi misi zRvruli Tvisebebis gamo. es gansakuTrebuloba 

gamomdinareobs ori momentidan: a) mravali SemTxveviTi sidide, 

romlis formirebazec moqmedebs didi raodenobis sxvadasxva faqtori da 

maTi gavlena daaxloebiT Tanabaria, ganawilebuli arian normaluri kanonis 

Tanaxmad; aseTebia, magaliTad, umravlesi gazomvis Sedegebi; b) xSirad, 

rodesac SemTxveviTi sidide ar aris ganawilebuli normaluri kanoniT, 

misi ganawilebis kanoni, garkveul pirobebSi, SeiZleba aproqsimirebuli 

iqnas normaluri kanoniT. statistikuri kriteriumebis didi umravlesoba 

damuSavebulia albaTobebis ganawilebis normaluri kanonisaTvis. normaluri 

kanoni aris yvelaze ufro Seswavlili ganawilebis yvela sxva kanonebTan 

SedarebiT. normaluri ganawilebis kanonis simkvrives aqvs Semdegi 

saxe 

2 ⎧ ⎫ 

1 ( x − a) 

ϕ( 

x) = exp ⎨− , x 

2 ⎬ − ∞ < < +∞ . 

2π 

⋅σ ⎩ 2⋅σ 

⎭ 

normaluri ganawilebis kanons aqvs ori a da 

a aris normalurad ganawilebuli SemTxveviTi sididis maTematikuri 

2 

molodini, xolo σ – saSualo kvadratuli gadaxra. normaluri ganawilebis 

kanonis simkvrivis grafiki naCvenebia nax. 2.4 – ze 

30 

p( 

x, 

θ ) = θ ⋅ e 

−θ 

⋅x 

2 

σ parametri. parametri

nax. 2.4. albaTobebis ganawilebis normaluri kanonis simkvrive 

rogorc grafikidan Cans ϕ ( x) 

miiswrafis nolisaken rodesac x → −∞ an 

x → +∞ . simkvrive simetriulia a wertilis mimarT. amasTan a wertilSi 

funqcia ϕ ( x) 

aRwevs Tavis maqsimums, romelic tolia 1/( 2 π ⋅ σ ) . 

parametri a axasiaTebs ganawilebis simkvrivis grafikis mdebareobas 

ricxviT RerZze. parametri σ > 0 axasiaTebs simkvrivis grafikis gaSlis an 

SekumSvis xarisxs. 

gansakuTrebuli adgili ukavia normaluri ganawilebis kanons, romlis 

maTematikuri molodini nolis, xolo dispersia erTis tolia. aseT Sem- 

TxveviT sidides normirebuli SemTxveviTi sidide ewodeba da mis simkvrives 

aqvs saxe 

2 ⎧ ⎫ 

1 x 

ϕ( 

x) = exp ⎨− ⎬, 

− ∞ < x < +∞ . 

2π 

⎩ 2 ⎭ 

is faqti, rom X SemTxveviTi sidide gnawilebulia normalurad maTema- 

2 

tikuri molodiniT a da dispersiiT σ , miRebulia Caiweros Semdegnairad 

2 

X ~ N( a, σ ) . normirebuli SemTxveviTi sididisaTvis gvaqvs X ~ N (0,1) . 

2 

ξ − a 

vTqvaT adgili aqvs ξ ~ N( a, 

σ ) , maSin samarTliania η = ~ N(0,1) 

. 

σ 

normirebuli SemTxveviTi η sididisaTvis adgili aqvs: albaToba imisa, 

rom SemTxveviTi sidide Rebulobs mniSvnelobebs intervalidan (-2; +2), e.i. 

p( −2 ≤η ≤ + 2) = 0,94 , agreTve samarTliania p( −3 ≤η ≤ + 3) = 0,9933 . ukanasknels 

ewodeba sami sigmas kanoni, radgan ara normirebuli SemTxveviTi 

sididsaTvis mas aqvs saxe p( a − 3⋅σ ≤η ≤ a + 3 ⋅ σ ) = 0,9933 . 

normirebuli normalurad ganawilebuli SemTxveviTi ξ ~ N(0,1) 

sididisaTvis 

ganawilebis funqcia Φ ( x) 

– iT aRiniSneba. adgili aqvs pirobas 

Φ ( x) = 1 − Φ( − x) 

. am Tvisebis safuZvelze yovelTvis SegviZlia gamovTvaloT 

normirebuli ganawilebis funqciis mniSvneloba ricxviTi RerZis marcxena 

naxevarze misi mniSvnelobebiT ricxviTi RerZis marjvena naxevridan. am 

Tvisebis gamoyenebiT cxrilebSi yovelTvis mocemulia ganawilebis funqciisa 

da simkvrivis mniSvnelobebi mxolod x ≥ 0 - Tvis. es cxrilebi ama Tu im 

moculobiT mocemulia albaTobis Teoriisa da maTematikuri statistikis 

31

praqtikulad yvela wignSi [ix. magaliTad 3, 14, 20]. erT – erTi aseTi 

cxrili mocemulia winamdebare wignis boloSic (ix. danarTi 3). normirebuli 

normaluri SemTxveviTi sididis cxriliT advilad gamoiTvleba aranormirebuli 

normaluri SemTxveviTi sididis Sesabamisi mniSvnelobebi. 

2 

marTlac, vTqvaT η ~ N( a, 

σ ) da ξ ~ N(0,1) 

, maSin adgili aqvs 

a x a x a 

F( x) P( x) P η ⎛ − − ⎞ ⎛ − ⎞ 

= η < = ⎜ < ⎟ = Φ ⎜ ⎟ 

⎝ σ σ ⎠ ⎝ σ ⎠ . 

2.5. organzomilebiani normaluri ganawileba 

iseve rogorc erTganzomilebiani SemTxveviTi sididis SemTxvevaSi, zogadad 

n ganzomilebiani SemTxveviTi sididisaTvis ganawilebis funqcia 

calsaxad ganisaRvreba ganawilebis simkvriviT. 

ξ SemTxveviTi sidideebi ganawilebuli arian normalu- 

vTqvaT ξ 1 da 2 

2 2 

rad Sesabamisad maTematikuri molodinebiT a1, a 2 da dispersiebiT σ1 , σ 2 , 

2 

2 

anu ξ ~ N( a , σ ) da ξ ~ N( a , σ ) . maSin organzomilebiani SemTxveviTi sidi- 

1 1 1 

de ξ ( ξ1, ξ2 

) 

2 2 2 

= ganawilebulia organzomilebiani normaluri ganawilebis 

kanoniiT. Tu SemTxveviTi sidideebi ξ 1 da ξ 2 normirebuli da damoukideblebi 

arian, anu maTi maTematikuri molodinebi nolis tolia, dispersiebi – 

erTis, xolo korelacia maT Soris nolis tolia, maSin ξ = ( ξ1, ξ2 

) organzomilebiani 

SemTxveviTi sididis ganawilebis normalur simkvrives aqvs saxe 

32 

2 2 ⎧ ⎫ 

1 x + y 

p( x, y) 

= exp ⎨− ⎬. 

2π ⎩ 2 ⎭ 

erTganzomilebiani SemTxveviTi sididis ganwilebis funqcia aris x 

cvladis (ganawilebis funqciis argumentis) mocemuli mniSvnelobidan marcxniv 

moxvedris albaToba. analogiurad ganisazRvreba organzomilebiani, 

samganzomilebiani da a.S. SemTxveviTi sididis ganawilebis kanoni. 

avRniSnoT, magaliTad X - iT raime are organzomilebian sivrceSi. maSin am 

areSi moxvedris albaToba tolia 

P( X ) = ∫∫ p( x, y) dxdy . 

X 

organzomilebiani SemTxveviTi ξ sidids ganawilebis simkvriviT advilad 

SeiZleba ganvsazrvroT ξ 1 da ξ 2 erTganzomilebiani SemTxveviTi sidideebis 

ganawilebis simkvriveebi Semdegnairad 

p ( x) p( x, y) dy 

1 

+∞ 

= ∫ , 2 

−∞ 

am formulebiT adviled gamoiTvleba 1 

maTematikuri molodinebi 

1 

+∞ 

p ( y) = ∫ p( x, y) dx . 

−∞ 

ξ da 2 

+∞ +∞ 

+∞ +∞ 

a = ∫ ∫ xp( x, y) dxdy , a2 = ∫ ∫ yp( x, y) dxdy . 

−∞ −∞ 

−∞ −∞ 

ξ SemTxveviTi sidideebis 

zogadad ξ 1 da ξ 2 korelirebuli SemTxveviTi sidideebisaTvis organzomilebian 

gnawilebis simkvrives aqvs Semdegi saxe

2 

1 ⎧⎪ 1 ⎡( x1 − a1) ( x1 − a1)( x2 − a2) ( x2 − a2) 

⎤⎫⎪ 

p( x, y) = × exp ⎨− 2 ρ 

, 

2 

2 ⎢ − + ⎥⎬ 

2 π σ 2(1 ) 

11σ 22(1 

− ρ ) ⎪ − ρ σ11 σ σ 

⎩ ⎢⎣ 11σ 22 

22 ⎥⎦ 

⎪⎭ 

sadac a 1 da a 2 arian Sesabamisad ξ 1 da ξ 2 SemTxveviTi sidideebis maTemati- 

2 2 

kuri molodinebi, xolo σ , σ - dispersiebi; ρ aris korelaciis koefici- 

enti ξ 1 da ξ 2 Soris. 

1 2 

2.6. normalur kanonTan dakavSirebuli ganawilebebi 

normalurad ganawilebuli SemTxveviTi sidideebis garkveuli arawrfivi 

gardaqmniT warmoiSoba axali ganawilebebis simravle, romelTagan mravals 

gansakuTrebuli adgili ukavia albaTobis Teoriasa da maTematikur 

2 

statistikaSi. maT Soris gansakuTrebiT aRniSvnis Rirsia stiudentis, χ da 

fiSeris ganawilebebi, radgan am ganawilebebs ukaviaT gansakuTrebuli 

adgili maTematikur statistikaSi da maTi kvantilebi anu procentuli wertilebi 

gamoiyeneba mraval kriteriumSi, romelTagan zogierTs qvemoT ganvixilavT. 

normalurad ganawilebuli SemTxveviTi sidideebis wrfivi gardaqmnisas 

2 

ganawilebis kanoni normaluri rCeba, anu Tu ξ ~ N( a , σ ), i = 1,..., n , maSin Sem- 

TxveviT sidides 

n 

i= 

1 

33 

i i i 

η = ∑ ( bi ⋅ ξi 

+ ci 

) , sadac i b da c i namdvili ricxvebia, aqvs 

normaluir ganawilebis kanoni maTematikuri molodiniT 

dispersiiT 

sidideebi. 

n 

2 2 2 

= bi 

⋅ i 

i= 

1 

n 

a = ∑ ( b ⋅ a + c ) da 

i= 

1 

i i i 

σ ∑ σ , Tu ξ i arian arakorelirebuli SemTxveviTi 

2.6.1. 

2 

χ ganawileba 

vTqvaT ξ1, ξ2,..., ξ n arian damoukidebeli normirebuli normalurad gana- 

wilebuli SemTxveviTi sidideebi, anu ~ (0,1) N ξ . ganvixiloT SemTxveviTi 

2 2 2 

2 

sidide η = ξ1 + ξ2 + ... + ξn 

, romelic ganawilebulia χ - ganawilebis kanoniT 

n – is toli Tavisuflebis xarisxiT. am kanonis ganawilebis simkvrives aqvs 

saxe 

1 n / 2−1 x / 2 

x e rodesac x > 0, 

n / 2 

2 Γ( 

n / 2) 

sadac Γ( ⋅ ) aris gama funqcia. simkvrivis es gamosaxuleba, iseve rogorc stiudentisa 

da fiSeris ganawilebis kanonebis SemTxvevebSi, praqtikaSi uSualod 

sakmaod iSviaTad gamoiyeneba, radgan, didi mniSvnelobis gamo, maTi 

mniSvnelobebi cxrilebis saxiT mocemulia praqtikulad maTematikuri statistikis 

yvela wignSi da, amitom, uSualo gamoTvlebisas simkvrivis analitikuri 

saxe ar gvWirdeba. 

i

η SemTxveviTi sidids maTematikuri molodini da dispersia Sesabamisad 

tolia M ( η ) = n da D( η ) = 2n 

. 

naxaz 2.5 – ze sqematurad naCvenebia η SemTxveviTi sididis ganawilebis 

simkvrivis saxe sxvadasxva Tavisuflebis xarisxebisaTvis 

nax. 2.5. 

34 

n = 1 

n = 2 

n = 3 

n = 6 

2 

χ - ganawileba farTod gamoiyeneba sxvadasxva statistikur kriteriumebSi, 

amitom, rogorc ukve vTqviT, misi ganawilebis funqciis da kvantilebis 

mniSvnelobebi cxrilebis saxiT mocemulia albaTobis Teoriisa da maTematikuri 

statistikis mraval wignSi. winamdebare wignis bolos, danarT 4 – Si 

mocemulia erT – erTi aseTi cxrili. 

2.6.2 stiudentis ganawileba 

es ganawilebac farTod gamoiyeneba sxvadasxva statistikur kriteriumebSi. 

vTqvaT ξ0, ξ1, ξ2,..., ξ n damoukidebeli normirebuli normalurad ganawilebuli 

SemTxveviTi sidideebia. SemovitanoT SemTxveviTi sidide 

0 

1 2 

1 

n 

ξ 

η = . 

∑ξi 

n i= 

mis albaTobebis ganawilebas hqvia stiudentis ganawilebis kanoni n Ta- 

n 

visuflebis xarisxiT. adgili aqvs M ( η ) = 0 da D( 

η ) = . stiudentis gana- 

n − 2 

wilebis simkvrivis saxe sqematurad mocemulia naxaz 2.6 – ze.

nax. 2.6. 

ganawilebis simkvrive simetriulia x = 0 mimarT. 

Sesabamis cxrilebSi mocemulia stiudentis ganawilebis mqone SemTxvevi- 

Ti sididis sxvadasxva donis kvantilebi da ganawilebis funqciis mniSvnelobebi. 

maTi didi praqtikuli mniSvnelobis gamo es mniSvnelobebi mocemulia 

maTematikuri statistikis praqtikulad yvela wignSi. winamdebare wignis 

bolos, danarT 5 – Si mocemulia erT – erTi aseTi cxrili. 

2.6.3. fiSeris ganawileba 

vTqvaT SemTxveviTi sidideebi ξ1, ξ2,..., ξ n da η1, η2,..., η m (sadac n da m naturaluri 

ricxvebia) damoukidebeli SemTxveviTi sidideebia, romelTgan 

TiToeuli ganawilebulia standartuli normaluri kanonis Tanaxmad. SemovitanoT 

SemTxveviTi sidide 

1 2 2 2 

( η1 + η2 + ... + ηm 

) 

Fm 

, n = m 

. 

1 2 2 2 

( ξ1 + ξ2 + ... + ξn 

) 

n 

mas aqvs fiSeris ganawilebis kanoni m da n Tavisuflebis xarisxebiT. 

2 

n 

2 n ( m + n − 2) 

adgili aqvs MFm, 

n = roca n > 2 da DFm, 

n = 

roca n > 4 . 

2 

n − 2 

m( n − 2) ( n − 4) 

ganawilebis simkvrives sqematurad aqvs nax. 2.7 – ze naCvenebi saxe sxvadasxva 

Tavisuflebis xarisxebisaTvis. 

35 

Φ 

Φ 

Φ 

1, 

4 

( x) 

10, 

50 

4, 

100 

( x) 

( x) 

n = 100 

n = 4 

n = 2 

n 

= 1

nax. 2.7. 

2 

cxadia rom iseve rogorc χ ganawilebis kanonis mqone SemTxveviTi sidide, 

fiSeris ganawilebis mqone SemTxveviTi sididec gansazRvrulia [0, +∞ ) 

intervalSi. 

fiSeris ganawilebis gamoTvlili mniSvnelobebi sxvadasxva m da n - 

Tvis ama Tu im moculobiT mocemulia maTematikuri statistikis praqtikulad 

yvela wignSi. winamdebare wignis bolos, danarT 6 – Si mocemulia erT 

– erTi aseTi cxrili. 

2.7. Tanabari ganawilebis kanoni 

Tanabari ganawilebis kanoni aris uwyveti SemTxveviTi sididis ganawilebis 

kanoni. am kanoniT aRiwereba mravali SemTxveviTi sididis ganawileba 

bunebaSi da teqnikaSi. moviyvanoT magaliTebi: 1) vTqvaT saswors, romli- 

Tac awarmoeben garkveuli masis awonvas aqvs 1 gramis toli minimaluri danayofi. 

vTqvaT saswori aCvenebs, rom wona moTavsebulia or k da k + 1 

gramebs Soris. bunebrivia sxeulis wonad miiRon k + 1/ 2 grami. am dros da- 

⎡1 1 ⎤ 

Svebuli SemTxveviTi Secdomis sidide moTavsebulia intervalSi 

⎢ 

, 

⎥ 

36 

⎣ 2 2⎦ 

da 

yoveli Secdomis albaToba am intervalidan erTnairia. 2) vTqvaT metros 

Semadgenlobebi moZraoben 5 wuTiani intervaliT. mgzavri, romelic peronze 

gamodis, matarebels elodeba drois ganmavlobaSi, romlis sididec 

SemTxveviTi sididea da erTnairi albaTobebiT Rebulobs mniSvnelobebs 

[ 0,5 ] intervalidan. 

naqvamidan cxadia, rom Tanabarad ganawilebuli SemTxviTi sidide mniSvnelobebs 

Rebulobs sasrulo intervalidan, romelsac misi gansazRvris intervali 

hqvia da yoveli mniSvnelobis albaToba erTnairia. 

vTqvaT ξ Tanabrad ganawilebuli SemTxveviTi sididea, romelic mniS- 

vnelobebs Rebulobs [ a, b ] intervalidan. maSin misi ganawilebis simkvrives 

aqvs saxe

⎧ 

⎪0, 

at x < a, 

⎪ 1 

p( 

x) 

= ⎨ , at a ≤ x ≤ b, 

⎪b 

− a 

⎪ 

⎩0, 

at x > b. 

Zneli ar aris davrwmundeT, rom Tanabrad ganawilebuli SemTxveviTi 

sididis ganawilebis funqcia 

⎧ 

⎪0, 

at x < a, 

⎪ x − a 

F( 

x) 

= ⎨ , at a ≤ x ≤ b, 

⎪b 

− a 

⎪ 

⎩1, 

at x > b. 

Tanabradganawilebuli SemTxveviTi sididis maTematikuri molodini 

2 

a + b 

( b − a) 

Mξ = , xolo dispersia Dξ = . vTqvaT [ α, β ] intervali miekuTvne- 

2 

12 

β −α 

ba [ a, b ] intervals (ix. nax. 2.8), anu [ α, β ] ∈ [ a, b] 

, maSin p( 

ξ ∈ [ α, β ]) = 

b − a 

. 

ganawilebis simkvrivisa da ganawilebis funqciis grafikebi naCvenebia 

Sesabmisad naxazebze 2.8 da 2.9 – ze. 

nax. 2.8. nax. 2.9. 

37

Tavi 3. statistikuri hipoTezebis Semowmebis safuZvlebi 

yoveldRiur saqmianobaSi yovel CvenTagans Zalze xSirad uwevs gadawyvetilebis 

miReba ama Tu im movlenis, sakiTxis mdgomareobis Sesaxeb. 

magaliTad, daniSnulebis adgilamde transportis romeli saxeobiT ufro 

swrafad mivalT mocemul momentSi, raimes yidvisas romeli firmis nawarms 

mivceT upiratesoba da a.S. rogorc wesi gadawyvetilebebi efuZnebian Cvens 

gamocdilebas da codnas im sakiTxis Sesaxeb romelTan mimarTebaSic 

vRebulobT gadawyvetilebas. Tu sakiTxis arsi da/an informacia, romlis 

safuZvelzec vRebulobT gadawyvetilebas, Seicavs SemTxveviT mdgenels, 

maSin nebismier gadawyvetilebas Tan axlavs Secdomis daSvebis garkveuli 

riski. magaliTad, fexburTis matCis angariSis winaswari ganWvreta 

garkveuli codnis da informaciis safuZvelze SesaZlebelia, magram Tan 

axlavs riski imisa, rom Cveni gadawyvetileba iqneba mcdari imitom, rom 

matCis saboloo angariSze gavlenas axdenen mravali SemTxveviTi 

faqtorebi. vTqvaT gvinda sami gasrolis SedegiT SevadaroT ori msroleli 

erTmaneTs. srolis SedegebiT miRebul gadawyvetilebasac Tan axlavs 

Secdomis daSvebis garkveuli riski imis gamo, rom savsebiT SesaZlebelia am 

konkretul SemTxvevaSi ufro zustma msrolelma aCvenos Tavis SesaZleblobebze 

gacilebiT uaresi Sedegi da piriqiT, ufro uaresma msrolelma 

aCvenos ufro kargi Sedegi. aseT SemTxvevebSi gadawyvetilebis misaRebad gamoiyeneba 

maTematikuri statistikis meTodebi, romlebsac statistikuri 

hipoTezebis Semowmebis meTodebi hqvia. statistikuri hipoTeza aris 

maTematikuri statistikis enaze formalizebuli daSveba Sesaswavli 

movlenis arsis ama Tu im mxaris Seaxeb. statistikuri hipoTeza Cveulebrivi 

cxovrebiseuli hipoTezisagan ZiriTadad gansxvavdeba imiT, rom is 

alternatiul hipoTezebTan erTad mTlianad moicavs Sesaswavli movlenis 

SesaZlo mniSvnelobaTa simravles, maSin rodesac cxovrebiseuli hipoTezis 

dros es aucilebebli ar aris. Winamdebare TavSi SemovitanT statistikuri 

modelebis ganmartebas, SeviswavliT statistikuri hipoTezebis arss, maT 

saxeebs da statistikuri hipoTezebis Semowmebis zogierT, praqtikaSi 

farTod gavrcelebul, meTodebs. 

3.1. statistikuri modelebi 

rogorc zeviT araerTxel aRvniSneT SemTxveviTi movlenis Sesaxeb informaciis 

miRebis erTad – erTi gza aris amonarCevis anu sasrulo 

raodenobis damoukidebeli dakvirvebis Sedegebis miReba generaluri 

amonarCevidan. rac ufro srulyofilad warmoadgens generalur amonarCevs 

konkretuli amonarCevi miT ufro metia SesaZlebloba, rom mis safuZvelze 

miRebuli gadawyvetileba iqneba WeSmariti. amonarCevis, anu sasrulo 

raodenobis damoukidebeli dakvirvebis Sedegebis safuZvelze maTematikur 

statistikaSi xdeba gadawyvetilebis miReba anu statistikuri hipoTezebis 

Semowmeba. statistikuri hipoTezis Semowmebis idea yoveldRiur cxovrebaSi 

gadawyvetilebis miRebis logikis analogiuria. kerZod, yoveldRiur 

cxovrebaSi gadawyvetilebis miRebisas, rogoc wesi, vsargeblobT Semdegi 

38

pragmatuli wesiT. vRebulobT gamoTqmul daSvebas, Tu is ar ewinaaRmdegeba 

movlenis, romlis Sesaxebac gamoiTqmeba daSveba, damaxasiaTebel niSnebze 

dakvirvebis Sedegebs (amonarCevs) da ukuigdeba gamoTqmuli daSveba, Tu 

amonarCevs warmoadgens is mniSvnelobebi, romlebsac ar SeiZleboda 

adgili hqonodaT gamoTqmuli daSvebis samarTlianobisas. statistikuri 

hipoTezebis Semowmebisas viqceviT analogiurad. dakvirvebis Sedegebis 

safuZvelze gamoiTvleba X dakvirvebis Sedegebis (amonarCevis) T ( X ) 

funqcia , romelic statistikuri hipoTezis samarTlianobisas Rebulobs did 

(patara) mniSvnelobebs, xolo hipoTezis arasamarTlianobisas Rebulobs 

sawinaaRmdego, patara (did) mniSvnelobebs. T ( X ) funqciis mniSvelobebi 

arian SemTxveviTi sidideebi, radgan dakvirvebis Sedegebi, anu amonarCevi 

X aris SemTxeviTi. amitom T ( X ) funqciis mier didi Tu patara 

mniSvnelobebis miRebis faqtis dadgena Semdegnairad xdeba. amoirCeva iseTi 

C kritikuli mniSvneloba, rom T ( X ) > C xdomilebis albaToba iyos erTTan 

(nolTan) axlos Sesamowmebeli hipoTezis samarTlianobisas, xolo hipoezis 

arasamarTlianobisas, piriqiT, iyos nolTan (erTTan) axlos. realuri 

amonarCevis X real miRebis Semdeg mowmdeba piroba T ( X real ) > C da Tu mas 

adgili aqvs, miiReba Sesamowmebeli hipoTeza, winaaRmdg SemTxvevaSi – 

ukuigdeba. xdomilebas, romlis albaTobac axlosaa erTTan praqtikulad 

WeSmariti xdomileba hqvia, xolo xdomilebas, romlis albaTobac axlosaa 

nolTan, praqtikulad SeuZlebeli xdomileba hqvia. praqtikulad 

SeuZlebeli xdomilebis albaTobis SerCevis sakiTxi, romlis drosac 

Sesamowmebeli hipoTeza ukuigdeba, aris amocana, romelic formalizacias 

ar eqvemdebareba. misi SerCeva damokidebulia konkretuli amocanis 

specifikaze. magaliTad, avtomobilis samuxruWo sistemis saimedobis 

gaTvlisas ar SeiZleba, rom maTi umtyuno muSaobis albaToba 0.001 – is 

toli iyos. Tumca bevr praqtikul amocanebSi WeSmariti gadawyvetilebis 

ukugdebis albaToba aiReba 0.01; 0.05 – is toli. avtomobilebis muxruWebis 

SemTxvevaSi, umtyuno muSaobis albaTobis 0.001 – Tan toloba niSnavs, rom 

muxruWebi mwyobridan gamovlen yoveli aTasi daWerisas saSualod erTxel, 

rac gamoiwvevs mravalricxovan avariebs da msxverpls. amitom, am 

6 

SemTxvevaSi, albaToba unda iyos 10 − 

- ze ara naklebi. 

3.2. statistikuri hipoTezebis Semowmeba 

rogorc zemoT aRvniSneT statistikuri hipoTeza aris Sesaswavli 

movlenis Tvisebebis da am TvisebebTan dakavSirebuli daSvebebis 

formalizebuli Cawera. statistikur hipoTezas vadgenT im SemTxvevaSi, 

rodesac Sesaswavl movlenaze gavlenas axdenen SemTxveviTi faqtorebi, anu 

Sesaswavl movlenaze dakvirvebis Sedegebi warmoadgenen SemTxveviT 

sidideebs. viciT, rom SemTxveviTi sididis Tvisebebs mTlianad gansazRvravs 

misi ganawilebis kanoni. amitom, statisitkuri hipoTeza aris daSveba 

ganawilebis kanonis ama Tu im Tvisebis Sesaxeb. daSveba SeiZleba exebodes 

rogorc TviTon ganawilebis kanonis saxes da mis parametrebs, aseve am 

ganawilebis kanonis mxolod parametrebs. statistikur hipoTezas aRniSnaven 

39

H - iT. rogorc ukve vTqviT, hipoTeza SeiZleba iyos daSveba ganawilebis 

kanonis saxis da parametrebis Sesaxeb. magaliTad, hipoTeza 

2 

H : p( x) = N( x; a, σ ) gulisxmobs, rom SemTxveviT sidide ganawilebulia 

normaluri kanoniT maTematikuri molodiniT a da dispersiiT 

40 

2 

σ . meores 

mxriv, rodesac ganawilebis kanonis saxe cnobilia, statistikuri hipoTeza 

SeiZleba iyos ganawilebis kanonis parametrebis Sesaxeb daSvebebi. magali- 

Tad, vTqvaT cnobilia, rom ξ SemTxveviTi sidide ganawilebulia 

normaluri kanoniT, anu 

2 

ξ ~ N( ⋅ ; a, 

σ ) . maSin hipoTezas SeiZleba hqondes saxe 

2 

2 

H : a = a1 

, an H : σ = σ1 

, an H : a = a1, σ = σ1 

da a.S. 

Tu hipoTeza ganawilebis kanons gansazRvravs calsaxad, anu hipoTezas Seesabameba 

mxolod erTi ganawilebis kanoni, maSin aseT hipoTezas martivi hipoTeza 

ewodeba. martivi hipoTezis magaliTia H : a = a1 

. winaaRmdeg SemTxvevaSi, 

rodesac hipoTezas Seesabameba erTze meti ganawilebis kanoni, 

hipoTezas rTuli hipoTeza ewodeba. rTuli hipoTezis magaliTebia: 

H : a > a1 

, H : a < a1 

, H : a ≠ a1 

. pirvel hipoTezas marjvenamxrivi hipoTeza 

hqvia, meores – marcxenamxrivi, xolo mesame hipoTezas – ormxrivi. 

cxadia, rom martivi hipoTezis Semowmeba ufro advilia vidre rTulis. 

imisaTvis, rom SevamowmoT statistikuri hipoTeza, anu miviRoT 

gadawyvetileba misi WeSmaritebis Sesaxeb, saWiroa Sesamowmebel 

hipoTezasTan erTad CamovayaliboT alternatiuli hipoTeza, romelTan 

mimarTebaSic vamowmebT ZiriTad hipoTezas. ZiriTad hipoTezas, rogorc 

H – iT. moviyvanoT 

wesi, 0 H - iT aRniSnaven, xolo alternatiul hipoTezas 1 

ZiriTadi da alternatiuli hipoTezebis magaliTebi: 1) H0 : a = a1 

, 

H1 : a = a2, a1 ≠ a2 

; 2) H0 : a = a1 

, H1 : a > a1 

; 3) H0 : a = a1 

, H1 : a ≠ a1 

. pirvel 

SemTxvevaSi ZiriTadi da alternatiuli hipoTezebi martivi hipoTezebia, 

meore da mesame SemTxvevebSi alternatiuli hipoTezebi rTuli hipoTezebi 

arian. zogadad hipoTezebis raodenoba SeiZleba iyos orze meti. 

winamdebare kursSi am SemTxvevas ar ganvixilavT. 

rogorc ukve vTqviT, H 0 hipoTezis Sesamowmeblad saWiroa movnaxoT 

iseTi kritikuli xdomileba, romelic iqneba praqtikulad WeSmariti 

xdomileba Sesamowmebeli hipoTezis WeSmaritebis dros da praqtikulad 

SeuZlebeli xdomileba alternatiuli hipoTezis WeSmaritebisas. idealSi, 

ra Tqma unda, yvelaze kargi iqneboda gvepova iseTi A xdomileba, romelic 

iqneboda WeSmariti 0 H hipoTezis dros, anu misi albaToba P( A | H 0) 

= 1 da 

SeuZlebeli 1 H alternatiuli hipoTezis dros, anu P( A | H 1) 

= 0 . magram 

aseTi xdomilebis arseboba yovelTvis ar aris SesaZlebeli. aseT 

SemTxvevebSi pouloben iseT xdomilebebs, romlebic arian praqtikulad 

WeSmariti xdomilebebi ZiriTadi hipoTezis WeSmaritebis dros da 

praqtikulad SeuZlebeli xdomilebebi alternatiuli hipoTezis 

WeSmaritebis dros. 

rogorc wesi, kritikuli xdomilebis mosaZebnad Semdegnairad iqcevian. 

amonarCevi aRvniSnoT X - iT, xolo T (X ) – iT avRniSnoT amonarCevis 

raime funqcia, romelsac Semdegi Tvisebebi eqneba. Sesamowmebeli 0 

H 

hipoTezis samarTlianobisas did mniSvnelobebs Rebulobs, xolo

alternatiuli H 1 hipoTezis samarTlianobisas mcire mniSvnelobebs 

Rebulobs an piriqiT, H 0 - is samarTlianobisas Rebulobs mcire 

mniSvnelobebs da H 1 – is samarTlianobisas Rebulobs did mniSvnelobebs. 

T ( X ) funqcias kriteriumis statistikas eZaxian. sicxadisaTvis vTqvaT T ( X ) 

statistika did mniSvnelobebs Rebulobs H 0 hipotezis samarTlianobisas da 

mcire mniSvnelobebs Rebulobs H 1 alternatiuli hipoTezis 

samarTlianobisas. maSin H 0 hipoTezis Semowmebis kritikuli aris 

gansazrvrisaTvis saWioa iseTis C sidids gansazRvra, romlisaTvisac adgili 

aqvs P( T ( X ) ≥ C | H0 

) → 1 da P( T ( X ) ≥ C | H1) 

→ 0. 

orive pirobis erTdrouli 

Sesruleba SeuZlebelia, radgan pirveli albaTobis gazrda yovelTvis 

iwvevs meore albaTobis gazrdas da piriqiT, meore albaTobis nolTan 

miaxloebiT (C - s mniSvnelobis SerCeviT) mcirdeba pirveli albaToba. 

statistikuri hipoTezebis Semowmebis dros adgili aqvs ori saxis Secdomebs: 

1) ukuvagdoT hipoTeza, rodesac is WeSmaritia; 2) miviRoT hipoTeza, 

rodesac is ar aris WeSmariti. am Secdomebs pirveli da meore gvaris Secdomebs 

uwodeben. 

pirveli gvaris Secdomis albaToba Caiwereba Semdegnairad 

P ( T ( X ) < C | H ) . am albaTobas kriteriumis mniSvnelobis dones eZaxian da 

0 

aRniSnaven α asoTi, e.i. T ( X ) < C | H ) = α H hipoTezis miRebis 

P ( 0 . 0 

≥ , xolo T ( X ) < C aris 1 

kritikul ares aqvs saxe T ( X ) C 

H hipoTezis miRebis 

are. kritikul ares gansazRvraven Semdegnairad: afiqsireben pirveli tipis 

Secdomis albaTobis sidides, anu irCeven kriteriumis mniSnelobis 

maqsimalur dones da zRurblur mniSvnelobas C – s irCeven ise, rom meore 

tipis Secdomis albaToba iyos minimaluri. 

meore gvaris Secdomis albaTobas aRniSnaven β asoTi. 1− β sidides 

hqvia kriteriumis simZlavre. amrigad, H 0 hipoTezis Semowmebisas 

kritikuli are unda ganvsazRvroT ise, rom kriteriumis mniSvnelobis done 

iyos zemodan SemozRuduli, nolTan axlos myofi sidide, xolo kriteriumis 

simZlavre iyos rac SeiZleba didi, erTTan axlos myofi sidide. 

kriteriumis mniSvnelobis donis arCeva formalizacias ar eqvemdebareba da 

mis sidides iCeven konkretuli amocanis arsidan gamomdinare. xSir 

SemTxvevaSi mis mniSvnelobas irCeven Semdegi mniSvnelobebidan 

α = 0.1;0.05;0.01;0.001 . im SemTxvevaSi, rodesac pirveli gvaris Secdoma 

dakavSirebulia did danakargebTan, α - s mniSvnelobas Rebuloben 

−1 

gacilebiT ufro naklebs, magaliTad α = 10 da a.S. 

nax. 3.1 – ze mocemulia ori martivi hipoTezis Semowmebis zemoT 

aRwerili wesis grafikuli interpretacia. 

41

Γ 0 daA Γ 1 Sesabamisad arian 0 H da 1 

H hipoTezebis miRebis areebi. 

nax. 3.1. 

3.3. statistikuri modelebisa da hipoTezebis magaliTebi 

ganvixiloT magaliTebi, romlebic gviCveneben Tu rogor SeiZleba praqtikuli 

amocanebis formalizacia statistikuri modelebiT da bunebriv enaze 

dasmuli sakiTxebis Camoyalibeba statistikuri hipoTezebis saxiT. zogadad 

ar arsebobs formaluri aparati araformaluri amocanebis statistikuri 

modelebiTa da statistikuri hipoTezebis warmodgenisaTvis. ganxiluli 

magaliTebis mizania warmodgena mogvces am procesebze da gamogvimuSaos 

garkveuli minimaluri Cvevebi amocanebis aseTi formalizaciisaTvis. 

sammagi sammagi sammagi sammagi testi. testi testi testi es magaliTi ganvixiloT fsiqologiuri testis saxiT, Tumca 

aseTi magaliTebi mravlad gvxvdeba rogorc yoveldRiur cxovrebaSi ise 

teqnikuri amocanebis amoxsnisas. magaliTad, ori an ramodenime 

teqnologiuri procesis Sedarebisas, swavlebis sxvadasxva meTodikebis 

Sedarebisas da a.S. 

vTqvaT TiToeuls adamianebis jgufidan miewodeba sami Wiqa wyali romelTagan 

orSi Casxmulia sufTa wyali, xolo mesames damatebuli aqvs 

cotaodeni Saqari. amocana mdgomareobs daadginon adamianebis gamosacdel 

jgufs SeuZlia Tu ara Saqris mocemuli koncentraciis garCeva. vTqvaT 

adamianebis jgufi erTgvarovania, maTi testireba xdeba erTnair pirobebSi 

da erTi adamianis testirebis Sedegebi gavlenas ar axdenen meore adamianis 

testirebis Sedegebze. im faqtis dasadgenad, rom adamianebs mocemuli 

jgufidan SeuZliaT ganasxvaon Saqris mocemuli koncentracia CavataroT 

Semdegi msjeloba. vTqvaT koncentracia iseTia, rom gamosacdel pirebs ar 

SeuZliaT misi sufTa wylisagan garCeva. maSin yoveli maTgani SemTxveviT 

irCevs miwodebuli Wiqebidan erT – erTs. avRniSnoT swori arCevani 

erTianiT, xolo araswori – noliT. Tu arCevis Sedegze ar moqmedeben sxva 

1 

p = 

faqtorebi, garda sufTa SemTxveviTisa, maSin swore arCevis albaToba 3 . 

amrigad, sawyisi amocana miviyvaneT statistikur modelamde, romelsac 

Seesabameba bernulis sqema. marTlac, yoveli eqsperimentis Sedegi oridan 

erT mniSvnelobas Rebulobs: 1 – damtkbari wylis swori arCevisas da 0 – 

mcdari arCevisas. swori arCevanis albaToba yovel eqsperimentSi 

erTnairia da tolia 1/3 – is. imisda mixedviT Tu rogori alternatiuli 

daSvebis mimarT mowmdeba ZiriTadi daSveba imis Sesaxeb, rom gamosacdeli 

42

pirovnebebi ver arCeven Saqris mocemul koncentracias, statistikuri 

1 

hipoTezebi formirdeba sxvadasxvanairad. magaliTad, 1) H0 : p = roca 

3 

1 

H1 : p > , rodesac alternatiulad igulisxmeba, rom gamosacdeli jgufi 

3 

1 

ansxvavebs Saqris mocemul koncentracias sufTa wylisagan; 2) H0 : p = roca 

3 

1 

H1 : p < , rodesac alternatiulad igulisxmeba, rom gamosacdeli jgufi 

3 

ansxvavebs Saqris mocemul koncentracias sufTa wylisagan, magram 

1 

erTmaneTSi ereva sufTAda Saqriani wylebi; 3) H0 : p = roca H1 : p = 0.9 , 

3 

rodesac alternatiulad igulisxmeba, rom gamosacdeli jgufis aTi 

wevridan cxra sworad ansxvavebs Saqris mocemul koncentracias sufTa 

wylisagan. 

pirvel or SemTxvevaSi alternatiuli hipoTezebi arian rTuli 

hipoTezebi, xolo mesame SemTxvevaSi alternatiuli hipoTeza aris martivi. 

Sewyvilebuli Sewyvilebuli Sewyvilebuli Sewyvilebuli dakvirvebebi. dakvirvebebi 

dakvirvebebi 

dakvirvebebi praqtikaSi xSirad gvxvdeba amocanebi, rodesac 

saWiroa ori moqmedebis erTmaneTTan Sedareba maTi SedegebiT. magaliTad, 

swavlebis ori meTodi, ori teqnologia, ori wamali da a.S. magaliTis 

saxiT ganvixiloT adamianis reaqciis siswrafis Sedareba xmis da sinaTlis 

signalebze. amisaTvis SeirCa erTgvarovani pirovnebebis jgufi, romlebsac 

erTmaneTisagan damoukideblad awvdidnen ori saxis signals: bgeriTs da 

sinaTlis. signalis miRebisTanave gamosacdeli piri xels aWerda 

specialuri xelsawyos Rilaks, romelic afiqsirebda signalis miwodebis da 

x y i n 

Rilakze xelis daWeris momentebs Soris sxvaobas. avRniSnoT i , i , = 1,..., 

Sesabamisad, xmis da sinaTlis signalebis miwodebis momentebsa da Rilakze 

xelis daWeris momentebs Soris gansxvavebebi. n – aris gamosacdelTa 

raodenoba, anu pirovnebebis ricxvi jgufSi. amocana mdgomareobs 

davadginoT orive tipis signalze adamianebis reagirebis identuroba. 

ganvixiloT aRniSnuli amocanis statistikis enaze aRweris SesaZlebloba, 

anu statistikuri modelis agebis da am modelis Tvisebebis Sesaxeb hipoTezebis 

formirebis SesaZlebloba, romelTa samarTlianobac Semowmebuli 

unda iqnas dakvirvebis Sdegebis safuZvelze. statistikuri modelebis 

agebisas SesaZlebelia sxvadasxvanairad moqceva. jer ganvixiloT amocanis 

dasmis parametruli midgoma. aseTi midgomisas miiReba, rom xi, y i 

dakvirvebis Sedegebi emorCilebian albaTobebis ganawilebis garkveul 

kanons. termini „miiReba“ qveS igulisxmeba an am faqtis codna wina 

gamocdilebis safuZvelze, an xi, y i dakvirvebis Sedegebis gamokvleva 

specialuri testebis (kriteriumebis) gamoyenebiT. 

statistikuri modelis arCeva. dauSvaT, rom xi, y i dakvirvebis Sedegebi ganawilebulia 

normalurad, anu 

2 

xi ~ N( ai , σ ) , yi 2 

~ N( bi , σ ) . es niSnavs, rom 

bgeriT da sinaTlis signalebze yoveli gamosacdelis reaqciis 

xangrZliobis maTematikuri molodinebi erTmaneTisagan gansxvavdebian, 

43

xolo dispersiebi erTnairia. am SemTxvevaSi bgeriT da sinaTlis 

signalebze gamosacdeli pirovnebebis reaqciis identurobis hipoTezas aqvs 

saxe: 

H a = b a = b a = b , 

: 1 1, 2 2,..., 

n n 

2 

sadac ai , bi , i = 1,..., n , da σ arian normaluri ganawilebis kanonebis ucnobi 

parametrebi da maTi mniSvnelobebi Sefasebuli unda iqnas dakvirvebis 

Sedegebis safuZvelze. amrigad, movaxdineT dasmuli problemis 

formalizacia, anu amovirCieT statistikuri modeli, romliTac aRiwereba 

dasmuli teqnikuri amocana da movaxdineT gamoTqmuli daSvebis 

formalizaciac statistikuri hipoTezis saxiT. 

miRebuli formalizebuli amocana sakmaod rTulia imitom, rom dakvirvebis 

SedegebiT saWiroa 2n + 1 ucnobi parametris gansazRvra. davuSvaT 

arsebobs safuZveli miviRoT, rom gamosacdeli pirovnebebis jgufi imdenad 

erTgvarovania, rom erTidaigive tipis signalebze maTi reaqciis saSualo 

xangrZlioba erTnairia da Sesabamisad tolia a da b . am SemTxvevaSi orive 

tipis signalze reaqciis drois tolobis daSveba statistikuri hipoTezis 

saxiT Caiwereba Semdegnairad 

H : a = b . 

aseTi formalizaciisas amocana mniSvnelovnad martivdeba. dakvirvebis 

2 

Sedegebis safuZvelze saWiroa mxolod sami ( a, b, σ ) parametris gansazRvra 

da Semdeg hipoTezis Semowmeba. amocanis formalizaciisas yovelTvis 

saWiroa rac SeiZleba martivi statistikuri modelis miRebis mcdeloba, 

romelic miiReba Sesaswavli movlenis Sesaxeb sxvadasxva daSvebebis 

safuZvelze. magram yoveli daSveba saWiroa gakeTdes didi sifrTxiliT 

imitom, rom Tu daSveba mcdaria, maSin rac ar unda faqizi maTematikuri 

meTodebi iqnas gamoyenebuli formalizebuli amocanis gadasawyvetad, 

araswori gadawyvetilebis miRbis albaToba mainc didia. 

statistikuri modelebis agebis meore gza aris araparametruli. am Sem- 

TxvevaSi araviTari daSvebebi ar keTdeba ganawilebis kanonebTan 

dakavSirebiT, romlebsac emorCilebian dakvirvebis Sedegebi, anu xi, y i 

dakvirvebis Sedegebis albaTobebis ganawilebis kanonebTan dakavSirebiT. 

hipoTezebi formirdebian dakvirvebis Sedegebis urTierT mimarTebis 

Sesaxeb da gadawyvetileba miiReba ara uSualod xi, y i dakvirvebis Sedegebis 

safuZvelze, aramed maTi urTierT mimarTebis safuZvelze an maTi rangebis, 

anu monotourad dalagebul dakvirvebis yvela SedegSi maTi rigiTi 

nomrebis safuZvelze. parametrulTan SedarebiT araparametruli meTodebis 

dadebiTi mxare mdgomareobs imaSi, rom am SemTxvevaSi araviTari 

daSvebebis gakeTeba ar aris saWiro ganawilebis kanonebTan dakavSirebiT 

romlebsac emorCilebian dakvirvebis Sedegebi da amiT, TiTqos da, Tavidan 

aicileba Secdomis daSvebis erT erTi SesaZlo wyaro amocanis 

formalizaciisas. uaryofiTi mxare mdgomareobs imaSi, rom dakvirvebis 

Sedegebis ganawilebis kanonebTan dakavSirebiT gakeTebuli daSvebebis 

samarTlianobisas, is damatebiTi informacia, romelsac Seicaven es kanonebi, 

araparametruli kriteriumebis gamoyenebisas ikargeba, rasac mivyevarT 

miRebuli gadawyvetilebis sandoobis SemcirebasTan. 

44

3.4. statistikuri hipoTezebis Semowmeba (gamoyenebiTi amocanebi) 

3.4.1. bernulis gamocdebis sqema 

gadavideT formalizebuli statistikuri amocanebis gadawyvetis 

meTodebis Seswavlaze. daubrundeT zemoT ganxilul sammagi testis 

magaliTs, romlis arsic mdgomareobs SemdegSi: saWiroa obieqtis 

mdgomareobis Sesaxeb gadawyvetilebis miReba, rodesac mas SeuZlia sam 

sxvadasxva mdgomareobaSi yofna, romelTagan ori – erTnairia. 

igulisxmeboda, rom yovel cdaSi swori gadawyvetilebis albaToba 

erTnairia da p – s tolia. Sesamowmebel ZiriTad hipoTezas hqonda Semdegi 

1 

saxe: H0 : p = . es hipoTeza SeiZleba SevamowmoT sxvadasxva alternatiul 

3 

hipoTezebTan mimarTebaSi. erT – erT SesaZlo alternatiul hipoTezas aqvs 

H 

1 

: p > . damtkbari wylis ganxilvisas alternatiuli hipoTeza 

saxe: 1 

3 

niSnavs, rom gamosacdeli adamianebis jgufs SeuZlia SeigrZnos Saqris 

mocemuli koncentracia. alternatiul hipoTezas SeiZleba hqondes saxe: , 

romelis niSnavs, rom rom gamosacdeli adamianebis jgufs SeuZlia 

ganasxvaos Saqris mocemuli koncentracia sufTa wylisagan, magram maT 

urevs erTmaneTSi. alternatiuli hipoTeza H3 : p = 0.9 niSnavs, rom aTidan 

cxra SemTxvevaSi gamosacdeli pirebi iReben swor gadawyvetilebas. 

maTematikuri statistikis TvalsazrisiT mesame alternatiuli hipoTezis 

ganxilvis SemTxveva aris yvelaze martivi, radgan am SemTxvevaSi, rogorc 

ZiriTadi, ise alternatiuli hipoTezebi arian martivebi, anu isini Seicaven 

TiTo – TiTo albaTobebis ganawilebis kanonebs. gansaxilvel SemTxvevaSi 

aseTebi arian bernulis ganawilebis kanonebi. or sxva 1 H da 2 H 

alternatiul hipoTezebSi igulisxmeba ara TiTo albaTobebis ganawilebis 

kanonebi, aramed maTi simravleebi. Pirvel SemTxvevaSi isenia bernulis 

ganawilebis yvela kanoni, romelTaTvisac swori gadawyvetilebis 

albaToba p akmayofilebs pirobas 1/ 3 

meore SemTxvevaSi albaToba 

akmayofilebs pirobas 0 

am magaliTze ganvixiloT gadawyvetilebis miRebis algoriTmis ageba. 

aRvniSnoT: A aris xdomileba, romlis albaToba, H 0 hipoTezis samarTlianobisas, 

aris patara. aRvniSnoT es albaToba α asoTi, anu A 

xdomilebisaTvis adgili aqvs P( A | H0 ) ≤ α . sidide α SeirCeva ise, rom 

gansaxilveli amocanisaTvis, xdomileba, romlis albaToba ≤ α , iTvleba 

praqtikulad SeuZleblad, anu A xdomileba, H 0 hipoTezis 

samarTlianobisas, aris praqtikulad SuZlebeli xdomileba. Tu 

SesaZlebelia A xdomilebis arCeva ise, rom alternatiuli hipoTezis 

samarTlianobisas misi albaToba iyos didi, anu A iqneba praqtikulad 

WeSmariti xdomileba, maSin aseTi xdomilebiT SesaZlebeli iqneba 

SevamowmoT ZiriTadi hipoTeza Sesabamis alternatiul hipoTezasTan 

mimarTebaSi. marTlac, Tu adili aqvs A xdomilebas, es niSnavs, rom moxda 

praqtikulad SeuZlebeli xdomileba maSin, rodasc H 0 hipoTeza aris WeSma- 

riti. amitom, didi albaTobiT, H 0 hipoTezas ar SeiZleba hqondes adgili. 

45

meores mxriv, alternatiuli hipoTezis samarTlianobisas A xdomileba 

aris praqtikulad WeSmariti xdomileba. amitom bunebrivia miviRoT 

alternatiuli hipoTeza A xdomilebis WeSmaritebis dros. 

konkretuli SemTxvevis, sami Wiqa wylidan erTi damtkbaris arCevis magaliTze 

ganvixiloT, gadawyvetilebis misaRebad A xdomilebis arCeva. konkretulobisaTvis 

avirCioT α = 0.02 , anu xdomilebebi romelTa Sesabamisi 

albaTobebic α – ze naklebia iTvlebian praqtikulad SeuZlebel xdomilebebad. 

gamoTvlebis simartivisa da Sedegebis TvalsaCinoebisaTvis dauSvaT 

n = 10 . zogadad, 10 – is toli amonarCevis moculoba ar aris sakmarisi seriozuli 

daskvnebisaTvis, magram aq n = 10 avirCieT dasaxelabuli mizezebis gamo. 

cxril 3.1 – Si mocemulia xdomilebebis albaTobebi, rom k ≤ n gamosacdelma 

swored airCia damtkbar wyliani Wiqa H 0 hipoTezis 

samarTlianobisas. cxril 3.2 – Si mocemulia xdomilebebis albaTobebi, rom 

k pirovnebaze metma miiRo swore gadawyvetileba ZiriTadi hipoTezis 

samarTlianobis pirobebSi. cxril 3.3 – Si mocemulia xdomilebebis 

albaTobebi, rom k gamosacdeli iZleva swor pasuxs H 3 alternatiuli 

hipoTezis samarTlianobisas. 

ZiriTadi hipoTezis Sesamowmeblad ganvixiloT A xdomileba, rom 

swori pasuxebis raodenoba S ≥ C . zRvruli mniSvneloba C avirCioT 3.1 da 

3.2 cxrilebis safuZvelze. 

k 0 1 2 3 4 5 

P( S = k | H ) 0.0173 0.0868 0.1950 0.2602 0.2276 0.1365 

0 

k 6 7 8 9 10 

P( S = k | H ) 0.0569 0.0163 0.0030 0.0004 0.0000 

0 

k 0 1 2 3 4 5 

P( S k | H0 

) 

k 6 7 8 9 10 

P( S ≥ k | H ) 0.0766 0.0197 0.0034 0.0004 0.0000 

≥ 1.000 0.9827 0.8959 0.7009 0.4407 0.2131 

0 

46 

cxrili 3.1. 

cxrili 3.2. 

cxrili 3.3. 

k 0 1 2 3 4 5 

P( S ≥ k | H3 

) 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 

k 6 7 8 9 10 

P( S ≥ k | H ) 0.9984 0.9872 0.9298 0.7361 0.3487 

3 

cxrili 3.1 – dan Cans, rom xdomilebebis didi raodenoba praqtikulad 

SeuZlebeli xdomilebebia kriteriumis mniSvnelobis donis α = 0.02 

SerCeuli mniSvnelobisaTvis. cxrili 3.2 – dan Cans, rom kriteriumis

mniSvnelobis donis arCeuli mniSvnelobiT xdomilebebi S ≥ 7, S ≥ 8 , S ≥ 9, 

S ≥ 10 arian praqtikulad SeuZlebeli xdomilebebi, anu nebismieri maTgani, 

principSi, SeiZleba arCeuli iqnas ZiriTadi hipoTezis SesarCevad. 

ganvixiloT am xdomilebebis albaTobebi H 3 alternatiuli hipoTezis 

WeSmaritebisas. cxrili 3.3 – dan Cans, rom H 3 - is WeSmaritebisas S ≥ 7 

xdomilebis albaToba tolia 0.9872, anu aris praqtikulad WeSmariti 

xdomileba. amrigad, ganxiluli konkretuli magaliTisaTvis SevZeliT 

mogveZebna xdomileba A ≡ S ≥ 7 romlis albaToba, H 0 - is WeSmaritebisas, 

praqtikulad nolis tolia, xolo H 3 alternatiuli hipoTezis 

WeSmaritebisas – erTTan axlos aris. amitom xdomileba A ≡ S ≥ 7 warmoadgens 

gadawyvetilebis miRebis wess, romlis arsic mdgomareobs SemdegSi: Tu 

eqvs kacze meti iZleva swore pasuxs, maSin didi darwmunebiT SeiZleba 

vTqvaT, rom gansaxilveli koncentracia garCevadia adamianebis mocemuli 

jgufisaTvis. dauSvaT α = 0.005. 

am SemTxvevaSi praqtikulad SeuZlebel 

xdomilebebad iTvlebian is xdomilebebi, romelTa albaTobebic ≤ 0.005 . 

cxrili 3.2 – dan Cans, rom A xdomilebad unda aviRoT S ≥ 8 . cxrili 3.3 – 

dan Cans, rom am xdomilebis albaToba, H 3 alternatiuli hipoTezis 

WeSmaritebisas, sakmaod maRalia. rTuli 1 H da H 2 alternatiuli 

hipoTezebis ganxilvisas A xdomileba unda avirCioT iseTnairad, rom A 

xdomilebis albaTobebi iyvnen SesaZleblobis farglebSi maqsimalurebi 

alternatiuli hipoTezebis Sesabamisi nebismieri ganawilebis kanonis 

SemTxvevaSi. 

gansaxilvel konkretul SemTxvevaSi H 1 alternatiuli hipoTezis dros 

gadawyvetilebis miRebis wesi S ≥ 7 rCeba misaRebi (kargi) kriteriumis mniSvnelobis 

donisas α = 0.02 . 

alternatiuli hipoTezebi 1 H da H 2 arian calmxrivi alternatiuli 

hipoTezebi. alternatiuli hipoTeza SeiZleba iyos omxrivi H 4 : p ≠ p0 

ZiriTadi hipoTezis SemTxvevaSi H0 : p = p0 

. am SemTxvevaSi gadawyvetilebis 

miRebis wesic unda iyos ormxrivi, anu unda moiZebnos ori ricxvi 1 C da C 2 , 

romlebisaTvisac adgili aqvs pirobas P( S ≤ C1 | H0 ) + P( S ≥ C2 | H0 ) ≤ α . 

3.4.2. niSnebis kriteriumi erTi amonarCevisaTvis 

es kriteriumi erT – erTi umartivesTagania da saWiroebs dakvirvebis SedegebTan 

dakavSirebiT minimalur informacias. karZod, isini unda iyvnen 

damoukideblebi da mediana unda iyos gansazRvruli calsaxad. kriteriumi 

dafuZnebulia bernulis sqemaze, romelic gavarCieT wina paragrafSi. is 

saSualebas gvaZlevs SevamowmoT hipoTeza ganawilebis kanonis medianasTan 

dakavSirebiT. viciT, rom ξ SemTxveviTi sididis albaTobebis ganawilebis 

madianas uwodeben iseT θ ricxvs, romlisaTvisac samarTliania piroba 

P( ξ < θ ) = P( 

ξ > θ ) = 1/ 2. 

mravali praqtikuli amocanis miyvana xerxdeba medianasTan 

dakavSirebuli hipoTezis Semowmebis SemTxvevasTan. magaliTad, 

vTqvaT poliklinikaSi gmokvlevebiT dadginda pacientebis arterialuri 

47

wnevis medianuri mniSvneloba da vTqvaT es mniSvneloba θ tolia. saWiroa 

dadgindes Seicvala Tu ara arterialuri wnevis medianuri mniSvneloba 

mocemuli poliklinikis pacientebisaTvis zafxulis Svebulebebis Semdeg. 

amisaTvis ikvleven poliklinikis n pacients. aRvniSnoT x1, x2,..., x n - 

pacientebis arterialuri wnevis mniSvnelobebia da gamoviTvaloT sxvaobebi 

x − θ, 

i = 1,..., n . SemovitanoT Semdegi funqcia 

i 

gamoviTvaloT mniSvneloba 

⎧ 

⎪1, 

at 

s( 

x) 

= ⎨ 

⎪ 

⎩0, 

at 

n 

i= 

1 

48 

x > 0, 

x < 0. 

S = ∑ s( x −θ 

) . SemTxveviTi sidide s( x ) Rebulobs 

i 

or mniSvnelobas 0 an 1. hipoTezis samarTlianobisas, rom arteriuli mniSvneloba 

ar Seicvala, yoveli am mniSvnelobis albaToba erTnairia da 

tolia 1/2– is. SemTxveviTi sidide S aris s( xi − θ ) – is dadebiTi 

mniSvnelobebis raodenoba n gamocdaSi. amrigad, sawyisi amocana miviyvaneT 

bernulis sqemaze, romelSic S - iT aRniSnulia warmatebebis ricxvi da n 

eqsperimentSi misi mniSvnelobis mixedviT unda Semowmdes hipoTeza 

H0 : p = 1/ 2 . am hipoTezis Semowmeba sxvadasxva alternativebisas ganxiluli 

iyo zemoT. 

mocemuli meTodis Rirseba mdgomareobs mis simartiveSi da 

dasakvirvebeli SemTxveviTi sididis ganawilebis kanonTan dakavSirebuli 

moTxovnebis minimalurobaSi. 

3.5. hipoTezebis Semowmeba or amonarCevian amocanebSi 

mocemuli SemTxveva wina SemTxvevisagan gansxvavdeba imiT, rom aq erTi 

amonarCevis nacvlad gvaqvs ori amonarCevi da daskvna unda gamovitanoT 

maTi urTierT mimarTebis Sesaxeb. aseTi amocanebis magaliTad SeiZleba 

moviyvanoT Semdegi: swavlebis ori meTodis, ori teqnologiis, ori wamlis 

efeqtis, ori obieqtis mdgomareobis Sedareba da a.S. 

aRvniSnoT F( x ) aris ganawilebis funqcia pirveli amonarCevisaTvis, 

xolo G( x ) - meore amonarCevisaTvis. ori amonarCevis Sedarebisas SeiZleba 

gakeTdes sxvadasxva daSveba maTi albaTobebis ganawilebis kanonebTan 

dakavSirebiT. yvelaze martiv da gavrcelebul SemTxvevaSi igulisxmeba, rom 

ori Sesadarebeli meTodi iwvevs ricxviT RerZze ganawilebis kanonis 

mxolod mdebareobis cvlilebas, e.i. iwvevs SemTxveviTi sididis mxolod 

maTematikuri molodinis cvlilebas. SedarebiT rTul SemTxvevaSi 

Sesadarebel amonarCevebs Soris gansxvaveba SeiZleba gamoxatuli iqnas ara 

mxolod albaTobebis ganawilebis kanonis mdebareobis cvlilebaSi, aramed 

SemTxveviTi sididis gafantvis maxasiaTeblis, anu dispersiis 

cvalebadobaSic. da bolos, yvelaze rTul SemTxvevaSi, damuSavebis 

meTodebis cvalebadobam SeiZlba gamoiwvios mTlianad ganawilebis kanonis 

Secvla.

G( x ) albaTobebis ganawileba miekuTvneba F( x ) albaTobebis ganawilebis 

kanonTa daZrul ojaxs θ daZvris parametriT, Tu nebismieri x - Tvis 

adgili aqvs F( x) = G( x − θ ) . 

albaTobebis ganawilebis aseTi kanonebisaTvis dakvirvebebis Sedegebis 

ori simravlis erTgvarovnebis Sesaxeb hipoTeza Caiwereba Semdegnairad 

H0 : θ = 0 . 

amboben, rom F ≥ G , sadac F da G ganawilebis funqciebia, Tu nebismieri 

x ricxvisaTvis adgili aqvs F( x) ≥ G( x) 

. amboben, rom F ≤ G , Tu nebismieri x 

ricxvisaTvis adgili aqvs F( x) ≤ G( x) 

. 

am gansazRvrebis azri mdgomareobs SemdegSi. avRniSnoT: ξ SemTxveviTi 

sididea, romlis ganawilebis kanoni aris F( x ) , xolo η – SemTxveviTi 

sididea G( x ) ganawilebis kanoniT. maSin F( x) ≥ G( x) 

niSnavs, rom ξ – s aqvs 

tendencia miiRos ufro patara mniSvnelobebi, vidre η – m; anu nebismiei x – 

Tvis sruldeba P( ξ < x) ≥ P( η < x) 

. 

3.5.1. mani-uitnis kriteriumi 

mocemuli kriteriumi saSualebas iZleva erTgvarovnebaze SevadaroT 

ori damoukidebeli amonarCevi. avRniSnoT x1, x2, ..., x m - pirveli amonarCevia, 

xolo y1, y2,..., y n - meore amonarCevia. pirveli amonarCevis ganawilebis 

funqcia avRniSnoT F( x ) – iT, xolo meoresi G( x ) – iT. 

gansaxilveli kriteriumis gamosayeneblad saWiroa Semdegi daSvebebi: 

1. dakvirvebis Sedegebi x1, x2, ..., x m ; y1, y2,..., y n erTmaneTisagan damoukidebeli 

arian; 

2. ganawilebis kanonebi F( x ) da G( x ) arian uwyvetebi. aqedan gamomdinareobs, 

rom erTis toli albaTobiT dakvirvebis Sedegebi ar Seicaven 

erTmaneTis tol mniSvnelobebs. 

ori amonarCevis erTgvarovnebis Sesaxeb hipoTeza Caiwereba 

Semdegnairad: H0 : F( x) = G( x) 

. es hipoTeza unda SevamowmoT alternatiul 

hipoTezasTan mimarTebaSi, romelic SeiZleba iyos rogorc erTmxrivi, ise 

ormxrivi, anu alternativis saxiT SeiZleba gvhqondes Semdegi 

hipoTezebidan erT – erTi: 

1. marjvenamxrivi alternativa F( x) > G( x) 

, anu y dakvirvebis Sedegebs 

aqvT tendencia miiRon ufro didi mniSvnelobebi vidre x ; 

2. marcxenamxrivi alternativa F( x) < G( x) 

, anu x dakvirvebis Sedegebs 

aqvT tendencia miiRon ufro didi mniSvnelobebi vidre y ; 

3. ormxrivi alternativa F( x) ≠ G( x) 

. 

ganvixiloT pirveli SemTxveva. mani – uitnis kriteriumis arsi mdgomareobs 

i x da y j , i = 1,..., m ; j = 1,..., n , dakvirvebis Sedegebis wyvil – wyvilad 

SejerebaSi. dakvirvebis Sedegebis saerTo ricxvia mn . Sedarebis Sedegi Cav- 

TvaloT warmatebulad Tu xi < y j da warumateblad Tu xi > y j . imis gamo, rom 

F( x ) da G( x ) uwyvetebia, x da y Soris erTnairi dakvirvebis Sedegi ar 

49

unda iyos. magram realur monacemebSi, dakvirvebis Sedegebis SezRuduli 

sizutiT Caweris gamo, x da y Sori gvxvdebian erTnairi sidideebi. jer 

ganvixilavT mani - uitnis kriteriumi Teoriuli SemTxvevisaTvis, Semdeg ki 

SemovitanoT Sesabamisi Sesworeba. dakvirvebis x da y Sedegebis Sedarebis 

safuZvelze daiTvleba warmatebebis saerTo raodenoba. es sidide 

aRvniSnoT U набл . cxadia, rom mas SeuZlia miiRos nebismieri mniSvneloba 0 - 

dan mn -de. marjvenamxrivi alternatiuli hipoTezis Semowmebisas 

bunebrivia dauSvaT, rom rac ufro didia U набл - s mniSvneloba, miT metia 

imis albaToba, rom ZiriTadi hipoTeza ar aris WeSmariti da adgili aqvs 

alternatiul hipoTezas. U набл SemTxveviTi sidids ganawileba aris 

diskretuli ganawileba, romelic, H 0 hipoTezis samarTlianobisas 

gansazRvrulia mocemuli kriteriumis avtorebis mier. am ganawilebis 

procentuli wertilebi databulirebulia da mocemulia Sesabamis 

cxrilebSi (ix. danarTi 7). 

avirCioT kriteriumis mniSvnelobis done α . Sesabamisi cxrilebidan α , 

m da n - is saSualebiT vpoulobT mani – uitnis ganawilebis α donis 

kvantils, anu vxsniT albaTur gantolebas 

P( U ≥ U ( α, m, n) | H ) = α 

(3.1) 

пр. 

0 

da Tu adgili aqvs Uнабл ≥ U пр. 

( α, 

m, n) 

, miiReba alternatiuli hipoTeza, 

winaaRmdeg SemTxvevaSi miiReba ZiriTadi hipoTeza. 

marcxenamxrivi alternatiuli hipoTezis ganxilvisas Sesabamisi cxrilebidan 

moiZebneba Semdegi gantolebis amoxsna 

P( U ≤ U ( α, m, n) | H ) = α . (3.2) 

л. 

0 

Tu adgili aqvs Uнабл ≤ U л. 

( α, 

m, n) 

, miiReba alternatiuli hipoTeza, 

winaaRmdeg SemTxvevaSi – ZiriTadi hipoTeza. 

cxrilebidan U л. 

( α , m, n) 

mniSvnelobis gamosaTvlelad SeiZleba visargebloT 

damokidebulebiT 

. ( , , ) ( , , ) 

U л α m n + U пр α m n = mn , 

romelic gamomdinareobs U statistikis ganawilebis simetriidan Tavisi 

mn / 2 centris mimarT. 

ormxrivi alternatiuli hipoTezis Semowmebisas kritikul ares aqvs 

Semdegi saxe 

{ U набл ≤ U л ( α, m, n) } { U набл ≥ U пр ( α, 

m, n) 

} 

∪ , 

anu miiReba alternatiuli hipoteza, Tu U набл moxvdeba U л. 

( α , m, n) 

da 

. ( , , ) U пр α m n - is Sesabamisad marcxniv an marjvniv. winaaRmdeg SemTxvevaSi, anu 

Tu U набл moxvdeba U л. 

( α , m, n) 

da U пр. 

( α , m, n) 

Soris, miiReba ZiriTadi 

hipoTeza. am SemTxvevaSi kriteriumis mniSvnelobis done 2α – s tolia. Tu 

gvinda, ormxrivi alternativis dris, SevinarCunoT kriteriumis 

mniSvnelobis done α - s toli, maSin kritikul areSi unda aviRoT 

. ( / 2, , ) 

U л α m n da Uпр. ( α / 2, m, n) 

. 

imis gamo, rom U aris diskretuli SemTxveviTi sidide (3.1) da (3.2) gantolebebs 

SeiZleba ar hqondeT zusti amoxsna α , m da n – is mocemuli mniS- 

50

vnelobebisaTvis. amitom cxrilebSi moiZebneba an maTi miaxloebiTi amoxsnebi 

an α SeirCeva ise, rom maT hqondeT zusti amoxsnebi. 

rogorc zemoT avRniSneT, dakvirvebis Sedegebis Caweris sizustis SezRudulobis 

gamo, x da y Soris SeiZleba Segvxdes erTnairi mniSvnelobebi. am 

SemTxvevaSi, U statistikis daTvlisas, iTvlian erTnairi dakvirvebis Sedegebis 

raodenobas da U набл - s umateben am raodenobis ganaxevrebul mniSvnelobas. 

magaliTad, vTqvaT gvaqvs dakvirvebis Sedegebi x : 1, 7, 4; y : 2, 4. maSin 

U набл =1+1+0+0+0+1/2=2.5. 

3.5.2. uilkoksonis kriteriumi 

wina kriteriumi dafuZnebulia niSnebis kriteriumze. masSi gamoiyeneba 

ara dakvirvebis Sedegebis mniSvnelobebi, aramed maTi urTierT mimarTeba. 

misgan gansxvavebiT uilkoksonis kriteriumi dafuZnebulia dakvirvebis 

Sedegebis rangebze. is aris rangebze dafuZnebuli kriteriumebidan erT – 

erTi pirvelTagani. es kriteriumi gamoiyeneba igive pirobebSi, rogorc 

mani – uitnis kriteriumi. 

vTqvaT mocemulia dakvirvebebis ori amonarCevi x1, x2, ..., x m da y1, y2,..., y n . 

saWiroa ZiriTadi hipoTezis Semowmeba, rom 1, 2, ..., m 

51 

x x x , n y y y ,..., , 2 

1 miekuTvne- 

bian erT generalur amonarCevs. 

daSvebebi dakvirvebis Sedegebis Sesaxeb analogiuria wina SemTxvevisa: 

1. dakvirvebis Sedegebi x1, x2, ..., x m da y1, y2,..., y n arian erTmaneTisagan 

damoukidebeli; 

2. ganawilebis kanonebi F( x ) da G( x ) arian uwyvetebi. 

ZiriTadi hipoTeza Semdegnairad formirdeba: H0 : F( x) = G( x) 

. 

alternatiuli hipoTezebi wina SemTxvevis analogiuria: 

1. marjvenamxrivi F( x) > G( x) 

; 

2. marcxenamxrivi F( x) < G( x) 

; 

3. ormxrivi F( x) ≠ G( x) 

. 

kriteriumis arsi mdgomareobs SemdegSi. dakvirvebis yvela Sedegi 

x1, x2, ..., x m , y1, y2,..., y n lagdeba zrdadobis mixedviT. S1, S2,..., S n – iT aRvniSnoT 

y 

- bis rangebi am variaciul rigSi. gamovTvaloT sidide 

Wнабл = S1 + S2 + ... + Sn 

, 

romelsac uilkoksonis statistika hqvia. ganvixiloT rogor iqcevian es 

rangebi sxvadasxva alternativebis dros. 

1. marjvenamxrivi alternativis dros, cxadia, rom y – is mniSvnelobebi 

ricxviT RerZze daikaveben marjvena naxevars da maTi rangebis jami 

miiRebs did mniSvnelobebs. 

2. marcxenamxrivi alternativis dros, piriqiT, y - bis mniSvnelobebi 

moxvdebian x – bis marcxniv da maTi rangebis jami miiRebs patara 

mniSnelobebs.

Aam Tvisebazea dafuZnebuli uilkoksonis kriteriumi. avtorma SeZlo 

moenaxa W sididis ganawilebis funqcia ZiriTadi H 0 hipoTezis 

WeSmaritebisas. am ganawilebis kvantilebi gamoTvlilia α , m , n sxvadasxva 

mniSvnelobebisaTvis da mocemulia Sesabamis cxrilebSi (ix. danarTi 8), 

saidanac m , n da α mocemuli mniSvnelobebisaTvis vpoulobT Semdegi 

gantolebis amoxsnas 

P( W ≥ W ( α, m, n) | H ) = α . 

пр 

Tu adgili aqvs W ≥ W ( α, 

m, n) 

, maSin miiReba marjvenamxrivi alterna- 

набл пр 

tiuli hipoTeza, winaaRmdeg SemTxvevaSi – ZiriTadi hipoTeza. 

analogiurad, marcxenamxrivi hipoTezisaTvis gvaqvs: α , m , n mocemuli 

mniSvnelobebisaTvis vpoulobT gantolebis amoxsnas 

P( W ≤ W ( α, m, n) | H ) = α . 

л 

Tu Wнабл ≤ Wл ( α, 

m, n) 

miiReba marcxenamxrivi alternatiuli hipoTeza, winaaRmdeg 

SemTxvevaSi miiReba ZiriTadi hipoTeza. 

W - s ganawilebis simetriulobis gamo adgili aqvs 

W ( α, m, n) + W ( α, 

m, n) = n( m + n + 1) . (3.3) 

л пр 

amitom databulirebuli mxolod marjvenamxrivi kritikuli 

mniSvnelobebi da marcxenamxrivi mniSvnelobebi moiZebneba (3.3) 

damokidebulebidan. 

ormxrivi alternatiuli hipotezis Semowmebis dros kritikul ares aqvs 

Semdegi saxe: 

Wнабл ≤ Wл( α, m, n) ∪ Wнабл ≥ Wпр ( α, 

m, n) 

, (3.4) 

anu (3.4) pirobis Sesrulebisas miiReba ormxrivi alternatiuli hipoTeza, 

winaaRmdeg SemTxvevaSi miiReba ZiriTadi hipoTeza. 

rogorc mani – uitnis, aseve uilkoksonis kriteriumisaTvis Sesabamis 

cxrilebSi mocemulia kvantilebis mniSvnelobebi m da n - is SezRuduli 

mniSvnelobebisaTvis. maTi cxrilebSi mocemul mniSvnelobebze meti 

mniSvnelobebisaTvis sargebloben Sesabamisi ganawilebis kanonebis 

normaluri kanoniT aproqsimaciiT. damtkicebulia, rom ZiriTadi 0 H 

hipoTezis WeSmaritebisas da Semotanili daSvebebis samarTlianobisas, 

* 

W ( W MW ) / DW 

= − SemTxveviT sidides miaxloebiT aqvs standartuli 

* 

normaluri ganawileba, rodesac m → ∞ da n → ∞ , anu W ~ N( ⋅ ;0,1) . aq 

MW = n( m + n + 1) / 2 da DW = mn( m + n + 1) /12 . am SemTxvevaSi kritikul ares 

aqvs Semdegi saxe: 

1. marjvenamxrivi alternatiuli hipotezis dros 

2. marcxenamxrivi alternatiuli hipotezis dros 

52 

0 

0 

* 

Wнабл ≥ zα ; 

* 

Wнабл ≤ − zα ; 

* * 

3. ormxrivi alternatiuli hipotezis dros Wнабл ≤ −zα ∪ Wнабл ≥ zα 

. 

aq zα aris standartuli normaluri ganawilebis α donis kvantili. ormxrivi 

alternatiuli hipotezis dros kriteriumis mniSvnelobis done 2α - 

s tolia. Tu gvinda SevinarCunoT kriteriumis done α – s toli, maSin 

Sesabamis kritikul areebSi zα – s nacvlad unda aviRoT zα / 2 . 

mani – uitnis da uilkoksonis kriteriumebi erTmaneTTan dakavSirebulia 

damokidebulebiT W = U + n( n + 1) / 2 . es damokidebuleba gviCvenebs U da W

statistikebis eqvivalentobas. amitom maT gamoyenebas erTnair Sedegebamde 

mivyavarT. 

3.6. Sewyvilebuli dakvirvebebi 

wina paragrafebSi SeviswavleT niSnebis, mani-uitnis, uilkoksonis kriteriumebi, 

romlebic saSualebas iZlevian erTmaneTs SevadaroT ori 

amonarCevi, gavakeToT daskvnebi maTi identurobis Sesaxeb. amasTan 

avRniSneT, rom dakvirvebis obieqtebi arian erTgvarovnebi. dakvirvebis 

obieqtebis erTgvarovneba aris mniSvnelovani piroba. magram, saubedurod 

yovelTvis ar aris saSualeba SevamowmoT maTi erTgvarovneba. dakvirvebis 

obieqtebis araerTgvarovnebis Tavidan asacileblad, xSirad, gamocdebisas 

iyeneben erTi da igive obieqtebs. magaliTad: swavlebis ori meTodis 

Sedarebisas am meTodebs iyeneben adamianebis erTi da igive jgufze; ori 

teqnologiuri procesis Sedarebisas iyeneben erTi da igive danadgarebs da 

a.S. aseT SemTxvevebSi, erTi da igive obieqtisaTvis Rebuloben dakvirvebebis 

or simravles da Tu n dakvirvebebis ricxvia, maSin Sedegad Rebuloben 

( xi , yi ), i = 1,..., n , Sewyvilebul dakvirvebebs. SeviswavloT aseTi dakvirvebis 

Sedegebis damuSavebis meTodebi. 

3.6.1. niSnebis kriteriumi Sewyvilebuli amonarCevis analizisaTvis 

vTqvaT ( xi , yi ), i = 1,..., n , n moculos Sewyvilebuli dakvirvebebia. saWiroa 

damuSavebis efeqtis ar arsebobis Sesaxeb hipoTezis Semowmeba, romelic Caiwereba 

ase 

H : P( x < y ) = P( x > y ) = 0.5 yvela i = 1,..., n . 

0 

i i i i 

SemovitanoT sidide zi = yi − xi , i = 1,..., n . keTdeba daSvebebi: 1) yvela z i ur- 

TierT damoukidebali SemTxveviTi sidideebia. es daSveba ar niSnavs 

erTnairi indeqsis mqone i x da y i damoukideblobas. es Zalze 

mnniSvnelovania praqtikaSi, rodesac dakvirvebebi xdeba erTi da igive 

obieqtze da amdenad, SeiZleba damokidebulebi iyvnen. 2) yvela z i aqvs nolis 

toli medianebi, e.i. P( z < 0) = P( z > 0) = 1/ 2 . kidev erTxel gausvaT xazi, rom 

i i 

sxva da sxva z i – s ganawilebis kanoni SeiZleba erTmaneTs ar emTxveodes. 

avirCioT alternatiuli hipoTza, romelic SeiZleba iyos rogorc marjvenamxrivi, 

aseve marcxenamxrivi da ormxrivi. avirCioT marjvenamxrivi: 

H1 : P( xi > yi ) < P( xi < yi 

) . z i – Tan mimarTebaSi ZiriTadi da alternatiuli hipoTezebi 

SeiZleba CavweroT Semdegnairad: H0 : P( zi < 0) = P( zi 

> 0) = 0.5 ; 

H1 : P( zi < 0) < P( zi 

> 0) , e.i. ZiriTadi hipoTeza gulisxmobs, rom zi, i = 1,..., n – 

Tvis noli aris mediana. 

53

amrigad, gansaxilveli SemTxveva daviyvaneT seriis kriteriumis cnobil 

sqemamde, rodesac bernulis ganawilebis dros z i SemTxveviTi sididisaTvis 

unda Semowmdes hipoTeza medianis Sesaxeb. 

z -s Soris nulovani mniSvnelobebis arsebobisas saWiroa maTi gadagdeba 

i 

da Sesabamisad n -is mnniSvnelobis Semcireba z i –is aranulovan raodenobamde. 

n –is didi mniSvnelobisas, iseve rogorc seriis kriteriumSi, SeiZleba 

visargebloT binomialuri ganawilebis noraluri apoqsimaciiT. 

Tu saWiroa SevadaroT ara marto ori Sewyvilebuli ganmeorebadi dakvirvebebi, 

aramed, maT Soris gansxvavebis arsebiTobis SemTxvevaSi, 

SevafasoT am gansxvavebis sidide, saWiroa visargebloT Semdegi modeliT. 

uSveben, rom z = θ + ε , i = 1,..., n , sadac θ – raRac mudmivia, romelic 

i i 

axasiaTebs erTi ganawilebis meores mimarT mdebareobas, xolo ε i – 

SemTxveviTi sidideebia maTematikuri molodiniT M ( ε i ) = 0 . aseTi modelis 

SemTxvevaSi ZiriTadi da alternatiuli hipoTezebi Rebuloben saxes: 

0 : 0 H θ = da H1 : θ > 0 . alternatiuli hipoTezis samarTlianobisas arsebobs 

SesaZlebloba SevafasoT θ - ori amonarCevis gansxvavebis sidide. 

ganxiluli kriteriumis Riseba mdgomareobs imaSi, rom is ar adebs 

x y dakvirvebis Sedegebs. erTad erTi moTxovna mdgoma- 

mkacr moTxovnebs , 

i i 

reobs imaSi, rom z i unda iyvnen damoukideblebi erTmaneTis mimarT. amasTan 

x i da i y SeiZleba iyvnen damokidebulebi. agreTve z i SeiZleba 

emorCilebodnen sxvadasxva ganawilebis kanonebs. 

3.6.2. ganmeorebadi Sewyvilebuli dakvirvebebis analizi niSnebis 

rangebis mixedviT (uilkoksonis niSnebis rangebis jamebis kriteriumi) 

Tu SeiZleba dauSvaT, rom wina punqtSi Semotanili zi, i = 1,..., n, 

uwyveti, 

erTnairad ganawilebuli SemTxveviTi sidideebia, maSin erTgvarovnebis 

hipoTezis Sesamowmeblad SeiZleba gamoviyenoT ufro Zlieri uilkoksonis 

niSnebis rangebis jamebis kriteriumi, romlis arsic mdgomareobs SemdegSi. 

ZiriTad da alternatiul hipoTezebs aqvT wina punqtSi moyvanilis 

analogiuri saxe: 

H0 : P( xi < yi ) = P( xi > yi 

) = 0.5 ; H1 : P( xi > yi ) < P( xi < yi 

) . 

gamovTvaloT i z – aris i x – is y i - gan gadaxris mniSvnelobebi, anu 

zi = yi − xi 

. SemovitanoT damxmare funqcia ψ i, i = 1,..., n , sadac 

⎧ 

⎪1, 

если zi 

> 0; 

ψ i = ⎨ 

⎪ 

⎩0, 

если zi 

< 0. 

vTqvaT i R aris i z – is rangi z1 ,..., z n – is zrdadobis mixedviT mowes- 

rigebul mimdevrobaSi. gamovTvaloT statistika 

H 1 hipoTezis WeSmaritebisas statistikas набл 

54 

n 

T = ∑ ψ R . alternatiuli 

набл i i 

i= 

1 

T aqvs tendencia miiRos didi

mniSvnelobebi. amitom 0 H hipoTezis H 1 hipoTezis mimarT Semowmebis 

kriteriums aqvs saxe: Tu Tнабл ≥ t( α, 

n) 

, maSin miiReba alternatiuli hipoTeza, 

winaaRmdeg SemTxvevaSi miiReba ZiriTadi hipoTeza. aq α aris kriteriumis 

mniSvnelobis done, t( α , n) 

aris α donis kvantili, romelic akmayofilebs 

gantolebas P( T ≥ t( α, n) | H ) = α . 

marcxenamxrivi alternativis dros H 2 : P( xi > yi ) > P( xi < yi 

) kritikul ares 

n( n + 1) 

aqvs saxe Tнабл ≤ − t( α, 

n) 

. 

2 

ormxrivi alternatiuli hipoTezis dros H3 : P( xi > yi ) ≠ P( xi < yi 

) 

kritikul ares, anu H 3 hipoTezis miRebis ares aqvs saxe 

n( n + 1) 

Tнабл ≥ t( α, n) ∪ Tнабл ≤ − t( α, 

n) 

. 

2 

am SemTxvevaSi kriteriumis mniSvnelobis done 2α – s tolia. Tu gvinda 

SevinarCunoT α toli kriteriumis mniSvnelobis done, maSin kritikul 

areSi t( α , n) 

nacvlad unda aviRoT t( α / 2, n) 

. 

x i da y i SemTxveviTi sidideebis uwyvetobis gamo maT Soris erTnairi 

mniSvnelobebi Teoriulad ar unda iyos. Magram, damrgvalebis Secdomebis 

gamo, dakvirvebebis praqtikul SedegebSi gvxvdeba erTnairi mniSvnelobebi. 

cxadia maTi Sesabamisi z i = 0 . aseT SemTxvevaSi z i –is nulovani mniSvnelobebi 

ukuigdeba, dakvirvebebis saerTo ricxvi mcirdeba Sesabamisi sididiT da 

aRwerili kriteriumi gamoiyeneba dakvirvebis Sedegebis Semcirebuli 

raodenobisaTvis. 

Tu zi, i = 1,..., n , Soris gvxvdeba erTnairi mniSvnelobebi, maSi R i rangebis 

gamoTvlisas gamoiyeneba saSualo rangebi. dakvirvebis Sedegebis didi 

raodenobisas SeiZleba visargebloT normaluri aproqsimaciiT. H 0 hipoTezis 

samarTlianobisas statistikas 

* T − MT T − n( n + 1) / 4 

T = = 

DT n( n + 1)(2n + 1) / 24 

asimptoturad (roca n → ∞ ) aqvs standartuli normaluri ganawileba 

N( ⋅ ;0,1) . kritikul ares (magaliTad, marjvenamxrivi alternativisas) aqvs 

* 

saxe Tнабл ≥ zα , sadac zα aris standartuli normaluri ganawilebis α 

procentuli wertili. 

55

Tavi 4. SefasebaTa Teoriis safuZvlebi 

4.1. Sesavali 

albaTobis Teoria da maTematikuri statistika swavloben kanonebs, romlebsac 

emorCilebian SemTxveviTi sidideebi, movlenebi, xdomilebebi; 

kerZod, sxvadasxva SemTxveviTi xdomilebebis albaTobebis gamoTvlis 

meTodebs. Zalze iSviaTad xerxdeba SemTxveviTi xdomilebis albaTobis 

gamoTvla. rogorc wesi, albaToba gamoiTvleba SemTxveviTi sidideebis 

albaTobebis ganawilebis kanonebis saSualebiT romlebzedac aris 

damokidebuli mocemuli xdomileba. albaTobebis ganawilebis kanoni aris 

gansazRvruli Tvisebebis mqone funqcia, romelic damokidebulia 

parametrebis simravleze. am parametrebis konkretuli mniSvnelobebi 

gamoyofen erT konkretul ganawilebas erTnairi tipis ganawilebebis 

usasrulo simravlidan. magaliTad, vTqvaT ξ SemTxveviT sidides aqvs 

2 

normaluri ganawilebis kanoni m maTematikuri molodiniT da σ 

2 

2 

dispersiiT, anu ξ ~ N( ⋅ ; m, 

σ ) . m da σ - is mniSvnelobebisagan damokidebulebiT 

miiReba normaluri kanonebis usasrulo simravle. erTi ganawilebis 

kanonis konkretizaciisaTvis, romelsac sinamdvileSi emorCileba ξ 

SemTxveviTi sidide, saWiroa normaluri kanonebis usasrulo simravlidan 

2 

gamovyoT erTi ganawileba, anu davakonkretoT m da σ parametrebis 

mniSvnelobebi, romlebic Seesabamebian ξ SemTxveviT sidides, anu vipovoT 

albaTobebis ganawilebis kanonebis parametrebis ucnobi mniSvnelobebi. es 

SesaZlebelia gakeTdes ξ SemTxveviTi sididis dakvirvebis Sedegebis 

safuZvelze, anu x1, x2,..., x n dakvirvebis Sedegebis safuZvelze. amitom 

ganawilebis parametrebis napovni mniSvnelobebi arian ara zusti, aramed 

maTi miaxloebiTi mniSvnelobebi. maTematikur statistikaSi im faqtis 

2 

aRsaniSnavad, rom dakvirvebis Sedegebis safuZvelze m da σ parametrebis 

napovni mniSvnelobebi arian miaxloebiTi mniSvnelobebi, gamoiyeneba sityva 

2 

2 

Sefaseba, anu a ≡ ( m, σ ) parametris Sefaseba aris aˆ ≡ ( mˆ , ˆ σ ) , romelic aris a 

parametrebis veqtoris ucnobi namdvili mniSvnelobis miaxloebiTi 

mniSvneloba, anu aˆ( x1, x2,..., xn) ≈ a . dakvirvebis Sedegebis safuZvelze â 

mniSvnelobis povnis process Sefasebas uwodeben. arseboben a parametris 

sxvadasxa miaxloebiTi mniSvnelobebis povnis didi raodenobis meTodebi. 

problema mdgomareobs a – s yvela SesaZlo mniSvnelobebidan garkveuli 

TvalsazrisiT saukeTeso Sefasebis gamoyofaSi. SeiZleba movifiqroT bevri 

kriteriumi terminis „saukeTeso“ gansazRvrisaTvis. qvemoT SevCerdebiT 

yvelaze ufro gavrcelebul da sxvebze ufro xSirad gamoyenebul 

kriteriumebze, rogorebic arian: 

1. sizustis kriteriumebi (parametris WeSmarit mniSvnelobasTan 

siaxlove); 

2. waunacvleblobis kriteriumebi (Sefasebis maTematikuri molodinis 

toloba parametris WeSmarit mniSvnelobasTan); 

3. safuZvlianobis kriteriumebi (dakvirvebebis raodenobis gazrdisas 

Sefasebis albaTurad miswrafeba parametris WeSmariti 

mniSvnelobisaken); 

56

4. efeqturobis kriteriumi (Sefasebis dispersiis minimaluroba sxva Sefasebebis 

dispersiebTan SedarebiT) da a.S. 

Sefasebis povna, romelsac eqneboda yvela zemoT CamoTvlili Tvisebebi, 

sakmaod rTuli amocanaa. amitom, dasmuli miznebidan gamomdinare, 

konkretuli amocanebis gadawyvetisas, upiratesoba eZleva Sefasebebis ama 

Tu im Tvisebebs. 

zogierT wina paragrafSi ukve gvqonda Sexeba albaTobebis ganawilebis 

parametrebis SefasebebTan. magaliTad, saSualo ariTmetikulis gamoyenebiT 

vafasebdiT maTematikuri molodinis mniSvnelobas, amonarCevis dispersiiT 

vpoulobdiT dispersiis ucnob mniSvnelobas. isini iyvnen SemTxveviTi 

sididis Sesabamisi ricxviTi maxasiaTeblebis Sefasebebi, Tumca terminiT 

„Sefaseba“ ar vsargeblobdiT. amonarCevis albaTobebis ganawilebis n F 

funqciiT vafasebdiT ucnob F albaTobebis ganawilebis funqcias; cdebSi A 

xdomilebis gamosvlis n( A) / n sixSiriT vafasebdiT am xdomilebis P 

albaTobas. Yyvela isini iyvnen e.w. wertilovani Sefasebebi, radgan 

iZleodnen Sesabamisi a parametris ucnobi mniSvnelobis erT â SefasebiT 

mniSvnelobas. Aam SemTxvevaSi SeuZlebelia miuTiTo ramdenad zustia a 

parametris â Sefaseba. am amocanis gadasawyvetad saWiroa miuTiTo ucnobi 

parametris ara marto erTi miaxloebiTi mniSvneloba, aramed raRac 

intervali, romelSic mocemuli albaTobiT imyofeba parametris ucnobi 

WeSmariti mniSvneloba. Tu xerxdeba aseTi intervalis moZebvna, mas 

uwodeben intervalur Sefasebas. 

Sefasebebis moZebnisas garkveuli daSvebebi keTdeba ξ SemTxveviTi 

sididis Tvisebebis Sesaxeb da am Tvisebebis safuZvelze moinaxeba zemoT 

CamoTvlili optimaluri Tvisebebis mqone Sefasebebi. Tu ξ - s Tvisebebis 

Sesaxeb gakeTebuli daSvebebi ar arian samarTliani, maSin gamoTvlili 

Sefasebebi iqnebian araoptimaluri. magaliTad, saSualo ariTmetikuli 

1 

1 

n 

x = ∑ xi 

aris normalurad ganawilebuli SemTxveviTi sididis 

n i= 

maTematikuri molodinis optimaluri Sefaseba, romelsac aqvs optimalobis 

yvela zemoT CamoTvlili Tviseba. AlbaTobebis ganawilebis kanonis 

normalurobis daSvebis darRvevisas (magaliTad, Tu x1, x2,..., x n amonarCevi 

Seicavs uxeS Secdomebs) saSualo ariTmetikuli ar aris maTematikuri 

molodinis optimaluri Sefaseba da miT ufro cud Sefasebebs iZleva, rac 

ufro uxeSia Secdomebi dakvirvebis SedegebSi. ukanaskneli ramodenime 

aTeuli weliwadia intensiurad viTardeba Sefasebebis specialuri 

meTodebi, romlebic iZlevian mdgrad Sefasebebs dakvirvebis Sedegebis TvisebebTan 

dakavSirebiT gakeTebuli sawyisi daSvebebis darRvevisas 

(magaliTad, dakvirvebis SedegebSi uxeSi Secdomebis arsebobisas). aseT 

meTodebs uwodeben Sefasebis robastul meTodebs. winamdebare kursSi ar 

ganvixilavT aseT Sefasebebs. 

57

4.2. did ricxvTa kanoni 

rogorc zemoT avRniSneT, Zalze iSviaTad xerxdeba xdomilebis 

albaTobis uSualo gamoTvla. es SesaZlebelia Tu eqsperimentis sqema daiyvaneba 

Sedegebis sasrulo ricxamde, romelTagan nebismieris moxdena aris 

SesaZlebeli eqsperimentebis Sedegad. aseT SemTxvevaSi atareben n 

eqsperiments da iTvlian im eqsperimentebis raodenobas, romlebSiac adgili 

hqonda CvenTvis saintereso A xdomilebas. avRniSnoT is n( A ) – Ti. A 

xdomilebis p albaTobis Sefasebad Rebuloben n( A) / n sixSires. 1 – is 

gazrdisas, anu rodesac n → ∞ adgili aqvs n( A) / n → p . maTematikuri 

analizis enaze es faqti SeiZleba Caiweros ase: arseboben iseTi N da ε > 0 , 

rom rodesac n > N adgili aqvs n( A) / n − p < ε . albaTobis TeoriaSi da 

maTematikur statistikaSi aseTi gansazRvra ar aris samarTliani imitom, 

rom nebismieri n - Tvis albaToba imisa, rom n( A) / n rac ar unda didad 

gansxvavdeba p - gan ar udris nols. amitom, aseTi uzustobebis Tavidan 

asacileblad, albaTobis TeoriaSi Semotanilia zRvris aseTi gansazRvra: 

n( A) / n sixSire ikribeba p albaTobisaken rodesac n → ∞ , Tu adgili aqvs 

⎛ n( A) 

⎞ 

P ⎜ − p < ε ⎟ →1 

⎝ n ⎠ 

nebismieri ε > 0 . 

istoriulad xdomilebis sixSiris krebadoba Sesabamisi albaTobisaken 

pirvelad daamtkica bernulma 17 – e saukunis bolos da mis sapativsacemod 

am Teoremas qvia bernulis bernulis Teorema Teorema. Teorema Teoremis arsi mdgomareobs SemdegSi: 

dauSvaT, rom n eqsperimentidan yovelSi A xdomilebis p albaToba 

ucvleli rCeba da yoveli eqsperimentis Sedegi danarCenisagan 

damoukidebelia, maSin n - is didi mniSvnelobisaTvis A xdomilebis 

moxdenis n( A) / n sixSire miaxloebiT A xdomilebis albaTobis tolia, anu 

p = n( A) / n . bernulis Teoremis Semdgomi ganviTareba aris did ricxvTa 

kanoni, romlis umartivesi variantic aris CebiSevis Teorema. avRniSnoT 

x1, x2,..., x n SemTxveviT ξ sidideze damoukidebeli dakvirvebis Sedegebia. 

maSin CebiSevis Teoremas aqvs saxe: 

⎛ x1 + x2 + ... + xn 

⎞ 

P ⎜ − M ( ξ ) < ε ⎟ →1 

roca n → ∞ , 

⎝ n 

⎠ 

anu damoukidebeli, erTnairad ganawilebuli didi raodenobis SemTxveviTi 

ricxvebis saSualo ariTmetikuli albaTurad miiswrafis maTi maTematikuri 

molodinisaken. 

n - is didi mniSvnelobisaTvis amonarCevis maxasiaTeblis miswrafeba Sesabamisi 

Teoriuli maxasiaTeblisaken (misi namdvili mniSvnelobisaken) 

samarTliania ara marto saSualo ariTmetikulisaTvis. ganawilebis F 

funqciis Tvisebebze da CvenTvis saintereso maxasiaTeblebze sakmaod susti 

daSvebebisas, didi n –Tvis, amonarCevis maxasiaTeblis mniSvneloba 

miiswrafis Sesabamisi Teoriuli maxasiaTeblis Teoriuli 

mniSvnelobisaken. es mtkiceba Zalze mniSvnelovania albaTobis 

TeoriisaTvis da maTematikuri statistikisaTvis da ewodeba did ricxvTa 

kanoni. 

58

did ricxvTa kanonis Semdgom ganviTarebas warmoadgens centraluri 

zRvruli Teorema. avRniSnoT θ = ( θ1,..., θr 

) ganawilebis parametrebis Teoriuli 

mniSvnelobebi. rodesac r = 1 parametri θ aris skalaruli sidide, rodesac 

r ≥ 2 parametri θ aris veqtoruli sidide. magaliTad, normaluri ka- 

2 

nonisaTvis θ = ( θ1, θ2 ) = ( m, 

σ ) . avRniSnoT θ n aris θ parametris Sefasebuli 

mniSvneloba, gamoTvlili dakvirvebis Sedegebis x1, x2,..., x n safuZvelze. maSin 

centraluri zRvruli Teorema amtkicebs: ganawilebis F funqciaze da θ 

parametrze Zalze susti daSvebebisas SemTxveviT sidides 

n( 

θ n −θ 

) 

aqvs asimptoturad (rodesac n → ∞ ) normaluri ganawileba garkveuli 

2 

( m, σ ) parametrebiT, anu 

n θ −θ N ⋅ m σ . 

2 

( n ) ~ ( ; , ) 

4.3. statistikuri parametrebi 

maTematikur statistikaSi terminis “statistikuri modelebis 

parametrebis” qveS igulisxmeba ori azriT axlos mdgomi, magram mainc sxva 

mniSvneloba: 1) albaTobebis ganawilebis parametrebi da 2) modelebis 

parametrebi. 

ganawilebis kanonebis parametrebi aris ricxviTi maxasiaTeblebis sasrulo 

erToblioba, romelTa gansazRvruli mniSvnelobebic gamoyofen erT 

konkretul ganawilebas mocemuli tipis albaTobebis ganawilebis kanonebis 

usasrulo simravlidan. magaliTad, albaTobebis ganawilebis normaluri 

kanonis simkvrives aqvs saxe 

59 

2 ⎧ ⎫ 

1 ( x − a) 

f ( x) 

= exp ⎨− 2 ⎬ . 

2πσ 

⎩ 2σ 

⎭ 

2 

mocemuli ganawilebis parametrebia a da σ . maTi mocemuli 

mniSvnelobebi gamoyofen erT konkretul ganawilebas albaTobebis 

ganawilebis normaluri kanonebis usasrulo simravlidan. 

parametrebis meore tipi aris statistikuri modelebis ricxviTi mniSvnelobebi, 

romlebic aRweren sxvadasxva statistikur kanonzomierebebs. magaliTad, 

parametri θ niSnebis kriteriumSi erTi amonarCevisaTvis, agreTve 

niSnebis kriteriumSi Sewyvilebuli ganmeorebadi dakvirvebebis 

analizisaTvis; regresiuli damokidebulebebis parametrebi, romlebic 

aRweren damokidebulebebs SemTxveviT sidideebs Soris (ganxiluli iqneba 

qvemoT); faqtorebis parametrebi faqtorul analizSi, romlis kerZo 

SemTxvevas warmoadgens dispersiuli analizi (ganxiluli iqneba qvemoT) da 

a.S. 

statistikuri parametrebi avRniSnoT θ . Tu parametrebis ricxvi erTia, 

maSin θ aris skalaruli sidide. Tu parametrebis ricxvia r , maSin 

θ = ( θ1,..., θr 

) aris veqtoruli sidide. 

statistikuri modelebis nebismieri maxasiaTeblebi SeiZleba gamosaxuli 

iqnas maTi parametrebis saSualebiT. amitom maTematikuri statistikis erT – 

erTi ZiriTadi amocana mdgomareobs SemTxveviT sidideze dakvirvebis Sede-

gebis, anu amonarCevis daxmarebiT am parametrebis mniSvnelobebis monaxvaSi. 

imis gamo, rom amonarCevi Sedgeba dakvirvebis Sedegebis sasrulo raodenobisagan, 

maTi saSualebiT gamoTvlili Sedegebi arian Sesabamisi 

maxasiaTeblebis miaxloebiTi mniSvnelobebi. sityvaTa Sexamebis 

“miaxloebiTi mniSvnelobebi” magier statistikaSi ixmareba termini 

“Sefaseba”. statistikuri modelis θ parametrebis Sefasebebi moinaxeba 

damoukidebeli erTnairad ganawilebuli x1, x2,..., x n dakvirvebis Sedegebis 

safuZvelze, anu pouloben iseT t( x1,..., x n) 

funqcias, rom adgili hqondes 

t( x ,..., x ) ≈ θ . 

1 

n 

4.4. ganawilebis parametrebis Sefaseba amonarCeviT 

rogorc zemoT ukve aRvniSneT, statistikuri parametrebis Sefasebebi 

arian am parametrebis miaxloebiTi mniSvnelobebi gamoTvlili dakvirvebis 

Sedegebis safuZvelze. arsebobs statistikuri parametrebis gamoTvlis ori 

ZiriTadi midgoma: 1) Sefasebebi gamoiTvleba dakvirvebebis mocemuli 

sasruli raodenobiT, anu sasrulo x1, x2,..., x n amonarCevis safuZvelze; 2) 

Sefasebebi moinaxeba mudmivad zrdadi raodenobis dakvirvebis Sedegebis 

safuZvelze; yoveli Semdgomi dakvirvebis SedegiT, wina dakvirvebis 

Sedegebis safuZvelze parametrebis ukve gamoTvlili Sefasebebi, zustdebian 

manmade, sanam miRebul Sefasebebs ar eqnebaT saWiro sizuste mocemuli 

albaTobiT. ukanasknelebs ewodebaT mimdevrobiTi analizis meTodebi. 

mimdevrobiTi analizis meTodebi arian ufro mniSvnelovani rogorc 

praqtikuli, aseve Teoriuli TvalsazrisiT, vidre sasrulo amonarCevze 

dafuZnebuli meTodebi imitom, rom isini saSualebas iZlevian ufro srulad 

gamoviyenoT dakvirvebis SedegebiT miRebuli informacia. mocemuli 

sizustis Sefasebebis misaRebad isini saSualod iyeneben ufro mcire sigrZis 

amonarCevs, vidre sasrulo amonarCevze dafuZnebuli meTodebi. 

zemoT viswavleT SemTxveviTi sidideebis zogierTi statistikuri maxasiaTeblebis 

gamoTvlis formulebi. magaliTad, viciT, rom saSualo ariTmetikuli 

x , gamoTvlili x1, x2,..., x n dakvirvebis Sedegebis safuZvelze, iZleva am 

ganawilebis a maTematikuri molodinis miaxloebiT mniSvnelobas (maSin 

saSualo ariTmetikuls Sefasebas ar veZaxodiT). Semotanili terminologiis 

Tanaxmad x n aris maTematikuri molodinis Sefaseba gamoTvlili n 

dakvirvebis Sedegebis safuZvelze, anu xn ≈ a . CebiSevis kanonis Tanaxmad 

по вероятности 

xn ⎯⎯⎯⎯⎯⎯→ a rodesac n → ∞ . zustad aseve, zemoT naswavli 

gamosaxuleba, romelic saSualebas iZleva gamoviTvaloT dispersiis 

2 1 n 

2 

miaxloebiTi mniSvneloba S = ∑ ( xi − x) 

aris am dispersiis Sefaseba, anu 

n i= 

1 

2 

S 

2 

2 по вероятности 2 

≈ σ da CebiSevis Teoremis Tanaxmad S ⎯⎯⎯⎯⎯⎯→ σ rodesac n → ∞ . 

zogadad, ganawilebis yoveli kanoni F( θ ) damokidebulia parametrebis 

sasrulo raodenobisagan. am ganawilebis nebismieri maxasiaTebeli 

gamoTvlili F( θ ) ganawilebis kanonis daxmarebiT damokidebulia am 

60

ganawilebis θ parametrze. avRniSnoT es maxasiaTebeli T – s saSualebiT. 

maSin T damokidebulia θ parametrze, anu T = T ( θ ) . dakvirvebis Sedegebis 

x1, x2,..., x n safuZvelze gamovTvliT mocemuli maxasiaTeblis Sefasebas. 

avRniSnoT is T n – is daxmarebiT. viciT, rom did ricxvTa kanonis Tanaxmad 

Tn ≈ T ( θ ) . Tu am gantolebas amovxsniT θ - s mimarT, miviRebT θ ucnobi 

parametris Sefasebas. es midgoma aris statistikuri parametrebis 

Sefasebebis miRebis saerTo meTodika. imisgan damokidebulebiT, Tu romeli 

maxasiaTeblebi amoirCeva, miiReba θ parametrebis sxvadasxva Sefasebebi. Tu 

amocana koreqtulad aris dasmuli da amoxsnili, maSin yvela isini arian θ 

parametris ucnobi WeSmariti mniSvnelobis miaxloebiTi mniSvnelobebi. 

magaliTis saxiT ganvixiloT egreT wodebuli momentebis da kvantilebis 

meTodebi. momentebis meTodSi statistikur maxasiaTeblebad gamoiyeneba Sem- 

TxveviTi sididis momentebi. arCeuli momentebis raodenoba damokidebulia 

albaTobebis ganawilebis kanonebis parametrebis ricxvze, anu θ - s ganzomilebaze. 

magaliTad, ganvixiloT normaluri ganawileba, romelic 

damokidebulia or parametrze – maTematikuri molodini da dispersia. 

amrigad, momentebis meTodSi maxasiaTeblebad unda amovirCioT romelime 

ori momenti. avirCioT pirveli da meore rigis sawyisi momentebi. viciT, 

2 2 2 

2 

rom normaluri ganawilebisaTvis Mξ = a da Mξ = a + σ . Mξ da Mξ - is 

1 

statistikuri analogebi Sesabamisad arian 

1 

n 1 2 

∑ xi 

da 

n i= 

1 

n 

∑ xi 

. SevadginoT 

n i= 

sistema ori gantolebisagan 

⎧ 1 n 

a = ∑ xi 

, 

⎪ n i= 

1 

⎨ 

. 

2 2 1 n 

⎪ 2 

a + σ = ∑ xi 

⎪⎩ n i= 

1 

amrigad, miviReT ori gantolebisagan Semdgari sistema ori ucnobiT a da 

2 

σ . 

am sistemis amoxsniT ucnobi parametrebis mimarT vRebulobT a = x da 

2 1 n 

2 

σ = ∑ ( xi − x) 

. sabolood miviReT, rom maTematikuri molodinisa da dis- 

n i= 

1 

persiis gamoTvlis zemoT naswavli formulebi arian Sefasebis 

gamosaTvleli formulebi, miRebuli momentebis meTodiT albaTobebis 

ganawilebis normaluri kanonis Sesabamisi parametrebisaTvis. Tu 

maxasiaTeblad avirCevdiT SemTxveviTi sididis ara pirvel or moments, 

2 

aramed romelime sxva or moments, miviRebdiT sxva formulebs a da σ 

2 

Sefasebebis gamosaTvlelad. isinic iqnebodnen a da σ – s ucnobi 

mniSvnelobebis Sefasebebi, magram mogvcemdnen sxva mniSvnelobebs, vidre 

zemoT moyvenili formulebi. 

kvantilebis kvantilebis meTodi meTodi. meTodi meTodis arsi analogiuria momentebis meTodis, magram 

am SemTxvevaSi T maxasiaTeblad da mis T n statistikur analogad airCevian 

kvantilebi. es meTodic ganvixiloT normaluri ganawilebis magaliTze. 

2 

vTqvaT ξ normalurad ganawilebuli SemTxveviTi sididea, anu ξ ~ N( a, 

σ ) , 

xolo η ~ N(0,1) 

, maSin ξ = a + σ ⋅ η . SemTxveviTi η sididis zeda da qveda 

61

kvantilebi ewodebaT Sesabamisad sidideebs 

62 

−1 

Φ (0.75) da 

−1 

Φ (0.25) , sadac Φ - 

normirebuli normalurad ganawilebuli SemTxveviTi sidids ganawilebis 

funqciaa. SemTxveviTi ξ sididis zeda da qveda kvartilebi Sesabamisad 

iqnebian 

−1 

−1 

( a + σ ⋅Φ (0.75)) da ( a −σ ⋅Φ (0.25)) . (4.1) 

viciT, rom normalurad ganawilebuli SemTxveviTi sididisaTvis mediana da 

maTematikuri molodini erTmaneTs emTxveva. Mediana aris 0.5 donis 

kvantili, anu 

−1 

Φ (0.5) . amitom 

= Φ (0.5) = ( , ,..., ) = (0.5) , 

−1 

a med x 1 x2 xn ϑn 

sadac ϑ n aris 0.5 donis mqone amonarCevis kvantili, anu mediana. avRniSnoT 

ϑ n(0.75) 

da ϑ n(0.25) 

arian SemTxveviTi ξ sididis Ssabamisad zeda da qveda 

kvartilebis statistikuri analogebi. maSin (4.1) – is safuZvelze vwerT 

−1 

a + σ ⋅Φ (0.75) = ϑ (0.75) , 

−1 

a −σ ⋅Φ (0.25) = ϑn 

(0.25) . 

aqedan advilad vRebulobT saSualo kvadratuli gadaxris Sefasebis gamosaTvlel 

formulas 

ϑn (0.75) −ϑn (0.25) ϑn (0.75) −ϑn 

(0.25) 

ˆ σ = = 

, 

−1 −1 −1 

Φ (0.75) − Φ (0.25) 2 ⋅Φ (0.75) 

−1 −1 

radgan samarTliania toloba Φ (0.75) = −Φ (0.25) . amrigad, maTematikuri 

molodinis da saSualo kvadratuli gadaxris Sefasebebis gamosaTvlelad 

miviReT momentebis meTodisagan gansxvavebuli formulebi. 

4.5. Sefasebebis Tvisebebi. intervaluri Sefasebebi 

zemoT ukve avRniSneT, rom principSi SesaZlebelia statistikuri parametrebis 

Sefasebebis usasrulo raodenobis povna. imisaTvis, rom gamovyoT 

am usasrulo raodenobidan yvelaze ufro optimaluri Sefasebebi, saWiroa 

CamovayaliboT optimalurobis kriteriumebi, anu kriteriumebi, romlebic 

saSualebas iZlebian upiratesoba mivceT ama Tu im Sefasebebs. yvelaze 

gavrcelebuli kriteriumebia Semdegi. 

1. Sefasebis safuZvlianoba, romlis qveSac igulisxmeba Sefasebis 

по вероят. 

Semdegi Tviseba θn ⎯⎯⎯⎯→ θ , rodesac n → ∞ . 

2. 2. Sefasebis waunacvlebloba, romlis qveSac igulisxmeba Sefasebis 

Semdegi Tviseba Mθn = θ . 

3. 3. Sefasebis efeqturoba, romlis qveSac igulisxmeba Sefasebis Semdegi 

2 

Tviseba M ( θn −θ ) ⇒ min , anu Sefasebis dispersiis minimaluroba. 

zegjer efeqturobis maCveneblad gamoiyeneba ara dispersia, aramed gansazRvruli 

W ( θn , θ ) funqcia, romelic axasiaTebs risks, dakavSirebuls θn - 

SefasebasTan. am funqcias danakargebis funqcias uwodeben. Sefasebas 

uwodeben efeqturs, Tu misi Sesabamisi danakargebis maTematikuri molodini 

MW ( θn , θ ) minimaluria. 

n

aqamde vixilavdiT wertilovan Sefasebebs, romlebic dakvirvebis 

Sedegebis ( x1, x2,..., x n) 

safuZvelze iZlevian Sesafasebeli parametris erT 

miaxloebiT mniSvnelobas. amasTan ucnobia Sefasebebis gamoTvlisas 

daSvebuli Secdomebi, anu ar aris cnobili θ n ramdenad Sors aris θ – gan. 

xSirad, amocanebis gadawyvetisas, moiTxoveba iseTi ares povna (dakvirvebis 

Sedegebis safuZvelze), romelic moicavs parametri θ - s ucnob namdvil 

mniSvnelobas mocemuli albaTobiT, anu saWiroa iseTi, mocemul 

albaTobaze damokidebuli An ( α ) ares povna, rom adgili hqondes 

P( θ ∈ A ( α)) = 1− 

α , 

α 

n 

anu albaToba imisa, rom θ - s ucnobi mniSvneloba imyofeba am areSi 1− α 

- s tolia. indeqsi n miuTiTebs im faqtze, rom An ( α ) are napovnia n 

moculobis amonarCevis safuZvelze. 1− α albaTobas eZaxian ndobis 

albaTobas, xolo An ( α ) ares – ndobis ares. rac ufro didia 1− α ndobis 

albaToba, miT didia ndobis intervali. 

4.6. maqsimaluri (udidesi) SesaZleblobebis meTodi 

SeviswavloT statistikuri modelebis parametrebis monaxvis kidev erTi 

meTodi (garda momentebis da kvantilebis). es meTodi yvelaze ufro gavrcelebulia 

imitom, rom saSualebas iZleva yvelaze ufro srulad 

gaviTvaliswinoT dakvirvebis SedegebSi mocemuli informacia. 

vTqvaT x1, x2,..., x n aris dakvirvebis Sedegebi ξ SemTxveviT sidideze, 

romelsac aqvs p( x, θ ) albaTobebis ganawilebis simkvrive, anu X1, X 2,..., 

X n - 

urTierT damoukidebeli SemTxveviTi sidideebia, romelTagan nebismiers 

aqvs p( x, θ ) albaTobebis ganawilebis simkvrive. dakvirvebis Sedegebis 

damoukideblobis gamo maTi albaTobebis ganawilebis erToblivi simkvrive 

p( x , θ ) ⋅ p( x , θ ) ⋅⋅⋅ p( x , θ ) . parametr θ -s udidesi SesaZleblobebis 

tolia 1 2 

n 

Sefaseba hqvia mis м. п. ( ) n θ mniSvnelobas, romlisaTvisac adgili aqvs 

p( x , θ ) ⋅ p( x , θ ) ⋅⋅⋅ p( x , θ ) → max . 

1 2 

ganvixiloT udidesi SesaZleblobebis Sefasebebis miRebis magaliTi 

albaTobebis normaluri ganawilebis kanonisaTvis. vTqvaT x1, x2,..., x n aris ξ 

2 

SemTxveviT sidideze dakvirvebis Sedegebi, sadac ξ ~ N( a, 

σ ) , anu adgili 

aqvs 

2 / 2 1 n 

−n 

2 

p( x1, θ ) ⋅ p( x2, θ ) ⋅⋅⋅ p( xn, θ ) = (2 πσ ) ⋅exp{ − ( x ) } 

2 ∑ i − a . (4.2) 

2σ 

i= 

1 

2 

udidesi SesaZleblobebis Sefasebebis misaRebad a da σ 

parametrebisaTvis unda vipovoT maTi iseTi mniSvnelobebi, romelTaTvisac 

(4.2) gamosaxuleba Rebulobs maqsimalur mniSvnelobas. 

2 

dauSvaT σ - is mniSvneloba mocemulia. gamosaxuleba (4.2) Rebulobs 

udides mniSvnelobas a parametriT, rodesac eqsponentis maCvenebeli nolis 

tolia, anu 

63 

n 

{ θ }

1 n 

2 

( ) 0 

2 ∑ xi − a = . (4.3) 

2σ 

i= 

1 

a = x . 

(4.3) – is amoxsniT vRebulobT м. п. 

2 

σ - is udidesi SesaZleblobebis Sefasebis misaRebad visargebloT fun- 

qciis maqsimumis povnis analitikuri meTodiT, anu gavadiferencialoT (4.2) 

2 

2 

σ - iT da amovxsnaT miRebuli gantoleba σ - Tan mimarTebaSi. Sedeged 

vRebulobT 

2 1 n 

2 

σ м. п. = ∑ ( xi − x) 

. 

n i= 

1 

amrigad miviReT, rom albaTobebis ganawilebis normaluri kanonisaTvis 

parametrebis udidesi SesaZleblobebis Sefasebebi da momentebis meTodiT 

miRebuli Sefasebebi erTmaneTs emTxveva. saubedurod, analogiur faqts 

adgili aqvs albaTobebis ganawilebis ara yvela kanonisaTvis. 

64

Tavi 5. erTi da ori normaluri amonarCevis analizi 

winamdebare TavSi ganvixilavT wina or TavSi Seswavlil sakiTxebs, anu 

SefasebaTa Teoriisa da hipoTezebis Semowmebis ZiriTad sakiTxebs 

normaluri ganawilebisaTvis am ukanasknelis gansakuTrebuli adgilis 

gamo albaTobis Teoriasa da maTematikur statistikaSi. 

5.1. normaluri amonarCevis gamokvleva 

zemoT ukve aRvniSneT, rom parametrebis Sefasebis da hipoTezebis 

Semowmebis meTodebi iyofa or did jgufad: parametrul da araparametrul 

meTodebad. parametruli meTodebia is meTodebi, romlebic dafuZnebulia 

dakvirvebis Sedegebis ganawilebis kanonebze. araparametruli meTodebi 

ganawilebis kanonebs ar iyeneben. 

parametruli meTodebis didi umravlesoba damuSavebulia normaluri 

ganawilebis kanonisaTvis. Tu parametruli meTodi damuSavebulia erTi 

romelime ganawilebis kanonisaTvis da mas viyenebT iseTi dakvirvebis SedegebisaTvis, 

romlebic ar emorCilebian am ganawilebis kanons, maSin 

miRebuli gadawyvetilebis sandooba, zogadad, naklebi iqneba saWiro sandoobaze. 

amitom, parametruli meTodebis gamoyenebis win, saWiroa Semowmdes 

hipoteza, rom dakvirvebis Sedegebi emorCilebian mocemul ganawilebis 

kanons. kerZod, normaluri ganawilebisaTvis damuSavebuli kriteriumebis 

gamoyenebisas, saWiroa davrwmundeT, rom dakvirvebis Sedegebi emorCilebian 

normalur ganawilebas. aseTi hipoTezis Semowmeba sakmaod rTuli 

amocanaa im TvalsazrisiT, rom sando gadawyvetilebis misaRebad saWiroa 

didi raodenoba dakvirvebis Sedegebi (asobiT, aTasobiT). Tumca ki dakvirvebebis 

mcire ricxvis dros, rodesac ar aris SesaZlebloba gavzardoT 

dakvirvebebis raodenoba, ganawilebis normalurobaze Semowmebas mainc 

axdenen, magram am dros miRebuli gadawyvetilebebis sandooba sakmaod dabalia 

da dakvirvebis Sedegebis ganawilebis kanonis normalurobis hipoTeza 

uariyofa im SemTxvevaSi, rodesac dakvirvebis Sedegebis ganawilebis 

kanoni mkveTrad gansxvavdeba normaluri ganawilebisagan. dakvirvebis Sedegebis 

ganawilebis kanonebis tipis Sesaxeb gadawyvetilebis miReba xdeba 

kriteriumebiT, romlebsac Tanxmobis kriteriumebi hqvia. 

2 

cnobilia Tanxmobis kriteriumebi: xi – kvadrati ( χ ), kolmogorov – 

2 

smirnovis, omega – kvadrati ( ω ) da sxva. es kriteriumebi universaluri 

kriteriumebia da gamoiyenebian nebismieri tipis ganawilebis kanonisaTvis. 

isini iZlevian saSualebas miviRoT swori gadawyvetileba mocemuli alba- 

TobiT dakvirvebaTa didi ricxvisaTvis. ganawilebis kanonis normalurobaze 

SemowmebisaTvis, rodesac dakvirvebaTa ricxvi mcirea, gamoiyenebian specialuri, 

normaluri ganawilebisaTvis damuSavebuli, kriteriumebi. normalurobaze 

Semowmebisas agreTve gamoiyenebian asimetriisa da eqsesis koeficientebze 

dafuZnebuli kriteriumebi. 

saerTod, normaluri amonarCevis ganxilvisas, SeiZleba ganxiluli iqnas 

ori tipis amocanebi: 

65

1) dakvirvebis SedegebiT moinaxos normaluri ganawilebis parametrebis, 

2 

anu a da σ - is Sefasebebi; 

2) Semowmdes am parametrebis garkveul mniSvnelobebTan tolobis hipo- 

2 2 

2 

Teza. magaliTad, H 0 : a = a0; 

H 0 : σ = σ 0 , sadac a 0 da σ 0 mocemuli mniSvnelobebia. 

an ori amonarCevis parametrebis erTmaneTTan toloba. Magali- 

Tad, 0 : 1 a2 

a H = , sadac a 1 da a 2 , Sesabamisad, pirveli da meore amonarCevis 

maTematikuri molodinebia. winamdebare TavSi SeviswavliT aseTi amocanebis 

gadawyvetis meTodebs. 

mokled moviyvanoT normaluri ganawilebis zogierTi Tviseba, romelic 

dagvWirdeba winamdebare TavSi. 

1) normaluri ganawilebis funqcia F (x) 

maTematikuri molodiniT a da 

2 

dispersiiT σ dakavSirebulia normalizebul normaluri ganawilebis 

Φ(x) funqciasTan Semdegnairad 

x − a 

F( 

x) 

= Φ( 

) . 

σ 

2) Tu ξ aris normalurad ganawilebuli SemTxveviTi sidide maTematiku- 

2 

ri molodiniT a da dispersiiT σ , xolo η aris normalizebuli normalurad 

ganawilebuli SemTxveviTi sidide, maSin ξ = a + σ ⋅η 

. 

ξ arian normalurad ganawilebuli SemTxveviTi sidideebi, 

3) Tu ξ 1 da 2 

2 2 

maTematikuri molodinebiT da dispersiebiT Sesabamisad a 1,a 2 da 1 2 ,σ σ , ma- 

Sin SemTxveviTi sidide ξ = ξ1 

+ ξ 2 ganawilebulia normaluri kanoniT mate- 

2 2 

matikuri molodiniT a 1 + a2 

da dispersiiT σ 1 + σ 2 . 

5.2. normalurobis Semowmebis grafikuli meTodi 

SeviswavloT dakvirvebis Sedegebis ganawilebis kanonis normalurobaze 

Semowmebis umartivesi grafikuli meTodi, romelic dafuZnebulia adamianis 

Tvalis SesaZleblobaze gamoarCios wrfivi damokidebuleba sxva saxis damokidebulebisagan. 

vTqvaT mocemulia dakvirvebis Sedegebi n x x x ,..., , 1 2 . davalagoT 

dakvirvebis es Sedegebi variaciuli mwkrivis saxiT x ( 1) 

, x( 

2) 

,..., x( 

n) 

. Tu 

amonarCevi aris normalurad ganawilebul SemTxveviT sidideebze dakvirvebis 

Sedegebi, maSin maT normirebul mnisvnelobebs aqvT standartuli 

normaluri ganawileba. movaxdinoT amonarCevis gardaqmna Semdegnairad 

x − a 

y = Φ( 

) , sadac a da σ arian Sesabamisad maTematikuri molodini da 

σ 

−1 

dispersia Sesabamisad. SemovitanoT sidide z = Φ ( y) 

. Cans, rom z da x So- 

x − a 

ris arsebobs wrfivi kavSiri, radgan z = . visargebloT am damokidebu- 

σ 

lebiT da yoveli dakvirvebis SedegisaTvis x ( 1) 

, x( 

2) 

,..., x( 

n) 

gamovTvaloT Sesabamisi 

z . 

66

ganawilebis statistikuri funqcia Fn (x) 

, rogorc viciT, aris safexurovani 

funqcia, romelic variaciul mwkrivis yovel wertilSi izrdeba 1 / n si- 

didiT, amasTan roca x < x Fn 

( x) 

= 0, 

xolo roca > x F ( x) 

= 1. 

Gamoviye- 

( 1) 

67 

x ( n ) n 

noT variaciuli mwkrivis yvela wertilis Sesabamisi Fn (x) 

funqciis naxto- 

mis SuawertilisaTvis funqcia 

−1 

Φ . Sedeged ( , z) 

x koordinatTa sibrtyeSi 

⎛ ⎛ − ⎞⎞ 

miviRebT wertilebs ⎜ Φ ⎜ ⎟⎟ 

⎝ ⎝ ⎠⎠ 

−1 

2i 

1 

x( i) 

, 

, romlebic ganlagdebian wrfeze, Tu 

2n 

dakvirvebis Sedegebi ganawilebuli arian normaluri kanonis Tanaxmad, 

winaaRmdeg SemTxvevaSi isini ar yofilan normalurad ganawilebulni. 

5.3. normaluri ganawilebis parametrebis Sefaseba da maTi Tvisebebi 

normalur ganawilebas, rogorc viciT, aqvs ori parametri: maTematikuri 

molodini da dispersia. zemoT viswavleT am parametrebis Sefasebebis 

gamoTvlis meTodebi da kerZod, vnaxeT, rom rogorc momentebis meTodiT, 

aseve udidesi SesaZleblobis meTodiT, am parametrebis Sefasebebi gamoiTvlebian 

Semdegnairad 

n 

n 

1 

2 1 

2 

x = ∑ xi 

, S = ∑ ( xi 

− x) 

. 

n i= 

1 n i= 

1 

SesaZlebelia am parametrebis Sefasebebi movnaxoT sxva formulebiTac, 

kerZod, maTematikuri molodinis Sefaseba movnaxoT formuliT 

∑ − n 1 1 

x = x( 

i) 

, dispersiis Sefasebisas movnaxoT formuliT 

n − 2 

i= 

2 

2 

n 

2 ⎡1 

⎤ 

S = ⎢ ∑ xi 

− x ⎥ da a.S. 

⎣n 

i= 

1 ⎦ 

wertilovani Sefasebis monaxvisas yovelTvis ismeba kiTxva: ramdenad 

zustad Seesabameba es Sefaseba saZiebel ucnob sidides? 

ganvixiloT maTematikuri molodinis SemTxveva. vTqvaT x aris ucnobi 

a maTematikuri molodinis Sefaseba. yoveli sasrulo amonarCevisaTvis 

sainteresoa x – is a - gan gadaxris Sefaseba. viciT, rom roca n → ∞ x 

albaTurad miiswrafis a - ken, anu albaToba utolobisa x − a < ε , sadac ε 

aris ragind mcire dadebiTi ricxvi, miiswrafis erTisken, rodesac n → ∞ . 

sainteresoa mocemuli n moculobis amonarCevs rogori ε Seesabameba 

mocemuli albaTobiT. viciT, rom Tu n x x x ,..., , 1 2 ganawilebulia normaluri 

2 

ganawilebis kanoniT maTematikuri molodiniT a da dispersiiT σ , maSin 

SemTxveviTi sidide x , rogorc dakvirvebis Sedegebis wrfivi kombinacia, 

ganawilebulia normaluri kanoniT maTematikuri molodiniT a da 

2 

2 

σ 

dispersiiT σ , anu x ~ N( 

⋅ ; a, 

) . amitom, SemTxveviTi sidide 

n 

η = n( x − a) 

/ σ ~ N( 

⋅; 

0, 

1) 

. aqedan gamomdinareobs P ( η < z) 

= 1− 2α 

, sadac z 

α 

1−

aris standartuli normaluri ganawilebis 1 −α 

donis kvantili. am formulis 

samarTlianoba cxadad Cans nax. 5.1 – ze moyvanili grafikidan. 

nax. 5.1. 

CavsvaT bolo formulaSi η mniSvneloba. miviRebT, rom 

⎛ σ ⎞ 

( ( − ) / σ < 1− 

) = 1− 2α 

, anu P ⎜ ( x − a) < z1−α 

⎟ = 1− 2 

P n x a z α 

⎝ n ⎠ 

68 

α . es imas niSnavs, 

rom x - is a - Tan miaxloebis sizute σ ⋅ z 1−α / n sidideze uaresi ar aris 

1− 2α 

- s toli albaTobiT. amovweroT es intervali maTematikuri molodinis 

ucnobi a mniSvnelobisaTvis. miviRebT 

σ 

σ 

x − z1− 

α < a < x + z1−α 

. (5.1) 

n 

n 

amrigad, miviReT intervali, romelic Seicavs normaluri ganawilebis 

maTematikur molodins mocemuli albaTobiT. am intervals maTematikuri 

molodinis ndobis intervali hqvia, Sesabamisad ndobis albaTobiT 1− 2α 

. 

gamovikvlioT rogor moqmedebs ndobis intervalis sidideze dakvirvebis 

Sedegebis moculoba n , SemTxveviTi sididis saSualo kvadratuli gadaxra 

σ da albaToba α . 

(5.1) - dan Cans, rom n - is gazrdiT, anu dakvirvebebis ricxvis gazrdiT, 

mcirdeba intervalis sidide saSualo kvadratuli gadaxris da α – s mocemuli 

mniSvnelobisaTvis. magram intervalis Semcireba xdeba ara n - is 

pirdapirproporciulad, aramed n -is proporciulad. magaliTad, 

TuUgvinda wminda statistikuri meTodebiT 10-jer gavzardoT Sefasebis sizuste, 

dakvirvebebis moculoba unda gavzardoT 100-jer. 

SemTxveviTi sididis saSualo kvadratuli gadaxris gazrdiT ndobis intervalis 

sidide izrdeba, anu uaresdeba Sefasebis sizuste. α – s SemcirebiT, 

anu ndobis albaTobis gazrdiT (erTTan miaxloebiT) ndobis intervalis 

sidide izrdeba, radgan α – s SemcirebiT z 1−α 

izrdeba. 

amrigad, ganvixileT SemTxveva, rodesac normalurad ganawilebuli Sem- 

TxveviTi sididis saSualo kvadratuli gadaxra iyo cnobili. ganvixiloT 

SemTxveva, rodesac saSualo kvadratuli gadaxra aris ucnobi da vsargeb-

lobT misi S SefasebiT. am SemTxvevaSi ganvixiloT Semdegi SemTxveviTi 

sidide 

( x − a) 

t = n . 

S 

Tu gavixsenebT zemoT ganxilul stiudentis ganawilebis kanons, advilad 

mivxvdebiT, rom t SemTxveviTi sidide ganawilebulia stiudentis ganawilebis 

kanoniT n − 1 - is toli Tavisuflebis xarisxiT. amitom adgili aqvs Sem- 

deg pirobas P ( t −α 

) 

η < 1 = 1− 2α 

, sadac t 1−α 

aris stiudentis ganawilebis kanonis 

1 −α 

procentuli wertili, anu 1 −α 

donis kvantili. TuUukanaknel 

⎛ 

formulaSi CavsvavT t - s mniSvnelobas miviRebT P ⎜ 

⎝ 

( x − a) 

⎞ 

n < t1−α 

⎟ = 1− 2α 

. 

S ⎟ 

⎠ 

saidanac maTematikuri molodinis ndobis intervalisaTvis vRebulobT 

x − 

S 

t1− 

α < a < x + 

n 

S 

t1−α 

. (5.2) 

n 

am SemTxvevaSic ndobis intervalis damokidebuleba n - is, σ - s da α - 

gan analogiuria wina SemTxvevis. 

movnaxoT ndobis intervali normaluri ganawilebis dispersiisaTvis. 

viciT, rom dispersiis waunacvlebeli Sefaseba gamoiTvleba formuliT 

n 

2 1 

S = ∑ ( xi 

n −1 

i= 

1 

2 

− x) 

. 

TuU gamoviyenebT im faqts, rom xi = a + σ ⋅η 

i , sadac η i ~ N( 

⋅; 

0, 

1) 

, maSin dispersiis 

Sefasebis formula Caiwereba Semdegnairad 

2 n 

2 σ 

2 

S = ∑ ( ηi 

−η 

) . 

n −1 

i= 

1 

aqedan vRebulobT, rom 

2 

( n −1) 

S 

= 2 

σ 

n 

2 

( ηi 

−η 

) . (5.3) 

Tu gavixsenebT zemoT Seswavlil 

∑ 

i= 

1 

2 

rwmundebiT, rom SemTxveviTi sidide ∑ 

i= 

69 

χ ganawilebis kanons, advilad dav- 

n 

1 

2 

( η −η 

) ganawilebulia 

i 

2 

χ ganawi- 

2 2 

lebis kanoniT n −1 

Tavisuflebis xarisxiT, anu ∑( η i −η ) ~ χ n−1 

( x) 

. aRvniS- 

2 

χ −1, 

α 

χ − 

noT n da 

2 

n −1, 

1 α n −1 

Tavisuflebis xarisxiani 

1 −α 

procentuli wertilebi. cxadia, rom adgili aqvs Semdeg pirobas 

2 

P( χ 

2 

< χ 

2 

( x) 

< χ ) = 1− 2α 

. 

n−1, α n−1 n−1,1−α 

n 

i= 

1 

2 

χ ganawilebis α da 

2 

ukanasknel formulaSi SevitanoT 1( ) x χ n− - is mniSvneloba (5.3) – dan, 

miviRebT 

2 

2 ( n −1) 

S 2 

P( χn−1, α < < χ 2 

n−1,1−α 

) = 1− 2α 

, 

σ 

saidanac advilad vRebulobT dispersiis ndobis intervals 

2 

2 

( n −1) 

S 2 ( n −1) 

S 

< σ < . 

2 

2 

χ 

χ n−1, 

1−α 

n−1, 

α

amrigad, miRebuli gamosaxuleba aris normalurad ganawilebuli Sem- 

TxveviTi sididis dispersiis ndobis intervali 1− 2α 

- s toli ndobis 

albaTobiT. 

5.4. normaluri ganawilebis parametrebTan dakavSirebuli 

hipoTezebis Semowmeba 

5.4.1. erTi amonarCevi 

vTqvaT n x x x ,..., , 1 2 aris normalurad ganwilebul SemTxveviT sidideze 

2 

dakvirvebis Sedegebi maTematikuri molodiniT a da dispersiiT σ . Ganvi- 

2 

2 

xiloT ori SemTxveva: 1) dispersia σ cnobilia; 2) dispersia σ ucnobia. 

dispersia dispersia cnobilia cnobilia. cnobilia vTqvaT, gvinda SevamowmoT hipoteza 0 : a0 

a H = , sadac 

a aris maTematikuri molodinis raRac garkveuli mniSvneloba. Ganvixi- 

0 

loT ormxrivi alternatiuli hipoteza 1 : a0 

a H ≠ . SemovitanoT SemTxveviTi 

sidide η 

( 0 ) 

σ 

a x 

= 

− 

n . Tu ZiriTadi hipoteza samarTliania, anu dakvirvebis 

Sedegebis maTematikuri molodini a 0 – is tolia, maSin SemTxveviTi sidide 

η emorCileba standartul normalur ganawilebas, anu 

η = 

( x − a0 

) 

n ~ N( 

⋅; 

0, 

1) 

. SevirCioT kriteriumis mniSvnelobis done 0 < α < 1. 

σ 

H hipotezis samarTlianobisas adgili aqvs utolobas 

maSin 0 

0 

1 / 2 

) ( x − a 

n < z −α 

, (5.4) 

σ 

sadac z 1−α / 2 aris standartuli normaluri ganawilebis 1− α / 2 donis kvantili. 

amrigad, miviReT H 0 hipotezis miRebi are. Tu adgili aqvs (5.4) tolobas, 

miiReba ZiriTadi hipoteza, winaaRmdeg SemTxvevaSi miiReba ormxrivi alternatiuli 

hipoteza. 

2 

dispersia dispersia ucnobia ucnobia. ucnobia am SemTxvevaSi vsargeblobT dispersiis S Sefase- 

( x − a0 

) 

biT. η statistikis nacvlad ganvixiloT statistika t = n . im Sem- 

S 

TxvevaSi, rodesac samarTliania H 0 ZiriTadi hipoteza, t SemTxveviTi sidide 

ganawilebulia stiudentis kanonis Tanaxmad n −1 

- is toli Tavisuflebis 

xarisxiT. Tu kriteriumis mniSvnelobis dones aviRebT isev α - s tols, 

H hipotezis miRebis ares aqvs saxe 

maSin 0 

sadac 1−α / 2 

0 

1 / 2 

) ( x − a 

n < t −α 

, (5.5) 

S 

t aris n −1 

- is toli Tavisuflebis xarisxis mqone stiudentis 

ganawilebis 1− α / 2 donis kvantili. amrigad, Tu adgili aqvs (5.5) pirobas 

70

miiReba ZiriTadi hipoTeza, winaaRmdeg SemTxvevaSi _ ormxrivi alternatiuli 

hipoteza. 

5.4.2. ori amonarCevi 

ganvixiloT ori normaluri amonarCevis maTematikuri molodinebis Sedarebis 

amocana. 

vTqvaT n x x x ,..., , 1 2 da y 1 , y2 

,..., ym 

damoukidebeli amonarCevebia Sesabami- 

2 

2 

sad ( a 1, σ 1 ) da ( a 2 , σ 2 ) parametrebis mqone normaluri ganawilebis kanonebidan. 

SevamowmoT 0 : 1 a2 

a H = ZiriTadi hipoteza, rom ori SemTxveviTi 

sididis maTematikuri molodinebi erTmaneTis tolia, H1 : a1 

≠ a2 

ormxrivi 

2 

2 

alternatiuli hipotezis winaaRmdeg. σ 1 da σ 2 parametrebTan dakavSirebiT 

SesaZlebelia oTxi SemTxveva: 

2 

a) dispersiebi cnobilia da erTmaneTis tolia σ 

2 2 

= σ = σ ; 

2 2 

b) dispersiebi cnobilia da erTmaneTisgan gansxvavdebian σ 1 ≠ σ 2 ; 

g) dispersiebi ucnobia da erTmaneTis tolia; 

d) dispersiebi ucnobia da erTmaneTisgan gansxvavdebian. 

x − y 

pirvel pirveli pirvel i SemTxveva SemTxveva. SemTxveva SemovitanoT SemTxveviTi sidide 

1 1 

σ + 

n m 

Tezis samarTlianobis dros es SemTxveviTi sidide ganawilebulia standar- 

x − y 

tuli normaluri ganawilebis kanoniT, anu ~ N( 

⋅; 

0, 

1) 

. SevirCioT 

1 1 

σ + 

n m 

kriteriumis mniSvnelobis done α , xolo z 1−α / 2 - iT aRvniSnoT standartuli 

normaluri ganawilebis 1− α / 2 donis kvantili. maSin cxadia, rom Ziri- 

H hipotezis miRebis ares eqneba Semdegi saxe 

Tadi 0 

x − y 

< z1−α 

/ 2 , 

1 1 

σ + 

n m 

winaaRmdeg SemTxvevaSi miiReba H 1 alternatiuli hipoteza. 

71 

1 

2 

. H 0 hipo- 

meor meore meor meore 

e SemTxveva SemTxveva. SemTxveva 

am SemTxvevaSi SemovitanoT Semdegi saxis SemTxveviTi 

x − y 

sidide 

, romelic, H 0 hipotezis samarTlianobis dros, gana- 

2 

2 

σ / n + σ / m 

1 

2 

wilebuli iqneba standartuli normaluri kanoniT. avRniSnoT: α - kriteriumis 

mniSvnelobis done; z 1−α / 2 - standartuli normaluri ganawilebis 

1 α / 2 

H hipotezis miRebis ares eqneba Semdegi saxe 

− donis kvantili. maSin 0 

x − y 

< z 

σ / n + σ / m 

2 

1 

2 

2 

1−α 

/ 2 

,

winaaRmdeg SemTxvevaSi miiReba H 1 alternatiuli hipoteza. 

x − y 

mesame mesamee mesame mesame e SemTxveva SemTxveva. SemTxveva SemovitanoT SemTxveviTi sidide 

, sadac 

1 1 

S + 

n m 

2 

2 

2 ( n −1) 

S1 

+ ( m −1) 

S 2 

S = 

aris gaerTianebuli amonarCeviT gamoTvlili amo- 

n + m − 2 

2 

2 

narCevebis toli dispersiebis Sefaseba, xolo S 1 da S 2 arian Sesabamisad 

pirveli da meore amonarCevebiT gamoTvlili erTi da igive dispersiis Sefasebebi. 

Tu gavixsenebT zemoT Seswavlil stiudentis ganawilebis kanons, 

advilad davrwmundebiT, rom gansaxilveli SemTxveviTi sidide ganawilebulia 

n + m − 2 Tavisuflebis xarisxis mqone stiudentis kanoniT. 

am SemTxvevaSi H 0 ZiriTadi hipotezis miRebis ares aqvs Semdegi saxe 

x − y 

< t1−α 

/ 2 , 

1 1 

S + 

n m 

t aris n + m − 2 Tavisuflebis xarisxis mqone stiudentis ganawi- 


1− α / doniani kvantili. winaaRmdeg SemTxvevaSi miiReba 1 

lebis 2 

natiuli hipoteza. 

72 

H alter- 

meoTx meoTxe meoTx e SemTxveva SemTxveva. SemTxveva 

am SemTxvevaSi SemovitanoT Semdegi saxis SemTxveviTi 

sidide 

x − y 

, 

2 2 

S / n + S / m 

sadac 

S da 

2 

1 

1 

2 

2 

S 2 arian Sesabamisad pirveli da meore amonarCevis dispersi- 

ebis Sefasebebi. am SemTxveviTi sididis zusti ganawilebis kanoni ucnobia, 

magram miaxloebiT is ganawilebulia stiudentis ganawilebis kanoniT, Tavi- 

suflebis xarisxiT 

2 2 2 

( S1 

/ n + S2 

/ m) 

2 2 2 

( S / n) 

( S / m) 

d = 

. 

1 

n −1 

+ 

2 

m −1 

am SemTxvevaSi ZiriTadi H 0 hipotezis miRebis ares aqvs Semdegi saxe 

S 

2 

1 

x − y 

/ n + S 

2 

2 

/ m 

< t 

2 

1−α 

/ 2 


1− α / 2 procentuli wertili. winaaRmdeg SemTxvevaSi miiReba 1 

t aris d Tavisuflebis xarisxis mqone stiudentis ganawilebis 

tiuli hipoteza. 

, 

H alterna- 

ganvixiloT ori amonarCevis dispersiebis tolobis tolobis hipo hipoTez hipo 

ez ezis ez is Semow Semowme Semow 

me me- me 

2 2 

bis bis amocana amocana. amocana 

ganvixiloT ZiriTadi hipoteza H : σ = σ ormxrivi alterna- 

2 2 

tiuli hipoTezis winaaRmdeg H : σ ≠ σ . 

1 

1 

2 

0 

1 

2

2 2 

ganvixiloT SemTxveviTi sidide 

1 / S2 

S F = , romelsac uwodeben fiSeris 

statistikas. ZiriTadi hipoTezis samarTlianobisas F SemTxveviTi sidide 

ganawilebulia fiSeris ganawilebis kanoniT Tavisuflebis xarisxebiT 

( n −1, m −1) 

, sadac n da m Sesabamisad pirveli da meore amonarCevebis 

moculobebia. 

avirCioT kriteriumis mniSvnelobis done α . ZiriTadi pipoTezis mirebis 

ares, ormxrivi alternatiuli hipoTezis winaaRmdeg Semowmebisas, aqvs 

saxe 

Fα / 2; 

n− 

1, 

m−1 

< F < F1−α 

/ 2; 

n−1, 

m−1 

, 

F α ( n −1, m −1) 

Tavisuflebis xarisxis mqone fiSe- 

sadac Fα / 2; 

n−1, 

m−1 

da 1− / 2; 

n−1, 

m−1 

ris ganawilebis α / 2 da 1− α / 2 donis kvantilebia Sesabamisad. winaaRmdeg 

SemTxvevaSi miiReba H 1 alternatiuli hipoTeza. 

5.4.3. Sewyvilebuli monacemebi 

viciT, rom Sewyvilebul monacemebs aqvs Semdegi saxe ( , y ), i = 1,..., 

n , 

73 

xi i 

sadac n aris dakvirvebebis raodenoba. i x da y i arian dakvirvebebi erTi 

da imave obieqtze sxvadasxva pirobebSi. SemovitanoT SemTxveviTi sidideebi 

= y − x , i = 1,..., 

n , da maT davadoT Semdegi moTxovnebi: 

zi i i 

zi 1) , i = 1,..., 

n erTmaneTisgan damoukidebeli SemTxveviTi sidideebia; 

2) z i - s warmodgena SeiZleba Semdegnairad zi = θ + ε i , sadac ε 1 , ε 2 ,..., ε n - 

damoukidebeli SemTxveviTi sidideebia, θ - ucnobi mjudmivi sididea; 

2 

3) SemTxveviTi sidide ε ~ N( 

⋅; 

0, 

σ ), i = 1,..., 

, sadac, rogorc wesi, dis- 

persia 

2 

σ ar aris cnobili. 

i n 

amrigad, Semotanili aRniSvniT sawyisi Sewyvilebul monacemebiani amocana 

miviyvaneT erTi normaluri amonarCevis amocanaze, romelic zemoT 

ganvixileT, sadac formirebuli hipotezebi SeiZleba CamovayaliboT θ parametris 

mimarT da maT Sesamowmeblad gamoviyenoT miRebuli kritikuli 

areebi. magaliTad, am SemTxvevaSi Sewyvilebuli monacemebis maTematikuri 

molodinebis tolobis hipoTezas eqneba Semdegi saxe H : θ = 0 , xolo or- 

mxrivi alternatiuli hipoTezas - H : θ ≠ 0 . 

1 

0

Tavi 6. dispersiuli analizi 

6.1. amocanis dasma 

aqamde vswavlobdiT SemTxveviTi faqtorebis gavlenas dakvirvebis Sedegebze. 

meore, ara nekleb mniSvnelovani SemTxvevaa dakvirvebis Sedegebze 

ara SemTxveviTi faqtorebis gavlena, anu SemTxveva, rodesac dakvirvebis 

Sedegebze moqmedeben ara marto romeliRac SemTxveviTi faqtorebi, aramed 

faqtorebi, romlebic icvlebian ara SemTxveviTad. magaliTad, nebismieri 

warmoebis mizania gamouSvas erTgvarovani produqcia. produqciis ara 

erTgvarovnebaze gavlenas axdenen warmoebis sxvadasxva etapebi da maTi 

gavlena, rogorc wesi, sxvadasxva nairia. araerTgvarovnebis faqtis 

aRmoCenisas saWiroa misi warmoSobis mizezis dadgena, anu im etapebis 

dadgena, romlebic ganapirobeben produqciis araerTgvarovnebas da maT 

Soris gamoiyos iseTi etapebi, romlebic ZiriTadad ganapirobeben am 

araerTgvarovnebas. es imisaTvis aris saWiro, rom kapitaldabandeba, 

pirvel rigSi, ganxorcieldes im etapebis gasaumjobeseblad, romlebsac 

SeaqvT yvelaze meti araerTgvarovneba. amave dros SeiZleba aRmoCndes, 

rom warmoebis romeliRac etapebi saerTod ar auareseben produqciis 

erTgvarovnebas da maTi modelizacia ar aris saWiro. aseTi amocanebis 

gadawyvetis saSualebas iZlevian maTematikuri statistikis specialuri 

meTodebi, romlebic gaerTianebuli arian saerTo saxelwodebiT “faqtoruli 

analizi”. winamdebare TavSi ganvixilavT mxolod im SemTxvevebs, rodesac 

SemTxveviTi cvalebadoba emorCileba normaluri ganawilebis kanons. 

normaluri ganawilebis kanons aqvs ori parametri: maTematikuri molodini 

da dispersia. amitom aseT SemTxveviT sidideze ara SemTxveviTi faqtoris 

gavlena SeiZleba aisaxos rogorc maTematikuri molodinis, aseve 

dispersiis cvalebadobaze. Tu dakvirvebebi xorcieldeba erTi da igive me- 

TodikiT, erTi da igive xelsawyoebiT, maSin dispersia SeiZleba CavTvaloT 

ucvlelad da ara SemTxveviTi faqtoris gavlena aisaxeba dakvirvebis 

Sedegebis maTematikuri molodinis cvalebadobaze. 

SemdgomSi ganvixilavT mxolod aseT SemTxvevebs, rodesac ara SemTxveviTi 

faqtorebis gavleniT SeiZleba Seicvalos dakvirvebis Sdegebis matematikuri 

molodini. imisaTvis, rom dadgindes araSemTxveviTi A faqtoris 

gavlena x1, x2,..., x n dakvirvebis Sedegebze, saWiroa SemovitanoT am gavlenis 

maxasiaTebeli. vTqvaT A faqtoris gavlena Seiswavleba am faqtoris 

A1, A2 ,..., A k doneebze. Sesabamisi dakvirvebis Sedegebi avRniSnoT a1, a2,..., a k . 

maSin A faqtoris gavlenis Sesaswavlad SeiZleba gamoviyenoT sidide 

2 1 k 

2 

1 

σ A = ∑ ( ai − a) 

, sadac 

k i= 

1 

1 

k 

2 

a = ∑ ai 

. σ A – s A faqtoris dispersias eZaxian. 

k i= 

2 

A Ffaqtori ar aris SemTxveviTi, amitom σ A ar aris dispersia klasikuri 

gagebiT. is gansaxilvelad Semotanili iqna ori mizezis gamo. jer erTi 

imitom, rom dispersia aris gabnevis umartivesi maxasiaTebeli. meorec, 

SemTxveviTi A faqtoris gavlena dakvirvebis Sedegebze am SemTxvevaSi xasiaTdeba 

SemTxveviTi faqtoris analogiurad. es saSualebas iZleva erTmaneTs 

SevadaroT SemTxveviTi da ara SemTxveviTi A faqtoris gavlena. 

74

dispersiuli analizi ewodeba meTodebis erTobliobas, romlebic 

saSualebas iZlevian maTi dispersiebis daxmarebiT gavacalkevoT 

SemTxveviTi da ara SemTxveviTi faqtorebis gavlena Sesaswavl movlenaze. 

dispersiuli analizis amocana aris: dakvirvebis SedegebiT gamovTvaloT 

SemTxveviTi da ara SemTxveviTi faqtorebis dispersiebi da SevadaroT 

isini erTmaneTs. ganvixiloT dispersiuli analizis umartivesi SemTxveva. 

vTqvaT x1, x2,..., x n dakvirvebis Sedegebze moqmedeben SemTxveviTi 

faqtori da A faqtori. vTqvaT SemTxveviTi faqtoris dispersia cnobilia 

2 

da σ - is tolia. dakvirvebis SedegebiT gamoiTvleba am dakvirvebis 

2 1 n 

2 

Sedegebis gafantvis dispersia S = ∑ ( xi − x) 

. es dispersia ganpirobebu- 

n −1 

i= 

1 

lia ori faqtoris gavleniT: SemTxveviTi faqtoriT, romelic xasiaTdeba 

2 

2 

σ dispersiiT da A faqtoriT, romelic xasiaTdeba σ A dispersiiT. es faqtorebi 

erTmaneTisgan damoukideblebi arian. im faqtis gaTvaliswinebiT, 

rom damoukidebeli SemTxveviTi sidideebis jamis dispersia tolia am 

2 2 2 

SemTxveviTi sidideebis dispersiebis jamis, vwerT S = σ + σ . aqedan vRebu- 

2 2 2 

lobT σ A = S − σ . 

praqtikaSi, rogorc wesi, SemTxveviTi faqtoris dispersia ucnobia da 

saWiroa dakvirvebis SedegebiT misi Sefaseba. gadavideT iseTi SemTxvevis 

ganxilvaze, rodesac dakvirvebis Sedegebze gavlenas axdens erTi faqtori. 

6.2. erTfaqtoruli dispersiuli analizi 

2 

ganvixiloT SemTxveva, rodesac SemTxveviTi faqtoris dispersia σ ucnobia. 

am SemTxvevaSi x1, x2,..., x n dakvirvebis SedegebiT saWiroa SemTxveviTi 

faqtoris 

2 

2 

σ dispersiis da ara SemTxveviTi faqtoris σ A dispersiis Sefa- 

seba. am problemis gadasawyvetad saWiroa 1, 2,..., 

k 

75 

A A A doneebze dakvirvebe- 

bis gameoreba. avRniSnoT xi,1 , xi,2 ,..., x i, n - iT A ( i = 1,..., k) 

faqtoris A i donis 

Sesabamisi dakvirvebis Sedegebi. am SemTxvevaSi SeiZleba sxvadasxva nairad 

moviqceT. SeiZleba A faqtoris erT doneze movaxdinoT dakvirvebebis didi 

2 

raodenoba, am dakvirvebebiT SevafasoT SemTxveviTi faqtoris σ dispersia, 

miviRoT is SemTxveviTi faqtoris dispersiis namdvil mniSvnelobad da 

gamoviyenoT zemoT moyvanili sqema. magram es ar aris saukeTeso 

gamosavali imitom, rom ar iZleva saSualebas A faqtoris doneebze SevamowmoT 

SemTxveviTi faqtoris dispersiis cvalebadoba da am cvalebadobis 

arsebobis SemTxvevaSi gaviTvaliswinoT is dispersiul analizSi. amitom 

Semdegnairad iqcevian. 

A faqtoris i doneze atareben dakvirvebebis erTnair n raodenobas: 

, , 1,..., ; 1,..., 

xi j i = k j = n . iTvlian saSualoebs A faqtoris yvela donisaTvis da 

dakvirvebebis yvela SedegisaTvis 

1 

, 

1 

n 

1 k n 

xi = ∑ xi 

j da x = ∑ ∑ xi, 

j . 

n j= 

n⋅ k i= 1 j= 

1 

A

yvela dakvirvebis Sedegis dispersia gamoiTvleba formuliT 

2 1 k n 

2 

S = ∑ ∑ ( xij − x) 

. 

nk −1 

i= 1 j= 

1 

gamoviTvaloT SemTxveviTi faqtoris dispersia. A faqtoris dafiqsirebuli 

donis dakvirvebis Sedegebze, anu gansaxilveli faqtoris A i doneze 

x , x ,..., x dakvirvebis Sedegebze gavlenas axdens mxolod SemTxveviTi 

i,1 i,2 i, n 

faqtori, romlis dispersiac fasdeba formuliT 

2 1 n 

S = ∑ ( x 

2 

− x ) , i = 1,..., k. 

i ij i 

n j= 

1 

2 

yoveli S i aris SemTxveviTi faqtoris dispersiis Sefaseba. SemTxveviTi 

faqtoris dispersiis ufro zust Sefasebas miviRebT, Tu gavasaSualebT maT 

mniSvnelobebs 

2 1 k 

2 1 k n 

2 

S0 = ∑ Si = ∑ ∑ ( xij − xi 

) . 

k i= 1 k( n −1) 

i= 1 j= 

1 

2 2 

gamoTvlili S da S 0 mniSvnelobebiT A faqtoris dispersia SeiZleba 

gamovTvaloT formuliT 

2 2 2 

σ A ≈ S − S0 

. (6.1) 

2 

2 2 

formula (6.1) iZleva σ A dispersiis uxeS Sefasebas, radgan S da S 0 arian 

dakvirvebis SedegebiT gamoTvlili Sefasebebi. ufro zusti Sefaseba 

SeiZleba miviRoT dakvirvebis Sedegebis A faqtoris doneebis mixedviT 

saSualo mniSvnelobebis, anu x1, x2,..., x k saSualo mniSvnelobebis gafantvis 

ganxilviT. am saSualo mniSvnelobebis gafantvis dispersia tolia 

2 

2 

1 k 

2 2 σ 2 S0 

∑ ( xi − x) 

= σ A + ≈ σ A + . (6.2) 

k −1 

i= 

1 

n n 

es disperisa ganpirobebulia ori faqtoriT: SemTxveviTi da A faqtoriT. 

(6.2) – dan vRebulobT 

2 

2 1 k 

2 S0 

σ A = ∑ ( xi − x) 

− . 

k −1 

i= 

1 n 

SemovitanoT aRniSvna 

2 n k 

2 2 2 

SA = ∑ ( xi − x) = σ A + S0 

, 

k −1 

i= 

1 

maSin 

2 2 

2 SA − S0 

σ A = . (6.3) 

n 

imisaTvis, rom davadginoT dakvirvebis Sedegebze A faqtis gavlena saWi- 

2 

2 

roa erTmaneTs SevadaroT S A da S 0 dispersiebi. Tu es dispersiebi erTmaneTisagan 

gansxvavdebian, maSin es gansxvaveba SeiZleba gamowveuli iqnas 

mxolod A faqtoris gavleniT, romelic gamoisaxeba S dispersiiT. Tu S 

2 

2 

da S 0 dispersiebi erTnairia, maSin es miuTiTebs imaze, rom σ A = 0 , anu A 

faqtori ar moqmedebs dakvirvebis Sedegebze. 

2 

S A da 

2 

S 0 arian Sesabamisi dispersiebis Sefasebebi. isini gamoTvlilia 

dakvirvebis SedegebiT, romlebic SemTxveviTi sidideebi arian. amitom 

Sefasebebic SemTxveviTi sidideebi arian. aqedan gamomdinare maTi 

76 

2 

A 

2 

A

erTmaneTTan Sedareba unda ganxorcieldes statistikuri kriteriumebis 

gamoyenebiT. kerZod, fiSeris kriteriumiT, radgan cnobilia, rom 

dispersiis ori Sefasebuli mniSvnelobis Sefardeba emorCileba fiSeris 

ganawilebis kanons. Sedarebis kriteriums aqvs saxe: Tu 

2 

S A > F 2 1−α 

, (6.4) 

S 

sadac F1− α aris l1 = k − 1, l2 = k( n − 1) Tavisuflebis xarisxebiani fiSeris ganawilebis 

α donis kvantili, maSin miiReba gadawyvetileba, rom dakvirvebis 

Sedegebze A faqtoris gavlena aris arsebiTi da am gavlenis dispersia 

gamoiTvleba (6.3) formuliT. winaaRmdeg SemTxvevaSi miiReba gadawyvetileba, 

rom dakvirvebis Sedegebze A faqtori gavlenas ar axdens. am SemTxvevaSi 

SemTxveviTi faqtoris dispersiis SefasebisaTvis 

moiyeneba 

77 

0 

2 

S 0 – is nacvlad ga- 

2 

S - i, romelic aris SemTxveviTi faqtoris dispersiis ufro zus- 

ti Sefaseba. 

konkretuli magaliTebis praqtikuli gamoTvlebis procedurebis gamartivebis 

mizniT moviyvanoT erTfaqtoruli dispersiuli analizis ganxorcielebis 

sqema. 

pirvel rigSi dakvirvebis yvela Sedegis Setana xdeba erTfaqtoruli 

dispersiuli analizis cxril 6.1 – Si. 

j - i dakvirvebis 

Sedegi 

A 1 

faqtoris doneebi 

A 2 

1 x 11 

x 21 

2 x 12 

x 22 

cxrili 6.1. 

… k A 

… x k1 

… x k 2 

... … … … … 

jami 

n x1 n 

X 1 

x2 n 

X 2 

… 

… 

x kn 

k X 

cxrilis bolo striqonSi X i - iT aRniSnulia i svetSi mocemuli dakvirvebis 

jami. 

dakvirvebis SedegebiT gamoiTvleba sidideebi: 

1) yvela dakvirvebis Sedegebis kvadratebis jami 

1 

k n 

ϑ = ∑ ∑ x ; 

i= 1 j= 

1 

2) svetebSi mocemuli dakvirvebis Sedegebis jamebis kvadratebis jami gayofili 

svetebSi dakvirvebebis raodenobaze 

1 2 

2 

1 

k 

ϑ = ∑ X i ; 

n i= 

3) dakvirvebis Sedegebis jamis kvadrati gayofili dakvirvebebis 

saerTo raodenobaze 

2 

ij

4) 

2 

S A da 

( X i ) 

1 

ϑ3 

= 

kn 

k 

∑ 

i= 

1 

2 

; 

S dispersiebis Sefasebebi formulebiT 

2 

0 

2 ϑ1 −ϑ2 

2 ϑ2 −ϑ3 

S0 = , SA 

= . (6.5) 

k( n −1) k −1 

fiSeris kriteriumebiT erTmaneTs dardeba dispersiebi 

78 

S da 

2 

A 

2 

S 0 . erTma- 

neTisagan maTi arsebiTi gansxvavebisas miiReba gadawyvetileba dakvirvebis 

Sedegebze A faqtoris arsebiTi gavlenis Sesaxeb da am faqtoris dispersia 

gamoiTvleba (6.3) formuliT. winaaRmdeg SemTxvevaSi miiReba gadawyvetileba 

dakvirvebis Sedegebze A faqtoris gavlenis ara arsebiTobis 

Sesaxeb da formuliT 

2 ϑ1 −ϑ3 

S = 

kn − 1 

gamoiTvleba SemTxveviTi faqtoris dispersiis ufro zusti Sefaseba vidre 

(6.5) - is pirveli formuliT gamoTvlili Sefasebaa. 

praqtikaSi xSirad gvxvdeba SemTxveva, rodesac A faqtoris sxvadasxva 

doneebze mocemulia dakvirvebis Sedegebis sxvadasxva raodenoba, anu faq- 

toris A i doneze mocemulia mniSvnelobebi 

x , x ,..., x , i = 1,..., k . am SemTxve- 

i1 i2 ini 

vaSi SeiZleba moviqceT Semdegnairad: A faqtoris yoveli donisaTvis davtovoT 

dakvirvebebis erTnairi raodenoba, romelic Seesabameba n i – is minimalur 

sidides, sadac i = 1,..., k ; danarCeni dakvirvebis Sedegebi ukuvagdoT. 

dakvirvebis sedegebis ukugdeba auaresebs analizis Sedegebis xarisxs. 

amitom, am SemTxvevaSi, axorcieleben erTfaqtorul dispersiul analizs, 

romelic aris zemoT moyvanili sqemis ubralo modifikacia da aqvs saxe: 

2 

1) ϑ = ∑ ∑ x ; 

1 

k 

ni 

i= 1 j= 

1 

ij 

k 

2 

X i 

2) ϑ2 

= ∑ ; 

i= 

1 n 

1 

3) 3 ( ) 

k 

ϑ X i 

i 

2 

k 

= ∑ , sadac N = ∑ ni 

. 

N i= 

1 

i= 

1 

am sidideebiT gamoiTvleba dispersiebis mniSvnelobebi 

2 ϑ1 −ϑ2 

2 ϑ1 −ϑ3 

S0 = , S A = . 

N − k k −1 

mowmdeba dispersiebis gamoTvlili mniSvnelobebis erTmaneTisagan 

gansxvavebis arsebiToba. Tu sruldeba (6.4) utoloba, sadac 1 F − α aris 

l1 = k − 1 da l2 = N − k Tavisuflebis xarisxebiani fiSeris ganawilebis kanonis 

α donis kvantili, maSin miiReba gadawyvetileba dakvirvebis Sedegebze A 

faqtoris gavlenis arsebiTobis Sesaxeb da am gavlenis dispersia 

2 ( k −1) 

N 2 2 

σ A = ( SA − S0 

) . 

k 

2 2 

N − ∑ n 

i= 

1 

i

winaaRmdeg SemTxvevaSi, anu Tu ar sruldeba (6.4) piroba, miiReba gadawyvetileba 

dakvirvebis Sedegebze A faqtoris gavlenis ara arsebiTobis 

Sesaxeb da dispersiis ufro zusti Sefaseba gamoiTvleba formuliT 

2 ϑ1 −ϑ3 

S = 

N − 1 

. 

6.3. orfaqtoruli dispersiuli analizi 

praqtikaSi, xSirad gvxvdeba amocanebi, rodesac dakvirvebis Sedegebze 

gavlenas axdenen mravali araSemTxveviTi faqtorebi. aseTi amocanebis 

gadasawyvetad gamoiyeneba mravalfaqtoruli analizis meTodebi, kerZod, 

mravalfaqtoruli dispersiuli analizis meTodebi. mravalfaqtoruli 

dispersiuli analizis magaliTad ganvixilavT orfaqtorul dispersiul 

analizs, anu semTxvevas, rodesac dakvirvebis sedegebze moqmedebs ori 

faqtori A da B . ori faqtoris gavlena SeiZleba Seswavlili iqnas erTfaqtoruli 

dispersiuli analizis saSualebiT Semdegnairad. davafiqsiroT 

erTi faqtoris (magaliTad, B faqtoris) mniSvneloba raRac doneze da am 

doneze SeviswavloT meore faqtoris (magaliTad, A faqtoris) gavlena 

dakvirvebis Sedegebze erTfaqtoruli dispersiuli analizis gamoyenebiT. 

Semdeg pirveli B faqtoris done davafiqsiroT meore doneze da am doneze 

(kvlav erTfaqtoruli dispersiuli analizis daxmarebiT) SeviswavloT 

meore A faqtoris gavlena da a.S. aseT midgomas gaaCnia Semdegi nakli: 1) 

B faqtoris yvela donisaTvis saWiroa axali dakvirvebis Sedegebi, romlebic 

B faqtoris sxva doneebisaTvis ar gamodgeba, anu gaumarTleblad izrdeba 

dakvirveis Sedegebis saerTo raodenoba; 2) SeuZlebelia A da B faqtorebis 

urTierTgavlenis faqtoris dakvirvebis Sedegebze gavlenis dadgena, 

im SemTxvevaSi Tu aseT gavlenas adgili aqvs. am naklisagan Tavisufalia 

orfaqtoruli dispersiuli analizi, romlis ganxilvazec gadavdivarT. 

avRniSnoT a1, a2,..., a k da b1, b2,..., b m Sesabamisad A da B faqtorebis doneebia. 

cxrili 6..2 warmoadgens orfaqtoruli dispersiuli analizis cxrils. 

B 

faqtorebi 

a 1 a 2 

b 1 x 11 x 21 

b 2 x 12 x 22 

79 

A 

… k a 

… x k1 

… x k 2 

cxrili 6.2. 

jami 

' 

X 2 

… … … … … … 

jami 

b m x1 m 

X 1 

x2 m 

X 2 

… 

… 

xkm 

k 

' 

X m 

X 

mivaqcioT yuradReba imas, rom nebismieri svetis da nebismieri striqonis 

gadakveTaze mocemulia dakvirvebis mxolod erTi Sedegi, anu A da B 

' 

X 1

faqtorebis doneebis nebismier wyvils Seesabamebs mxolod erTi dakvirvebis 

Sedegi. cxril 6.2 – Si X1, X 2,..., 

X k - iT aRniSnulia Sesabamis svetebSi 

' ' ' 

dakvirvebis Sedegebis jamebi, xolo X1, X 2,..., 

X m - iT aRniSnulia dakvirvebis 

Sedegebis jamebi Sesabamis striqonebSi. 

avRniSnoT x i i - r striqonSi mocemuli dakvirvebis Sedegebis saSualo 

X i 

ariTmetikulia, anu xi = , i = 1,..., k , xolo x j j - r striqonSi mocemuli 

m 

X j 

dakvirvebis Sedegebis saSualo ariTmetikulia, anu x j = , j = 1,.., m . x – 

k 

iT aRniSnulia yvela dakvirvebis Sedegis saSualo ariTmetikuli. 

ganvixiloT saSualo ariTmetikulebis gabneva svetebSi da striqonebSi. 

saSualo ariTmetikulebis gabnevaze svetebSi moqmedebs mxolod ori 

faqtori: A faqtori da SemTxveviTi faqtori, B faqtori ar moqmedebs, 

radgan misi moqmedeba yovel svetze saSualdeba. amitom SeiZleba davweroT: 

2 

1 k 

2 2 σ 

∑ ( xi − x) 

= σ A + . (6.6) 

k −1 

i= 

1 

m 

zustad aseve, saSualo ariTmetikulebis gafantvaze striqonebSi 

moqmedebs ori faqtori: B faqtori da SemTxveviTi faqtori, A faqtori ar 

moqmedebs, radgan misi moqmedeba saSualdeba yovel striqonze. Amitom 

samarTliania: 

2 

1 m 

2 2 σ 

∑ ( x j − x) 

= σ B + . (6.7) 

m −1 

j= 

1 

k 

2 

cnobili, rom iyos SemTxveviTi faqtoris dispersia σ , (6.6) da (6.7) – dan 

advilad gamovTvlidiT A da B faqtorebis Sesabamis dispersiebs 

80 

2 

2 

σ A da σ B 

da davamTavrebdiT orfaqtorul dispersiul analizs. mgram dispersia 

ucnobia da misi mniSvneloba saWiroa ganvsazRvroT arsebuli dakvirvebis 

Sdegebis safuZvelze. moviqceT Semdegnairad. gamovTvaloT i - i striqonis 

dakvirvebis Sedegebis dispersia 

2 1 m 

2 

Si = ∑ ( xij − xi 

) , i = 1,..., k . 

m −1 

j= 

1 

es dispersia ganapirobebulia ori faqtoriT: SemTxveviTi faqtori da 

B faqtori, radgan A faqtoris done dafiqsirebulia. amitom 

S = σ + σ , i = 1,..., k . (6.8) 

2 2 2 

i B 

toloba (6.8) ufro zusti iqneba, Tu 

gasaSualebuli mniSvnelobiT, anu 

2 2 1 k 

2 1 k m 

2 

σ B + σ = ∑ Si = ∑ ∑ ( xij − xi 

) . (6.9) 

k i= 1 k( m −1) 

i= 1 j= 

1 

Toloba (6.9) – s gamovakloT toloba (6.7). miviRebT 

2 

2 σ 1 k m 

2 1 m 

2 

σ − = ∑ ∑ ( xij − xi ) − ∑ ( x j − x) 

. 

k k( m −1) i= 1 j= 1 m −1 

j= 

1 

2 

aqedan advilad ganvsazRvravT σ – s: 

2 1 ⎡ k m m 

2 2 ⎤ 

σ = ∑ ∑ ( xij − xi ) − k ∑ ( x j − x) 

( k −1)( m −1) ⎢ 

⎣ 

⎥ 

i= 1 j= 1 j= 

1 ⎦ . 

2 

σ 

2 

S i nacvlad visargeblebT maTi

miRebuli gamosaxuleba iZleva saSualebas dakvirvebis yvela SedegiT 

2 

gamovTvaloT SemTxveviTi faqtoris dispersiis Sefaseba. aRvniSnoT is S 0 – 

iT. 

SemovitanoT aRniSvna 

2 m k 

2 2 2 2 2 

SA = ∑ ( xi − x) = mσ A + σ ≈ mσ A + S0 

, 

k −1 

i= 

1 

2 k m 

2 2 2 2 2 

SB = ∑ ( x j − x) = kσ B + σ ≈ kσ B + S0 

. 

m −1 

j= 

1 

am formulebidan cxadad Cans, rom dakvirvebis Sedegebze A faqtoris ga- 

2 

2 

vlenisas S A dispersia gansxvavebuli iqneba S 0 dispersiisagan. maTi 

statistikurad arsebiTi gansxvaveba unda dadgindes fiSeris kriteriumis 

2 2 

daxmarebiT, radgan dakvirvebis SedegebiT gamoTvlili Sefasebebi S A da S 0 

arian SemTxveviTi sidideebi. maTi Sedarebis kriteriums aqvs Semdegi saxe: 

Tu 

2 

S A > F 2 1−α 

, 

S 

0 

sadac 1 F − α aris k − 1 da ( k −1)( m − 1) Tavisuflebis xarisxebiani fiSeris 

ganawilebis kanonis 1− α donis kvantili, maSin miiReba gadawyvetileba, 

rom dakvirvebis Sedegebze A faqtoris gavlena arsebiTia da am gavlenis 

dispersia gamoiTvleba formuliT: 

2 2 

2 SA − S0 

σ A = . 

m 

2 2 

Tu A faqtori gavlenas ar axdens dakvirvebis Sedegebze, maSin S A da S 0 

arian SemTxveviTi faqtoris dispersiis Sefasebebi da ufro zusti Sefasebis 

sapovnelad saWiroa maTi saSualo mniSvnelobis gamoyeneba 

2 2 2 2 

2 ( k − 1) SA + ( k −1)( m −1) S0 ( k − 1) S A + ( k −1)( m −1) 

S0 

S = = 

. 

( k − 1) + ( k −1)( m −1) m( k −1) 

zustad aseve SeiZleba B faqtoris gavlenis arsebiTobis dadgena. Tu 

S 

S 

2 

B 

2 

0 

> F , 

sadac 1 F − α aris m − 1 da ( k −1)( m − 1) Tavisuflebis xarisxebiani fiSeris 

ganawilebis kanonis 1− α donis kvantili, maSin miiReba gadawyvetileba, 

rom dakvirvebis Sedegebze B faqtoris gavlena arsebiTia. am gavlenis 

dispersia gamoiTvleba formuliT: 

2 2 

2 SB − S0 

σ B = . 

k 

Tu B 2 2 

faqtori gavlenas ar axdens dakvirvebis Sedegebze, maSin S B da S 0 

arian SemTxveviTi faqtoris dispersiis Sefasebebi da ufro zusti Sefasebis 

gamoTvla SesaZlebelia maTi saSualo mniSvnelobebis gamoyenebiT 

2 2 2 2 

2 ( m − 1) SB + ( k −1)( m −1) S0 ( m − 1) SB + ( k −1)( m −1) 

S0 

S = = 

. 

( m − 1) + ( k −1)( m −1) k( m −1) 

81 

1−α

Tu dakvirvebis Sedegebze ar moqmedeben A da B faqtorebi, maSin 

2 

da S 0 dispersiebis daxmarebiT SesaZlebelia SemTxveviTi faqtoris 

dispersiis yvelaze zusti Sefasebis gamoTvla 

2 2 2 2 2 2 

2 ( k − 1) SA + ( m − 1) SB + ( k −1)( m −1) S0 ( k − 1) S A + ( m − 1) SB + ( k −1)( m −1) 

S0 

S = = 

. 

( k − 1) + ( m − 1) + ( k −1)( m −1) mk −1 

erTfaqtoruli dispersiuli analizis analogiurad gamoTvlebis avtomatizaciisaTvis 

moviyvanoT orfaqtoruli dispersiuli analizis gamutvlebis 

sqema: 

2 

1) ϑ = ∑ ∑ x ; 

1 

k 

ni 

i= 1 j= 

1 

1 2 

2) 2 

1 

k 

ϑ = ∑ X i ; 

m i= 

2 1 ' 

3) 3 

1 

m 

ϑ = ∑ X j ; 

k j= 

4) ϑ4 

( ) 

ij 

2 

1 k 1 m 

' 

∑ X i ∑ X j 

i= 1 j= 

1 

⎛ ⎞ 

= = ⎜ ⎟ 

km km ⎝ ⎠ . 

2 

gamoTvlili sidideebiT CvenTvis saintereso dispersiebi gamoiTvleba 

Semdegi formulebiT 

2 ϑ1 + ϑ4 −ϑ2 −ϑ3 

2 ϑ2 −ϑ4 

2 ϑ3 −ϑ4 

S0 

= 

; SA 

= ; SB 

= . 

( k −1)( m −1) 

k −1 

m −1 

ganxilul orfaqtorul analizSi igulisxmeboda, rom gansaxilveli ara 

SemTxveviTi A da B faqtorebi erTmaneTze ar moqmedeben. aseTi urTierT 

zemoqmedebis SemTxvevaSi saWiroa ganmeorebadi dakvirvebebi A da B faqtorebis 

dafiqsirebuli mniSvnelobebisaTvis urTierT gavlenis faqtoris 

arsebiTobis dasadgenad. avRniSnoT xij1, xij 2,..., 

x ijn dakvirvebis Sedegebia A 

da B faqtorebis a i da b i doneebze Sesabamisad. avRniSnoT A da B faqto- 

rebis urTierT moqmedebis faqtoris dispersia 

82 

2 

S A , 

2 

S B 

2 

σ AB - iT. SevinarCunoT 

aRniSvna x ij ganmeorebadi dakvirvebis Sedegebis saSualo ariTmetikulisa- 

Tvis. Sesabamisi dispersia gamoiTvleba formuliT 

2 1 n 

2 

Sij = ∑ ( xijl − xij 

) . 

n −1 

l= 

1 

is ganapirobebulia ori faqtoris gavleniT: urTierTgavlenis faqtori 

2 

dispersiiT σ AB da SemTxveviTi faqtori 

bac gamoiTvleba ase 

2 1 k m 

2 1 k m n 

2 

S = ∑ ∑ Sij = ∑ ∑ ∑ ( xijl − xij 

) . 

km i= 1 j= 1 km( n −1) 

i= 1 j= 1 l= 

1 

orive es dispersia Sedis 

bas 

sidanac 

2 

S dispersiiT, romlis mniSvnelo- 

2 

S 0 dispersiaSi da gansazRvravs mis mniSvnelo- 

S 

2 

S 

≈ σ + . 

n 

2 2 

0 AB

2 

2 2 S 

σ AB ≈ S0 

− . 

n 

2 

2 

aqedan cxadia, rom Tu n ⋅ S0 

statistikurad arsebiTad gansxvavdeba S - 

gan, anu Tu adgili aqvs 

2 

n ⋅ S0 

> F 2 1−α 

, 

S 

F − − − fiSeris ganawilebis kanonis 1− α donis kvanti- 

sadac F1− α aris ( k 1)( m 1), km( n 1) 

li, maSin miiReba gadawyvetuleba, rom dakvirvebis Sedegebze moqmedebs 

urTierT gavlenis faqtori da misi dispersia fasdeba formuliT 

2 2 

2 nS0 − S 

σ AB = , 

n 

winaaRmdeg SemTxvevaSi dakvirvebis Sedegebze urTierTgavlenis faqtori 

ar moqmedebs. am SemTxvevaSi 

2 

S da 

2 

S 0 arian SemTxveviTi faqtoris disper- 

siis Sefasebebi da maTi saSualebiT SeiZleba gamovTvaloT am ukanasknelis 

ufro zusti Sefaseba 

2 2 

2 km( n − 1) S + ( k −1)( m −1) 

S0 

σ ≈ 

. 

km( n − 1) + ( k −1)( m −1) 

urTierTgavlenis faqtoris gaTvaliswinebisas zemoT moyvanili orfaqtoruli 

analizis sqema Seivseba Semdegnairad. damatebiT gamoiTvleba 

k m n 

2 

2 ϑ5 − nϑ1 

ϑ5 

= ∑ ∑ ∑ xijl 

da S = 

i= 1 j= 1 l= 

1 

km( n − 1) 

. 

Semdgomi analizi xorcieldeba zemoT moyvanilis Sesabamisad. 

83

Tavi 7. regresiuli analizi 

7.1. Sesavali 

korelaciuri analizi saSualebas iZleva davadginoT erTi SemTxveviTi 

sididis meoreze gavlenis faqti. disperisuli analizi saSualebas iZleva 

davadginoT SemTxveviT sidideze ara SemTxveviTi faqtoris gavlena. kvlevis 

Semdgomi (ara mniSvnelobis mixedviT, aramed ganxilvis mimdevrobiT) 

etapi aris am gavlenis raodenobrivi aRwera misi arsebobis SemTxvevaSi. 

gadavideT regresiuli analizis ZiriTadi momentebis ganxilvaze. MaTematikuri 

statistikis am metad mniSvnelovani nawilis ufro Rrmad Seswavlis 

msurvelebma SeuZliaT isargeblon literaturiT [7, 15, 36, 37, 56]. 

ganvixiloT ori ξ da η SemTxveviTi sidideebi. vTqvaT isini gavlenas 

axdenen erTmaneTze, anu erTi maTganis cvlileba ganapirobebs meores 

cvlilebas. am SemTxveviTi sidideebis mniSvnelobebis Sepirispirebisas SesaZlebelia 

ori saxis Secdomebis daSveba: a) ξ - s SemTxveviTi fluqtuaciis 

gamo η - s mniSvnelobas uTanaddeba ξ - s ara is mniSvneloba, romelic 

sinamdvileSi Seesabameba η - s mocemul mniSvnelobas; b) Sesabamisobis 

Secdoma gamowveulia η - s SemTxveviTi fluqtuaciiT. formalurad orive 

Secdoma SeiZleba mivakuTnoT η - s da warmovidginoT, rom saerTo Secdomis 

sidide gamowveulia mxolod η - s SemTxveviTi gafantviT. avRniSnoT η 

- s ganawilebis funqcia F( y ) - iT. radgan η aris damokidebuli ξ SemTxveviTi 

sididisagan, ganawilebis funqcia F( y ) aris pirobiTi ganawilebis 

funqcia, romlis mniSvnelobac damokidebulia imaze Tu rogor mniSvnelobas 

Rebulobs ξ SemTxveviTi sidide , anu F( y ) funqcia aris ori argumentis 

funqcia F( y) = F( x, y) 

. Tu cnobilia F( x, y ) , maSin cnobilia η - s ξ - gan 

damokidebulebis mTliani aRwera, amasTan gamosaxuli zust funqcionalur 

damokidebulebebSi. F( x, y ) – is moZebna dakvirvebis Sedegebis safuZvelze 

sakmaod Zneli amocanaa, romelic moiTxovs didi raodenobis dakvirvebis 

Sedegebs. amitom maTi moZebna xexdeba sakmaod iSviaTad. dauSvaT ξ da η 

normalurad ganawilebuli SemTxveviTi sidideebia. maSin F( x, y ) iqneba or 

ganzomilebiani normaluri ganawilebis funqcia, romelic srulad 

ganisazRvreba maTematikuri molodinebiTa da dispersiebiT. amitom η - s ξ 

- gan damokidebulebis saCveneblad sakmarisia Cveneba Tu rogor icvleba η 

sididis maTematikuri molodini da dispersia ξ - s cvlilebisas. 

SemdegSi ξ parametris mniSvneloba avRniSnoT x - iT, η parametris 

mniSvneloba - y - iT. a y da 

2 

σ y - iT avRniSnoT η SemTxveviTi sididis saSu- 

alo da dispersia. maSin mivdivarT ori funqciis moZebnis aucileblobas- 

Tan: 

ay = f1 ( x) 

da 2 

σ y = f2( x) 

. 

meore damokidebuleba aRwers meTodikis sizustis cvalebadobas parametris 

cvalebadobisas. mas skedastikuri damokidebuleba ewodeba da iSviaTad 

gamoiyeneba. pirveli damokidebuleba aRwers η - saSualo mniSvnelobebis 

cvalebadobas ξ sididis mniSvnelobebis cvalebadobisas. am damokidebulebas 

regresiuli damokidebuleba hqvia da is TamaSobs did rols mrava- 

84

li amocanis gadawyvetisas, radgan aRwers ξ da η sidideebis WeSmarit, yvela 

SemTxveviTi damatebisagan Tavisufal, damokidebulebas. amitom yovelgvari 

damokidebulebis gamokvlevis mizania regresiuli damokidebulebis 

povna, xolo disperisia gamoiyeneba moZebnili Sefasebebis sizustis Sesafaseblad. 

amrigad, zogadad, regresiuli damokidebuleba miaxloebiT SeiZleba 

CavweroT Semdegnairad 

y = f ( x) + ε , (7.1) 

sadac ε - normalurad ganawilebuli SemTxveviTi sididea nolis toli ma- 

2 

Tematikuri molodiniT da σ dispersiiT. zogadad f ( x ) funqcia damokidebulia 

parametrebis garkveuli raodenobisagan, romelTa mniSvnelobebi 

unda iqnas monaxuli dakvirvebis Sedegebis safuZvelze. avRniSnoT 

x , y , i = 1,..., n , Sesabamisad ξ da η SemTxveviT sidideebze dakvirvebis Sedege- 

i i 

bia. (7.1) regresiuli damokidebulebiT x i - s yovel mniSvnelobas Seesabameba 

y i - s garkveuli mniSvneloba. am dros daSvebuli Secdoma tolia 

yi − f ( xi 

) – is. yvela i, i = 1,..., n - Tvis daSvebuli Secdomebi arian SemTxvevi- 

Ti sidideebi, amitom ucnobi parametrebis Sefasebebad bunebrivia aviRoT 

maTi iseTi mniSvnelobebi, romlebic minimizacias gaukeTeben am SemTxveviTi 

sidideebis dispersias 

1 n 

2 

D = ∑ ( yi − f ( xi 

)) . (7.2) 

n − l l= 

1 

imis gamo, rom f funqciis ucnobi parametrebi SezRudvebs adeben dakvirvebis 

Sedegebs romlebiTac isini fasdebian, D dispersiis Tavisuflebis 

xarisxis sidide mcirdeba Sesabamisi raodenobiT. amitom l aris f funqciis 

ucnobi parametrebis ricxvi. dakvirvebis Sedegebis erTnairi n raodenobisaTvis 

D dispersiis gazrda SeiZleba gamowveuli iyos ara mxolod f 

funqciis parametrebis mniSvnelobebis ara koreqtuli SerCeviT, aramed am 

parametrebis raodenobis gazrdiTac. amitom saWiroa regresiad SerCeuli 

iqnas SesaZleblobis farglebSi minimaluri raodenobis ucnobi 

parametrebis mqone funqcia. 

7.2. miaxloebiTi regresiis gamoTvla da analizi 

winamdebare Tavis SesavalSi vTqviT, rom regresiis ucnobi koeficientebis 

Sefasebebis monaxva xdeba Sesabamisi dispersiis minimizaciiT. iqve av- 

RniSneT, rom regresia moiZebneba erTnairi raodenobis parametrebis mqone 

funqciebs Soris. am SemTxvevaSi regresiis aRdgenis dispersiis minimizacia 

(7.2) tolfasia Semdegi gamosaxulebis minimizaciis 

n 

2 

∑ ( i ( i )) , (7.3) 

l= 

1 

S = y − f x 

sadac xi, yi, i = 1,..., n - dakvirvebis Sedegebia. (7.3) kriteriums ewodeba umcires 

kvadratTa kriteriumi. (7.3) – is minimizaciisaTvis gamoiyeneba matematikuri 

analizis kargad nacnobi meTodi. kerZod, gavadiferenciroT (7.3) gamo- 

85

saxuleba ucnobi parametrebiT, moRebuli gamosaxulebebi gavutoloT nols 

da amovxcnaT miRebuli gantolebaTa sistema ucnobi parametrebis mimarT. 

avRniSnoT a1, a2,.., a n regresiis ucnobi parametrebi, anu adgili aqvs 

damokidebulebas y = f ( x) = f ( a1, a2,.., an; x) 

. gavadiferenciroT (7.3) kriteriumi 

am parametrebiT da miRebuli gamosaxulebebi gautoloT nols 

∂S n 

∂f 

= ∑ ( yi − f ( xi 

)) = 0 , 

∂a1 i= 

1 ∂a1 

∂S n 

∂f 

= ∑ ( yi − f ( xi 

)) = 0 , 

∂a2 i= 

1 ∂a2 

…………………………….. 

∂S n 

∂f 

= ∑ ( yi − f ( xi 

)) = 0 . 

∂an i= 

1 ∂an 

miRebul gantolebaTa sistemas ewodeba normalur gantolebaTa sistema. 

imis gamo, rom S dadebiTad gansazRvruli funqciaa da f regresiad yovel- 

Tvis airCeva diferencirebadi funqcia, normalur gantolebaTa sistema 

yovelTvis arsebobs da aqvs amoxsna. misi amoxsnis meTodi damokidebulia 

funqciis saxisagan da ξ da η SemTxveviTi sidideebis maxasiaTeblebisagan. 

Tu f funqciaSi ucnobi parametrebi Sedian wrfivad, normalur gantoleba- 

Ta sistema iqneba wrfiv gantolebaTa sistema, winaaRmdeg SemTxvevaSi, anu 

rodesac f - Si koeficientebi Sedian ara wrfivad, normalur gantolebaTa 

sistema aris ara wrfiv gantolebaTa sistema da misi amoxsnisaTvis, yovel 

konkretul SemTxvevaSi, saWiroa Sesabamisi meTodis SerCeva. Tu normalur 

gantolebaTa sistemas aqvs erTze meti amonaxseni, maSin maT Soris 

amoirCeva is amonaxsni, romelic minimizacias ukeTebs (7.3) umcires 

kvadratTa kriteriums. 

ganvixiloT normalur gantolebaTa sistema, rodesac regresia aRsdgeba 

2 

meore rigis polinomis saxiT, anu regresias aqvs saxe y = a + a ⋅ x + a ⋅ x . am 

∂ y 

SemTxvevaSi = 1, 

∂a1 

Rebulobs saxes: 

∂ y 

= x , 

∂a 

2 

∂ y 

= x 

∂a 

3 

2 

n 

2 

( yi − ( a1 + a2 ⋅ xi + a3 ⋅ xi 

)) ⋅ 1 = 0 

i= 

1 

∑ , 

n 

2 

( yi − ( a1 + a2 ⋅ xi + a3 ⋅ xi )) ⋅ xi 

= 0 

i= 

1 

∑ , 

n 

2 2 

( yi − ( a1 + a2 ⋅ xi + a3 ⋅ xi )) ⋅ xi 

= 0 

i= 

1 

∑ . 

86 

1 2 3 

da normalur gantolebaTa sistema 

es sistema aris wrfivi a1, a2, a 3 parametrebis mimarT da misi amoxsana ar 

warmoadgens araviTar sirTules. 

y = f ( x) 

regresiis povnis Semdeg ismeba kiTxva: Seesabameba Tu ara 

napovni regresia unob WeSmarit damokidebulebas, Tu arsebobs iseTi ϕ ( x) 

Sesworeba, rom y = f ( x) + ϕ( 

x) 

regresia ukeTesad Seesabameba arsebul 

dakvirvebis Sedegebs? avRniSnoT l 1 - iT y = f ( x) 

regresiis ucnobi 

parametrebis raodenoba, xolo l 2 - iT avRniSnoT y = f ( x) + ϕ( 

x) 

regresiis

parametrebis 

dispersiebi 

raodenoba. bunebrivia l1 < l2 

. gamovTvaloT Sesabamisi 

1 n 

2 1 n 

2 

D1 = ∑ ( yi − f ( xi 

)) , D2 = ∑ ( yi − f ( xi ) −ϕ 

( xi 

)) . 

n − l 1 

i= 

1 

n − l2 

i= 

1 

imisaTvis, rom ϕ ( x) 

Sesworebas hqondes azri, anu gaaumjobesos sawyisi 

regresia, saWiroa, rom D 2 dispersia statistikurad arsebiTad iyos 

D F 

naklebi 1 D dispersiaze. Tu 1 > 1−α 

, sadac F1− α aris n − l1 

da n l2 

D2 

87 

− Tavisuf- 

lebis xarisxebis mqone fiSeris ganawilebis 1− α donis kvantili, miiReba 

gadawyvetileba, rom y = f ( x) + ϕ( 

x) 

regrsia ukeTesad Seesabameba dakvirvebis 

Sedegebs, vidre y = f ( x) 

regresia. winaaRmdeg SemTxvevaSi miiReba 

gadawyvetileba, rom ϕ ( x) 

ar aumjobesebs y = f ( x) 

sawyis regresias. 

ϕ ( x) 

Sesworebis ugulebelyofa ar niSnavs, rom y = f ( x) 

regresia 

saerTod ar saWiroebs Sesworebas. gadawyvetilebis misaRebad, rom y = f ( x) 

regresias esaWiroeba Sesworeba saWiroa SemTxveviTi faqtoris dispersiis 

codna. rogorc wesi es dispersia ucnobia da misi SefasebisaTvis saWiroa 

ganmeorebiTi dakvirvebebi. avRniSnoT yoveli x i - Tvis ganmeorebadi 

dakvirvebebi yi1, yi 2,..., 

y im - iT, maSin SemTxveviTi faqtoris dispersiis 

Sefaseba iqneba 

2 1 m 

2 

Si = ∑ ( yij − yi ) , i = 1,..., n. 

m −1 

j= 

1 

am dispersiis ufro zusti Sefaseba gamoiTvleba ase 

2 1 2 

n 

S = ∑ S . 

y f ( x) 

n i= 

1 

= regresiis dispersia, anu 1 

i 

D ganpirobebulia ori faqtoris gav- 

2 

leniT: S dispersiis mqone SemTxveviTi faqtoriT da y = f ( x) 

regresiis da 

ucnobi WeSmariti regresiis Seusabamisobis faqtoriT. avRniSnoT misi dispersia 

S - iT. imis gamo, rom es ori faqtori erTmaneTisagan damouki- 

2 

регр 

2 2 

deblebi arian, adgili aqvs D1 = S + S регр . Tu D 1 dispersia 

gan statistikurad arsebiTad gansxvavdeba, es SeiZleba gamowveuli iyos 

2 

mxolod da mxolod imiT, rom S ≠ 0, 

anu regresia y = f ( x) 

regresia ar 

Seesabameba ucnob namdvil regresias. D 1 da 

регр 

2 

S dispersiisa- 

2 

S gansxvavebis arsebiToba 

D1 mowmdeba ase: Tu > F 2 1−α 

, sadac 

S 

1 F − α aris n − l1 

da n( m − 1) Tavisuflebis 

xarisxebis mqone fiSeris ganawilebis 1− α donis kvantili, maSin miiReba 

gadawyvetileba, rom y = f ( x) 

regresia saWiroebs Sesworebas, winaaRmdeg 

SemTxvevaSi miiReba gadwyvetileba, rom 

dakvirvebis Sedegebs. 

y = f ( x) 

regresia Seesabameba 

y = f ( x) 

regresiis ucnobi parametrebis Sefasebebis monaxvas umcires 

kvadratTa kriteriumis minimizaciis gziT ewodeba regresiis identifikacia, 

xolo y = f ( x) 

regresiis dakvirvebis SedegebTan Sesabamisobis uSualo 

analizs ewodeba regresiuli analizi.

7.3. wrfivi regresia 

am SemTxvevaSi regresiul damokidebulebas aqvs Semdegi saxe 

y = a + b⋅ x , sadac y = a + b⋅ x da b ucnobi koeficientebia. umcires kvadratTa 

kriteriumi Caiwereba ase 

n 

2 

∑ ( i ( i )) , 

i= 

1 

S = y − a + b⋅ x 

sadac xi, yi, i = 1,..., n - dakvirvebis Sedegebia. normalur gantolebaTa sistemas 

aqvs Semdegi saxe 

n 

n 

i= 

1 

∑ ( y − ( a + b ⋅ x )) = 0 , 

i= 

1 

i i 

∑ ( y − ( a + b ⋅ x )) ⋅ x = 0. 

i i i 

gadavweroT es sistema Semdegnairad 

n n 

a ⋅ n + b∑ x = ∑ y , 

i i 

i= 1 i= 

1 

n n n n 

a∑ x + b∑ x ⋅ ∑ x = ∑ x ⋅ y . 

i i i i i 

i= 1 i= 1 i= 1 i= 

1 

b ricxvs ewodeba regresiis koeficienti. is SeiZleba advilad movZebnoT 

determinantebis daxmarebiT 

n n n 

n∑ x ⋅ y − ∑ x ⋅ ∑ y 

b = 

i i i i 

i= 1 i= 1 i= 

1 

n n 

2 

2 

n∑ xi − ( ∑ xi 

) 

i= 1 i= 

1 

a ricxvs ewodeba regresiis Tavisufali wevri. isic advilad ganisaz- 

Rvreba determinantebis daxmarebiT wrfiv gantolebaTa sistemidan, magram 

simartivisaTvis ganvsazRvroT sistemis pirveli gantolebidan ukve gansaz- 

Rvruli b koeficientis daxmarebiT 

n n 

∑ yi − b∑ xi 

i= 1 i= 

1 

a = = y − b⋅ x , 

n 

aqedan y = a + b ⋅ x . 

amrigad, aRmoCnda, rom wrfivi regresia gadis ( x, y ) wertilze. amitom, 

wrfivi regresiis asagebad saWiroa ( x, y ) wertilze gavavloT wrfe, romelic 

abscisTa RerZTan adgens kuTxes, romlis tangensic b koeficientis tolia. 

ganvsazRvroT are, romelsac miekuTvneba wrfivi regresia mocemuli 

albaTobiT. rogorc ukve iyo naTqvami, wrfivi regresia ganisazRvreba 

mniSvnelobiT y da b koeficientiT. y – Tvis ndobis intervalis ageba ganxiluli 

iyo zemoT. avRniSnoT y ' da y '' am intervalis ukiduresi marcxena 

da marjvena wertilebi. b koeficientis ndobis intervalis sapovnelad 

visargebloT bartletis SedegiT, romelmac daamtkica, rom statistika 

Sx n − 2 

t = ( b − b0 

) , 

S 1− 

r 

y 

88 

.

sadac x S da y S Sesabamisad i x da y i amonarCevebis TaviaanTi saSualoebis 

mimarT dispersiebidan kvadratuli fesvebia; r - amonarCevis korelaciis 

b - WeSmariti regresiis koeficientia, ganawilebulia n − 2 - 

koeficientia; 0 

Tavisuflebis xarisxis mqone stiudentis kanoniT. avRniSnoT t1 − α / 2 – sti- 

udentis ganawilebis (1 − α / 2) donis kvantili. maSin adgili aqvs 

S n − 2 

−t ≤ ( b − b ) ≤ t 

S 1− 

r 

x 

1 −α / 2 0 1 −α 

/ 2 

saidanac Zneli ar aris miviRoT b 0 - Tvis ndobis intervali 

y 

S y 1− r S y 1− 

r 

b − t1 −α / 2 ≤ b0 ≤ b + t1 

−α 

/ 2 

S n − 2 S n − 2 

. 

x x 

' 

b da '' 

b - iT avRniSnoT am ndobis intervalis Sesabamisad ukiduresi mar- 

2 

cxena da marjvena wertilebi. maSin ndobis are romelSiac (1 − α) 

albaTobiT 

imyofeba WeSmariti regresiis Sesabamisi wrfe aigeba Semdegnairad. 

' 

'' 

koordinatTa sibrtyeze ( x, y ) da ( x, y ) wertilebze atareben or – or 

' 

wrfes kuTxuri koeficientebiT b da 

luri are aris wrfivi regresiis ndobis are (ix. nax. 7.1). 

nax. 7.1. 

89 

, 

'' 

b . am wrfeebis mier moculi maqsima- 

cnobilia, rom SemTxveviT sidideebs Soris wrfivi kavSiris siZlieris 

maCvenebelia korelaciis koeficienti. ganvsazRvroT kavSiri wrfivi regresiis 

koeficientebs da korelaciis koeficients Soris. gavyoT regresiis ko- 

2 

eficientis b - s gamosaTvleli gamosaxulebis mricxveli da mniSvneli n - 

ze, miviRebT 

1 n 1 n 1 n 

∑ xi yi − ∑ xi ∑ yi 

n rS i 1 i 1 i 1 

xS y S 

= n = n = 

y 

b = = = r , 

2 2 

1 n 

2 ⎛ 1 n ⎞ Sx Sx 

∑ xi − ⎜ ∑ xi 

⎟ 

n i= 1 ⎝ n i= 

1 ⎠ 

saidanac

⋅ Sx 

r = . 

S y 

Tu korelaciis koeficienti gamoTvlili iqna adre, maSin is SeiZleba 

gamoyenebuli iqnas wrfivi regresiis gansasazRvrad 

S y 

y = a + r ⋅ x , 

Sx 

anda Tu a - s SevcvliT y − b ⋅ x - iT, miviRebT 

S y 

y − y = r ( x − x) 

. 

Sx 

aqedan cxadad Cans, rom rodesac korelaciis koeficienti r = 0 , maSin 

wrfivi regresia aris abscisTa RerZis paraleluri wrfe, anu y ar aris 

damokidebuli x – ze. 

7.4. arawrfivi regresia 

rogorc zemoT iyo aRniSnuli, arawrfivi regresiis identifikaciisaTvis 

saWiroa umcires kvadratTa kriteriumis minimizacia. regersiis saxisagan, 

anu y = f ( x) 

- gan damokidebulebiT miiReba sxvadasxva saxis arawrfiv normalur 

gantolebaTa sistemebi. maTi amoxsnisaTvis, yovel konkretul 

SemTxvevaSi, saWiroa avirCioT Sesabamisi meTodi. identifikaciis amocanis 

gasamartiveblad saWiroa vecadoT, Tu es SesaZlebelia, regresia avirCioT 

rac SeiZleba naklebi raodenobis ucnobi parameterebiT. zogjer 

SesaZlebelia cvladebis SecvliT movaxdinoT regresiis gawrfiveba ucnobi 

parametrebis mimarT. Tu regresiis yvela parametris gawrfiveba ver xerxdeba, 

unda vecadoT gawrfivebiT SevamciroT regresiaSi ara wrfivad Semavali 

parametrebis raodenoba. ganvixiloT ramodenime martivi magaliTi. 

dauSvaT saWiroa arawrfivi regresiis aRdgena xi, yi, i = 1,..., n dakvirvebis 

Sedegebis safuZvelze. vTqvaT regresias aqvs maCvenebliani funqciis saxe 

x 

y = a ⋅ b . galogariTmebiT SeiZleba mivaRwioT am damokidebulebis gawrfiveba 

ln y = ln a + x ⋅ ln b . Tu regresiuli damokidebulebis identifikaciis win 

movaxdenT sawyisi monacemebis gardaqmnas xi ,ln yi, i = 1,..., n , maSin gawrfivebuli 

regresiis aRsadgenad, anu ln a da ln b gansasazRvrad SeiZleba gamoviyenoT 

wrifvi regresiis identifikaciis formulebi. cxadia, rom amis Semdeg, 

sawyisi regresiis a da b koeficientebis gansazRvra ar warmoadgens aravi- 

Tar sirTules. 

b 

ganvixiloT xarisxobrivi damokidebuleba y = a ⋅ x . am SemTxvevaSi galogariTmeba 

iZleva ln y = ln a + b ⋅ ln x . sawyisi monacemebi unda gardavqmnaT 

Semdegnairad: ln xi ,ln yi, i = 1,..., n . ln a da b koeficientebi ganisazRvrebian 

rogorc wrfivi regresiis koeficientebi, xolo a koeficienti ganisazRvreba 

uku gardaqmniT. 

90

amocanebi amocanebi praqtikuli praqtikuli mecadineobisaTvis 

mecadineobisaTvis 

1. latariaSi 1000 bileTia; maTgan erTi bileTi igebs 500 lars, 10 bile- 

Ti igebs 100 – 100 lars, 50 bileTi igebs 20 – 20 lars, xolo 100 bile- 

Ti igebs 5-5 lars, danarCeni bileTebi aramomgebiania. ipoveT ara 

nakleb 20 laris mogebis albaToba [4]. 

2. sam iaraRis sawyobs esvrian erT bombs. pirvel sawyobze moxvedris 

albaToba tolia 0.01 – is; meoreze – 0.008- is; mesameze – 0.025. erT sawyobze 

moxvedrisas feTqdeba samive sawyobi. ipoveT albaToba imisa, 

rom sawyobebi iqnebian afeTqebuli [4]. 

3. wriuli samizne Sedgeba sami zonisagan: I, II da III. erTi gasrolisas 

pirvel zinaSi moxvedris albaToba tolia 0.15 – is, meoreSi – 0.23 – 

is, mesameSi – 0.17 – is. ipoveT acdenis albaToba [4]. 

4. yuTSi ori TeTri da sami Savi burTulaa. yuTidan mimdevrobiT iReben 

or burTulas. ipoveT albaToba imisa, rom orive burTula TeTria 

[4]. 

5. igive pirobaa, magram pirveli amoRebis Semdeg burTulas abruneben 

yuTSi da burTulebs ureven [4]. 

6. sami msroleli erTmaneTisagan damoukideblad esvrian mizanSi. mizan- 

Si moxvedris albaToba pirveli, meore da mesame msrolelisaTvis 

Sesabamisad tolia: p = 0.2 ; p = 0.5 ; p = 0.3 . ipoveT albaToba imisa, 

rom samive msroleli moaxvedrebs mizanSi [4]. 

7. samjer esvrian erTi da igive samiznes. moxvedrebis albaToba pirveli, 

meore da mesame gasrolisas Sesabamisad tolia: p = 0.4 ; p = 0.5 ; 

p = 0.7 . ipoveT albaToba imisa, rom am sami gasrolis Sedegad samiznes 

moxvdeba erTxel [4]. 

8. wina magaliTis pirobebSi ipoveT albaToba imisa, rom samiznes erTxel 

mainc moxvdeba [4]. 

9. samiznes esvrian erTxel. moxvedris albaToba tolia 0.3 – is. Sem- 

TxveviTi sidide X aris moxvedrebaTa raodenoba. aageT X sididis 

ganawilebis mwkrivi da ganawilebis mravalkuTxedi [4]. 

10. msrloleli samjer esvris samiznes. yoveli gasrolisas moxvedris 

albaToba tolia 0.4-is. yoveli moxvedrisaTvis msrolels ericxeba 5 

qula. aageT Segrovebuli qulebis ganawilebis mwkrivi [4]. 

11. raRac mizans esvrian pirvel moxvedrebamde. yoveli gasrolisas moxvedrebis 

albaToba p - s tolia. X SemTxveviTi sidide aris gasrolaTa 

ricxvi. aageT X sididis ganawilebis mwkrivi [4]. 

12. mizanSi isvrian erTxel. moxvedrebis albaToba tolia 0.3 – is. aageT 

moxvedrebaTa ricxvis ganawilebis funqcia [4]. 

13. mizanSi isvrian 4 – jer; yoveli gasrolisas moxvedrebis albaToba 

tolia 0.3 – is. aageT moxvedrebaTa ricxvis ganawilebis funqcia [4]. 

14. uwyveti SemTxveviTi sididis ganawilebis funqcia mocemulia gamosaxulebiT 

91

⎧ 

⎪0 

при x < 0, 

⎪ 2 

F ( x) 

= ⎨ax 

при 0 < x < 1, 

⎪ 

⎪1 

при x > 1. 

⎪⎩ 

a) ipoveT koeficienti a . 

b) ipoveT ganawilebis simkvrive f (x) 

. 

g) ipoveT SemTxveviTi sidide X -is 0.25-dan 0.5-de intervalSi moxvedris 

albaToba [4]. 

15. X SemTxveviTi sidide ganawilebulia kanoniT romlis simkvrivea: 

π π 

f ( x) = a ⋅ cos x roca − < x < ; 

2 2 

π π 

f ( x) 

= 0 roca x < − an x > . 

2 2 

a) ipoveT koeficienti a . 

b) aageT f (x) 

ganawilebis simkvrivis grafiki. 

g) ipoveT F (x) 

ganawilebis funqcia da aageT misi grafiki. 

π 

d) ipoveT SemTxveviTi sidide X -is 0-dan -de intervalSi moxvedris 

4 

albaToba [4]. 

16. X SemTxveviTi sididis ganawilebis simkvrive mocemulia formuliT: 

1 

f ( x) 

. 2 

π ( 1+ 

x ) 

a) aageT f (x) 

simkvrivis grafiki. 

b) ipoveT albaToba imisa, rom X SemTxveviTi sidide moxvdeba ( − 1 , + 1) 

intervalSi [4]. 

17. cxril 1.a. – Si mocemulia Carxze damzadebuli (romelic uSvebs aTasobiT 

aseT nakeTobebs) 200 samagris Tavakebis zomebi. yvela piroba, romelSiac 

muSaobda Carxi, ucvleli iyo. aageT ganawilebis empiriuli funqcia, histograma, 

gamoiTvaleT Sesabamisi SemTxveviTi sididis ricxviTi maxasiaTeblebi. 

gamoiTvaleT igive ricxviTi maxasiaTeblebi dajgufebuli monacemebiT. 

gamoTvlili mniSvnelobebiT imsjele SemTxveviTi sididis da misi 

ganawilebis kanonis Sesaxeb [1]. 

92 

cxrili 1.a. 

100 samagris Tavakis diametri, mm 

13.39 13.34 13.33 13.14 13.56 13.31 13.38 13.51 13.38 13.40 

13.28 13.23 13.34 13.37 13.50 13.64 13.38 13.30 13.42 13.34 

13.53 13.43 13.58 13.58 13.32 13.63 13.27 13.48 13.26 13.32 

13.57 13.38 13.36 13.33 13.43 13.57 13.38 13.28 13.39 13.28 

13.40 13.34 13.39 13.54 13.50 13.40 13.52 13.47 13.55 13.43 

13.29 13.28 13.33 13.46 13.38 13.37 13.61 13.53 13.44 13.26 

13.43 13.33 13.51 13.39 13.50 13.56 13.38 13.24 13.34 13.34 

13.41 13.43 13.49 13.51 13.42 13.51 13.45 13.48 13.48 13.54

13.55 13.52 13.44 13.23 13.50 13.48 13.40 13.66 13.48 13.32 

13.43 13.53 13.26 13.44 13.58 13.69 13.38 13.43 13.59 13.37 

13.45 13.58 13.47 13.24 13.62 13.32 13.45 13.52 13.39 13.50 

13.40 13.37 13.57 13.18 13.46 13.50 13.33 13.45 13.40 13.60 

13.52 13.40 13.35 13.40 13.29 13.33 13.48 13.20 13.43 13.44 

13.39 13.41 13.46 13.39 13.29 13.48 13.55 13.42 13.31 13.46 

13.40 13.30 13.20 13.45 13.31 13.40 13.46 13.45 13.13 13.40 

13.62 13.35 13.42 13.42 13.54 13.36 13.31 13.44 13.58 13.41 

13.47 13.28 13.48 13.37 13.59 13.54 13.20 13.43 13.56 13.35 

13.29 13.41 13.31 13.51 13.42 13.44 13.32 13.36 13.48 13.36 

13.45 13.26 13.29 13.51 13.32 13.38 13.24 13.46 13.38 13.34 

13.32 13.53 13.52 13.40 13.57 13.25 13.62 13.37 13.29 13.55 

18. cxril 2.a. – Si mocemulia 9 fermeris nakveTze moyvenili pomidorisa da 

kitris mosavlis raodenoba da amave nakveTebis niadagebSi nitratebisa da 

fosfatebis Semcvelobebi. gamoiTvaleT Sesabamis SemTxveviTi sidideebis 

ricxviTi maxasiaTeblebi, maT Soris korelaciis koeficientebi miRebuli 

mosavalis sidideebsa da niadagSi nitratebisa da fosfatebis Semcvelobebs 

Soris. 

fermeris 

nakveTis 

nomeri 

nitratebis 

Semcveloba. 

mg/kg 

fosfatebis 

Semcveloba. 

mg/kg 

pomidoris 

mosavali 

kg/0.01 ha 

kitris 

mosavali 

kg/0.01 ha 

cxrili 2.a. 

1 2 3 4 5 6 7 8 9 

61.1 24.3 21.7 17.3 16 13.2 12.4 12 3.5 

512.2 348.66 292 259.36 257.94 243 170.74 122.4 75 

- 253.33 130 104.5 101 139.33 72 90 71 

145 168 - - 192 63 80 63 76 

19. uwyveti SemTxveviTi X sidide ganawilebulia Semdegi simkvrivis mqone 

kanoniT 

f ( x) 

= Ae . 

ipoveT A koeficienti. gansazRvreT X sididis maTematikuri molodini, 

dispersia, saSualo kvadratuli gadaxra, asimetria da eqscesi [4]. 

20. X SemTxveviTi sidide ganawilebulia Semdegi simkvrivis mqone kanoniT: 

93 

− x 

⎧ 

⎪ax 

at 0 < x < 1, 

f ( x) 

= ⎨ 

⎪ 

⎩0 

at x < 0 or x > 1.

aageT albaTobebis gnawilebis simkvrivis grafiki, gansazRvreT a , maTematikuri 

molodini, dispersia, saSualo kvadratuli gadaxra da asimetriis 

koeficienti [4]. 

21. avtomaturi satelefono sadguri saaTSi saSualod Rebulobs K gamoZaxebas. 

gamoZaxebebis raodenobebi drois nebismier intervalSi ganawilebulia 

puasonis kanoniT. ipoveT albaToba imisa, rom or wuTSi sadgurSi mova 

zustad sami gamoZaxeba [4]. 

22. wina magaliTis pirobebSi ipoveT albaToba imisa, rom or wuTSi erTi 

gamoZaxeba mainc mova [4]. 

23. igive pirobebSi ipoveT albaToba imisa, rom or wuTSi mova ara nakleb 

sami gamoaxeba [4]. 

24. sarTavi dazgis muSaobisas Zafi wydeba saaTSi saSualod 0.375 – jer. ipoveT 

albaToba imisa, rom cvlaSi (8 saaTis ganmavlobaSi) Zafis gawyvetis 

raodenoba iqneba moTavsebuli 2-sa da 4-s Soris (ara nakleb orisa da ara 

umetes oTxis) [4]. 

25. samizneSi erTmaneTisagan damoukideblad esvrian 50-jer. erTi gasrolisas 

mizanSi moxvedrebis albaToba tolia 0.4-is. binomialuri ganawilebis 

zRvruli Tvisebis gamoyenebiT ipoveT miaxloebiTi albaToba imisa, 

rom mizanSi moxvdeba: arc erTi yumbara, erTi yumbara, ori yumbara [4]. 

26. mocemulia normalurad ganawilebuli SemTxveviTi sidide 1.2-is toli 

maTematikuri molodiniT da 0.8-is toli saSualo kvadratuli gadaxriT. 

ipoveT albaToba imisa, rom SemTxveviTi sidide xvdeba intervalSi [-1.6; 

+1.6] [4]. 

27. mocemulia normalurad ganawilebuli X SemTxveviTi sididis maTema- 

Tikuri molodini m da abscisTa RerZze intervali ( α , β ) . rogori unda 

iyos SemTxveviTi sididis saSualo kvadratuli gadaxra σ , rom mocemul 

intervalSi misi moxvedris albaToba iyos maqsimaluri [4]? 

28. gazomvis cdomilebis ganawilebis kanonis gamokvlevis mizniT radiomzomiT 

ganxorcielda daSorebis 400 gazomva. cdebis Sedegebi mocemulia statistikuri 

mwkrivis saxiT: 

I 20;30 30;40 40;50 50;60 60;70 70;80 80;90 

cxrili a.3. 

90;100 

i 

m 21 72 66 38 51 56 64 32 

i 

p 0.052 0.180 0.165 0.095 0.128 0.140 0.160 0.080 

i 

gaasworeT statistikuri mwkrivi Tanabari simkvrivis kanoniT [4]. 

94

29. ganxorcielda miwiszeda samizneze TviTmfrinavidan srolisas damiznebis 

gverdiTi Secdomis 500 gazomva. gazomvis Secdomebi (radianis meaTased nawilebSi) 

moyvenilia statistikuri mwkrivis saxiT: 

I i 

-4;-3 -3;-2 -2;-1 -1;0 0;1 1;2 2;3 3;4 

m 6 25 72 133 120 88 46 10 

i 

i 

p 0.012 0.050 0.144 0.266 0.240 0.176 0.092 0.020 

aRniSvnebi: i I - damiznebis Secdomis mniSvnelobebis intervalebi; m i - mo- 

mi 

cemul intervalSi dakvirvebebis ricxvi; pi 

= - Sesabamisi sixSireebi [4]. 

n 

statistikuri mwkrivis monacemebiT miaxloebiT aageT damiznebis Secdomis 

ganawilebis statistikuri mwkrivi. 

30. moaxdineT wina magaliTSi mocemuli statistikuri ganawilebis gagluveba 

normaluri kanonis daxmarebiT [4]: 

95 

2 

( x−m 

) 

− 

2 

2 

1 σ 

f ( x) 

= e . 

σ 2π 

31. cxrilSi moyvenilia [0,5] intervalSi Tanabrad ganawilebul SemTxveviT 

sidideze dakvirvebis Sedegebi. gamoTvaleT maTematikuri molodini. dispersia. 

aageT albaTobebis ganawilebis empiriuli funqcia da simkvrive. 

i 1 2 3 4 5 6 7 8 9 10 11 12 

x 2.128 0.410 2.373 0.352 4.204 0.298 1.466 4.586 1.839 3.873 1.639 3.488 

i 

i 13 14 15 16 17 18 19 20 21 22 23 24 

x 4.220 

i 

3.589 1.533 0.813 1.647 2.330 1.233 4.128 1.395 2.408 0.745 4.371 

i 25 26 27 28 29 30 

x 1.436 3.863 4.882 2.462 4.439 4.136 

i 

32. cxrilSi mocemulia normalurad ganawilebuli SemTxveviTi sidideebis 

generatoris muSaobis Sedegebi maTematikuri molodiniT 2 da dispersiiT 

4. generirebuli monacemebiT gamoTvaleT maTematikuri molodini da 

dispersia. aageT albaTobebis ganawilebis empiriuli funqcia da simkvrive. 

i 1 2 3 4 5 6 7 8 9 10 11 12 

x 0.178 

i 

1.169 1.830 0.523 1.65 

5 

-4.082 -1.646 13. 

24 

2.509 1.57 

3 

6.996 5.68 

2 

i 13 14 15 16 17 18 19 20 21 22 23 24 

x 2.914 

i 

1.169 -0.111 6.010 2.8 

03 

-0.313 3.473 6.7 

09 

-0.653 3.26 

3 

-4.756 1.74 

0 

i 25 26 27 28 29 30 

x 5.702 

i 

6.458 -1.074 10.05 

6 

3.74 

2 

4.106

33. Cvidmeti gamosacdelisagan TiToeuls SemTxveviTi mimdevrobiT miewodeboda 

ori signali: sinaTlisa da bgeriTi. signalebis intensivoba mTeli 

eqsperimentis dros ucvleli iyo. dro signalis warmoSobis momentsa da 

gamosacdelis reaqcias Soris fiqsirdeboda xelsawyoTi. eqsperimentis Sedegebi 

mocemulia cxrilSi. 

i 

x i 

i y 

96 

i 

x i 

i y 

1 223 181 10 191 156 

2 104 194 11 197 178 

3 209 173 12 183 160 

4 183 153 13 174 164 

5 180 168 14 176 169 

6 168 176 15 155 155 

7 215 163 16 115 122 

8 172 152 17 163 144 

9 200 155 

SeiZleba CavTvaloT adamianis sinaTlisa da bgeriT signalebze reaqciis 

dro erTnairi? 

34. cxrilSi mocemulia mdinaris wylis amiakiT dabinZurebis monacemebi or 

mezobel kveTSi TerTmeti Tvis ganmavlobaSi (TveSi TiTo gazomva). cnobilia, 

rom mdinaris wylis dabinZurebis parametris saSualo wliuri mniSvneloba 

zeda kveTSi 0.245 – is tolia. kontrolis qvemo kveTSi Sesrulebuli 

mocemuli parametris koncentraciis TerTmeti gazomvis SedegiT daadgineT 

misi saSualo wliuri mniSvnelobis gazrdis faqti im pirobiT, rom gazomvis 

Sedegebi ganawilebulia normaluri ganawilebis kanonis Tanaxmad. 

gazomvis 

dro 

kontrolis 

zeda 

kve- 

Ti 

kontro- 

lis 

qveda 

kveTi 

I II III IV V VI VII VIII IX X XI 

x 11 8 x 

0.3 0.32 0.6 0.34 0.06 0.13 0.34 0.4 0.02 0.1 0.09 0.245 0.311 

0.62 0.68 0.8 0.38 0.18 0.48 0.54 0.51 0.08 0.18 0.1 0.414 0.524 

35. gadawyviteT wina magaliTis amocana pirveli rva Tvis monacemebis safuZvelze. 

anu mdinaris zeda kveTSi dabinZurebis parametris saSualo mniSvneloba 

0.311 – is tolia. kontrolis qveda kveTSi rva gazomvis SedegiT 

daadgineT am kveTSi dabinZurebis saSualo mniSvnelobis gazrdis faqti. 

36. gadawyviteT 35 da 36 magaliTebSi dasmuli problemebi mani-uitnisa da 

uilkoksonis kriteriumebiT.

37. fermerebis xuT nakveTSi iyo ganxorcielebuli niadagSi nitratebis Semcvelobebis 

gazomva gazafxulze da Semodgomaze. daadgineT SemodgomiT niadagSi 

nitratebis Semcvelobis cvlilebis faqti gazafxulTan SedarebiT. 

s/s nakveTis 

nomeri 

gazafxulze 

niadagSi 

nitratebis 

Semcveloba 

(mg/kg) 

SemodgomiT 

niadagSi 

nitratebis 

Semcveloba 

(mg/kg) 

1 2 3 4 5 

18 17.3 9.5 4.6 5.3 

15.3 12.7 12.2 11.8 10.5 

38. SeamowmeT ξ SemTxveviTi sididis medianis mocemul mniSvnelobasTan 

tolobis θ = θ 0 hipoTeza am SemTxveviT sidideze 15 dakvirvebis SedegiT, 

romlebic mocemulia cxrilSi 

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

x 9.18 5 2.81 3.58 3.99 4.9 7.32 3.35 3.41 3.1 7.14 8.11 6.11 4.3 3.17 

i 

a) θ = 5 ; 

b) θ = 3; 

g) θ = 3.5 . 

39. fermeris nakveTis nitratebiT dabinZurebis donis cvlilebis dadgenis 

mizniT gazafxulze da SemodgomiT zomaven mis koncentracias niadagis xuT 

siRrmiT fenaSi (15 sm, 30 sm, 45 sm, 60 sm, 90 sm). Sedegebi mocemulia cxrilSi. 

mani-uitnis kriteriumis daxmarebiT daadgineT niadagis gabinZurebis donis 

cvlilebis faqti. 

siRrmiTi 

fenis nomeri 

gazafxulze 

niadagis 

fenebSi 

nitratebis 

Semcveloba 

(mg/kg) 

SemodgomiT 

niadagis 

fenebSi 

1 2 3 4 5 

18 17.3 9.5 4.6 5.3 

15.3 12.7 12.2 11.8 10.5 

97

nitratebis 

Semcveloba 

(mg/kg) 

40. wina magaliTi gadawyvite uilkoksonis kriteriumis daxmaebiT. 

41. mdinaris mocemuli kveTis dabinZurebaze gavlenas axdenen garkveuli 

sawarmos Camdinare wylebi, romlebic mdinaris wyals abinZureben amiakiT. 

sawarmoSi danerges axali, ekologiurad ufro usafrTxo, teqnologia. 

cxrilis pirvel striqonSi mocemulia mdinaris wyalSi amiakis koncentraciis 

gazomvis Sedegebi axali teqnologiis danergvamde, xolo meore striqonSi 

mocemulia gazomvis Sedegebi axali teqnologiis danergvis Semdeg. 

mani-uitnisa da uilkoksonis kriteriumebis daxmarebiT daadgineT mdinaris 

wylis dabinZurebis donis cvlilebis faqti sawarmoSi axali teqnologiis 

danergvis Semdeg. 

gazomvis 

rigiTi 

nomrebi 

amiakis 

Semcveloba 

mdinaris 

wyalSi 

axali 

teqnologiis 

danergvamde 

amiakis 

Semcveloba 

mdinaris 

wyalSi 

axali 

teqnologiis 

danergvis 

Semdeg 

1 2 3 4 5 6 7 8 

0.3 0.52 0.53 0.39 0.25 0.19 0.35 0.23 

0.02 0.2 0.07 

42. gamoiyeneT niSnebis kriteriumi sinaTlisa da bgeriT signalebze 

adamianebis reaqciis drois monacemebis analizisaTvis (ix. magaliTi 34) [1]. 

43. gadawyvite wina magaliTis amocana uilkoksonis niSnebis rangebis 

jamebis kriteriumis gamoyenebiT. roca: a) n = 15 ; b) n = 17 [1]. 

44. gamoTvaleT adamianis sinaTlisa da bgeriT signalebze reagirebis monacemebis 

ZiriTadi ricxviTi maxasiaTeblebi (maT Soris korelacia) (ix. magaliTi 

34) [1]. 

45. cxrilSi moyvanilia avadmyofebisaTvis limfuri jirkvalis gazomvis Sedegebi 

ultrasonografiisa da kompiuteruli tomografiis meTodebiT. er- 

Ti da imave avadmyofs orive meTodiT gazomvebi utardeboda erTi da igive 

98

dros. SeamowmeT orive meTodiT miRebuli gazomvis Sedegebis identuroba: 

a) niSnebis kriteriumis meTodiT Sewyvilebuli dakvirvebebisaTvis; b) 

uilkoksonis niSnebis rangebis jamis meTodiT. 

gazomvis 

nomeri 

ultrasonografiakompiuterulitomografia 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

2.3 1.5 1.4 1.5 1.8 1.2 1.0 1.5 1.5 1.7 0 0 0 0 0 

1.8 1.8 1.7 1.5 1.5 1.5 1.3 1.7 1.6 1.6 1.8 0.7 1.5 0.8 1.2 

46. grafikuli meTodiT SeamowmeT hipoTezebi imis Sesaxeb, rom adamianis 

reaqciis dro sinaTlisa da bgeris signalebze (ix. magaliTi 34) ganawilebuli 

arian normaluri kanoniT [1]. 

47. cxrilSi moyvanilia or ganzomilebiani normalurad ganawilebuli 

fsevdo SemTxveviTi veqtoris generaciis Sedegebi. ipoveT or ganzomilebiani 

ganawilebis parametrebis Sefasebebi, aageT am parametrebis ndobis intervalebi, 

SeamowmeT am parametrebTan dakavSirebuli hipoTezebi. 

maTematikuri molodinis veqtori: 

a = [ 1.0000. 3.0000]. 

dispersiuli matrica 

┌ ┐ 

W = │ 2.000000 0.700000│; 

│ 0.700000 4.000000│ 

└ ┘ 

Ggenerirebuli monacemabi 

┌────╥──────────┬──────────┐ 

x , 

x , 

│ j ║ 1 j │ 2 j │ 

├────╫──────────┼──────────┤ 

│1 ║-8.075323 │ 1.851354 │ 

│2 ║ 1.866408 │ 0.934127 │ 

│3 ║ 0.506295 │ 4.112154 │ 

│4 ║-0.208784 │ 1.254827 │ 

│5 ║ 0.470386 │ 2.764051 │ 

│6 ║-0.809748 │ 3.434635 │ 

│7 ║-0.256132 │ 5.545604 │ 

│8 ║-1.316983 │ 2.559858 │ 

│9 ║ 2.556789 │ 1.783427 │ 

│10 ║ 1.684340 │ 1.825249 │ 

│11 ║ 2.283028 │ 4.774898 │ 

│12 ║ 1.421519 │ 1.779352 │ 

│13 ║-0.523746 │ 2.530648 │ 

│14 ║ 0.468836 │ 1.692398 │ 

│15 ║ 1.830686 │ 1.477231 │ 

│16 ║ 0.299683 │ 0.980654 │ 

│17 ║ 2.113039 │ 1.440862 │ 

│18 ║ 3.177876 │ 6.591503 │ 

│19 ║ 1.727140 │ 5.389623 │ 

│20 ║ 0.935191 │-1.391321 │ 

99 

│21 ║-1.026028 │ 1.345145 │ 

│22 ║-0.245524 │-0.960215 │ 

│23 ║-0.146793 │-1.679388 │ 

│24 ║ 2.197345 │ 3.455451 │ 

│25 ║ 2.279845 │ 3.775871 │ 

│26 ║ 0.677116 │ 1.335242 │ 

│27 ║ 1.743165 │ 3.270833 │ 

│28 ║ 3.391363 │ 3.040776 │ 

│29 ║ 4.315069 │ 0.460287 │ 

│30 ║ 1.249540 │ 1.765827 │ 

│31 ║-0.277846 │ 5.010329 │ 

│32 ║ 1.715998 │ 5.323771 │ 

│33 ║ 1.145639 │ 3.351479 │ 

│34 ║ 3.430488 │ 3.269623 │ 

│35 ║-1.028074 │-1.590042 │ 

│36 ║ 3.099870 │ 7.267988 │ 

│37 ║ 2.372273 │ 4.016786 │ 

│38 ║-0.738203 │ 1.255776 │ 

│39 ║ 3.220242 │ 3.993731 │ 

│40 ║ 1.967129 │ 5.068442 │ 

│41 ║ 1.282559 │ 1.706056 │ 

│42 ║-0.474923 │ 4.172657 │ 

│43 ║ 0.976219 │ 5.508640 │

│44 ║-0.631014 │ 2.215554 │ 

│45 ║ 2.226529 │ 6.405292 │ 

│46 ║-0.813716 │ 1.403541 │ 

│47 ║-0.357754 │ 1.435465 │ 

│48 ║ 1.243116 │ 2.913436 │ 

100 

│49 ║-0.346079 │ 1.246020 │ 

│50 ║ 1.750597 │ 4.146453 │ 

│ ║ │ │ 

└────╨──────────┴──────────┘ 

48. erTfaqtoruli dispersiuli analizis gamoyenebiT gamoikvlieT ramodenime 

katalizatoris gavlena garkveuli qimiuri reaqciis gamosavalze. cxrilSi 

mocemulia qimiuri reaqciis gamosavali produqtis monacemebi gramebSi [3]. 

dakvirvebis 

nomeri 

katalizatorebi 

A 1 A 2 

3 A A 4 

5 A 

1 3.2 2.6 2.9 3.7 3.0 

2 3.1 3.1 2.6 3.4 3.4 

3 3.1 2.7 3.0 3.2 3.2 

4 2.8 2.9 3.1 3.3 3.5 

5 3.3 2.7 3.0 3.5 2.9 

6 3.0 2.8 2.8 3.3 3.1 

jami 18.5 16.8 17.4 20.4 19.1 

49. oTx sxvadasxva laboratoriaSi xdeboda sami sxvadasxva tipis zRvis wylis 

gamomxdeli aparatis gamocda (ix. cxrili). yoveli konkretuli gamocda meordeboda 

samjer. gamocdis Sedegebi gamosaxulia narCeni wylis marilianobis 

procentebSi. aparatebi avRniSnoT asoebiT A 1 , A 2 , A 3 , xolo laboratoriebi – 

asoebiT B 1 , B 2 , 3 B , B 4 . miviReT ori faqtori: “aparatis faqtori” A da 

“laboratories faqtori” B . imisaTvis, rom avirCioT saukeTeso aparati, unda 

SevafasoT A faqtori; miRebuli Sedegebis swori interpretacia saWiroebs B 

faqtoris gaTvaliswinebas. miT umetes, rom laboratoriebs Soris gansxvaveba 

SeiZleba gamowveuli iyos adgilobrivi pirobebis gansxvavebiT (zRvis wylis 

tipi), romelic gaTvaliswinebuli uda iqnas. Bolos, A da B faqtorebs Soris 

SeiZleba iyos urTierTqmedeba, radgan erTi laboratoriisaTvis kargi 

aparati SeiZleba cudi gamodges sxva laboratoriisaTvis [3]. 

A 1 

A 2 

3 A 

A 

B 

B 3.6 3.8 4.1 2.9 3.1 3.0 2.7 2.5 2.9 

1 

B 4.2 4.0 4.1 3.3 2.9 3.2 3.7 3.5 3.6 

2 

B 3.8 3.5 3.6 3.6 3.7 3.5 3.2 3.0 3.4 

3 

B 3.4 3.2 3.2 3.4 3.6 3.5 3.6 3.8 3.7 

4 

50. cxrilSi moyvanilia sxvadasxva sofelSi ganTavsebuli fermerebis nakveTebis 

niadagebSi nitratebisa da fosfatebis koncentraciebis (mg/kg) gazomvis Sedegebi 

erTi da igive wlis oTx sxvadasxva TveSi. gamoikvlieT niadagebSi nitratebisa 

da fosfatebis Semcvelobebze gazomvebis drois da teritorialuri 

mdebareobis faqtorebis gavlena. 

soflis 

rigiTi 

nakveTis 

rigiTi 

ivlisi 

niadagebSi nitratebis Semcveloba. cxrili a. 

seqtemberi 

oqtomberi 

noemberi

nomeri nomeri 

1 

1 343.61 10.3 38.5 30.2 

2 34.54 15.5 40 25.2 

3 121.78 17 30.2 30.5 

4 306.42 17 15.5 12.8 

5 13.19 13.2 22.5 25.5 

2 

6 34.54 15.5 27.4 24.3 

7 34.54 13.2 22.7 25.2 

8 43.39 15.2 20.5 12.5 

9 11.64 11.6 19.5 21.3 

3 

10 19.48 20 20.5 22.2 

11 54.46 13.95 27.6 25.7 

4 12 96.97 16.8 30.5 28.7 

5 13 136.8 15.2 34.5 25.2 

6 

14 121.77 15.6 153.6 17 

15 136.82 20.3 160.2 150 

16 108.93 20 136.8 100 

soflis 

rigiTi 

nomeri 

1 

2 

3 

nakveTis 

rigiTi 

nomeri 

ivlisi 

niadagebSi fosfatebis Semcveloba. cxrili b. 

seqtemberi 

101 

oqtomberi 

noemberi 

1 1071 200 642 471 

2 183.6 100 580.2 200 

3 104 70 100.2 102.2 

4 91.8 190 104 60.5 

5 318 183 70 70 

6 269 189 55 40 

7 171 61 70.2 80.2 

8 104 30 90.3 70.2 

9 122 30.6 147 100 

10 318 120 147 90.8 

11 91.8 257 120.8 100.2 

4 12 355 30.5 200 170.5 

5 13 844 35 514 305 

6 

14 336 53 924 800 

15 226 52 200 170 

16 330 52 134 120 

51. cxrilSi mocemulia monacemebi, romlebic miRebulia y = 2 x + 3 funqcionalur 

damokidebulebaze N (⋅; 

0, 

4) 

normalurad ganawilebuli aditiuri xmauris 

damatebiT. cxrilSi mocemuli monacemebiT aRadgineT da gamoikvlieT wrfivi 

regresia. 

i 1 2 3 4 5 6 7 8 9 10 

x 0.5 1 1.2 1.4 2.3 2.7 6 6.2 7 8 

y 5.33 2.92 3.94 8.15 8.73 8.09 19.11 13.37 13.49 17.21

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redaqciiT, Tbilisi, 2000, 661 gv. 

64. doWviri b. albaTobis Teoria da maTematikuri statistika. leqciebis kursi 

ekonomikuri fakultetis studentebisaTvis, nawili I da II, Tbilisis universitetis 

gamomcemloba, Tbilisi, 1984. 

65. mania g. albaTobis Teoria da maTematikuri statistika. saxelmZRvanelo 

ekonomikuri fakultetis studentebisaTvis, Tbilisis universitetis 

gamomcemloba, Tbilisi, 1976. 

104

danarTi danarTi 1. 1. binomialuri ganawilebis zeda boloebis albaTobebi: 

n = 2 ( 1) 

10, 

p = . 05 (. 05) 

. 45, 

1/ 

3; 

n = 2 ( 1) 

25, 

p = . 50 [2]. 

rodesac B – s aqvs binomialuri ganawileba n da p 0 parametrebiT, 

cxrilebi ikiTxeba Semdegnairad: 

1. rodesac p . 5 ) , ( n 

p p = - Tvis moyvanilia 

0 < 

Pp { B ≥ b} 

. Tu moncemuli ( , n) 

0 

b - Tvis cxrilSi 0 

b – Tvis cxrilSi 0 p 

dabeWdilia α , maSin b α , n, 

p ) = b . 

( 0 

105 

p = - Tvis 

2. rodesac p . 5 boloebis albaTobebi SeiZleba miviRoT 

0 > 

Pp { 0 1 p0 

b 

0 = 

. 7{ 

B ≥ 8} 

= P.. 

3{ 

B ≤ 2} 

= 1− 

P.. 

3{ 

B ≥ 3} 

= 1− 

. 

0 . = p ) , ( n 

= . 5 

P. 5{ 

B b} 

= P. 

5{ 

B ≤ ( n − 

gantolebidan B ≥ b} 

= P − { B ≤ n − } . magaliTad, rodesac n = 10, 

p . 7, 

8 = b vpoulobT 

P 

3. rodesac 5 

6172 = 

. 3828. 

b - Tvis Sesabamisi ujredis Semcveloba cxrilSi 

p - Tvis aris ≥ b)} 

. Tu ( b , n) 

- Tvis cxrilSi 

p = . 5 - Tvis dabeWdilia α , maSin b (α , n, 

1/ 

2) 

= b . 

p 

= . 05

p 

= . 15 

106

p 

= . 30 

107

108

109

i λ 

danarTi danarTi 22. 

2 

puasonis ganawileba (albaTobebi λ e / i! 

− 

, gamravlebuli 106 - ze) 

110

111

112

113

114

115

116

117

danarTi danarTi danarTi 3. standartuli normaluri ganawilebis zeda bolos albaTobebi. 

cxrilSi mocemuli x - Tvis moyvanilia P( X ≥ x) 

, sadac X aqvs ganawileba 

N ( 0, 

1) 

. amrigad, Tu x iseTia, rom P ( X ≥ x) 

= α , maSin z = x 

118 

(α ) [2].

danarTi danarTi 4. pirsonis ganawilebis kvantilebi χ [3]. 

rodesac f > 30 

2 

1− p 

miaxloebiTi formuliT 

119 

2 

1− p 

χ - is mniSvneloba SeiZleba gamovTvaloT 

χ 

2 1 2 

1− p 2 f −1 

+ u1− 

p ) 

= ( 

2 

sadac u1 − p - aris standartuli normaluri ganawilebis kvantili. 

am kanonis albaTobebis ganawilebis funqcias aqvs Semdegi saxe: 

1 n / 2−1 

x / 2 

x e , x > 0, 

n / 2 

2 Γ( 

n / 2) 

sadac Γ (⋅) 

aris gama funqcia da is gansazRvrulia nebismieri realuri 

a ricxvisaTvis Semdegi formuliT: 

∞ 

∫ 

a−1 −x 

Γ( 

a) 

= x e dx . 

0 

,

120

danarTi danarTi 5. stiudentis ganawilebis kvantilebi 

121 

1 

2 

p t 

− 

[3].

am kanonis albaTobebis ganawilebas aqvs SEmdegi saxe: 

⎛ n + 1⎞ 

n+ 

1 

Γ⎜ 

⎟ 2 

− 

1 

2 

2 ⎛ ⎞ 

( ) 

⎝ ⎠ x 

fn 

x = ⋅ ⎜ 

⎜1+ 

⎟ 

nπ 

⎛ n ⎞ 

Γ⎜ 

⎟ 

⎝ n ⎠ 

⎝ 2 ⎠ 

danarTi danarTi danarTi 6. fiSeris ganawilebis kvantilebi F − p 

122 

1 [3]. 

am kanonis albaTobebis ganawilebas aqvs SEmdegi saxe: 

⎛ m + n ⎞ m 

Γ⎜ 

⎟ 

−1 

2 

2 

( ) 

⎝ ⎠ x 

fmn x = ⋅ , ( x > 0). 

m+ 

n 

⎛ m ⎞ ⎛ n ⎞ 

Γ⎜ 

⎟⋅ 

Γ⎜ 

⎟ ( x + 1) 

2 

⎝ 2 ⎠ ⎝ 2 ⎠

123

124

125

danarTi danarTi danarTi 7. mani – uitnis kriteriumis U statistikis zeda kritikuli 

mniSvnelobebi [57]. 

126

127

128

129

130

danarTi danarTi danarTi 8. uilkoksonis kriteriumis W statistikis qveda kritikuli 

mniSvnelobebi [29]. 

131

132

133

134

danarTi danarTi 9. amonarCeviT gamoTvlili korelaciis koeficientis ganwilebis 

kvantilebi [3]. 

1 

2 

p r 

− 

135

danarTi danarTi 10 10. 10 

uilkoksonis niSnebis rangebis jamebis statistikis zeda da 

qveda procentuli wertilebi 

136

137

ibeWdeba avtoris mier 

warmodgenili saxiT 

gadaeca warmoebas 17.10.2004. xelmowerilia dasabeWdad 19.10.2004. beWdva 

ofseturi. qaRaldis zoma 210X297 1/4. pirobiTi nabeWdi Tabaxi 8,5. 

saaRricxvo-sagamomcemlo Tabaxi 5. tiraJi 200 egz. SekveTa # 

gamomcemloba “teqnikuri universiteti”, Tbilisi, kostavas 77

q. yaWiaSvili avtomatizebuli marTvis modelebi. statistikuri modelebi

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