
vakrata

probability sidhaanthathilum sthithivivarakkanakkukalim, oru yadhaartha moolyamulla kramarahita veriyabilinte sharashariyekkurichulla probability vitharanathinte asamathvathinte alavukolaanu vakrata. vakrathayude moolyam positteev, poojyam, negatteevu allengil nirvachikkappedaathathaayir.
oru unimodal vitharanathinu, negatteevu scue saadhaaranayaayi vitharanathinte idathuvashathaanu "wall" ennu soochippikkunnu, positteev scue wall valathuvashathaanennu soochippikkunnu. oru wall neelamullathum matte wall kattiyullathumaya saahacharyangalil, vakrata oru lalithamaaya niyamam anusarikkunilla. udaaharanathinu, poojya moolyam ennaal sharashari baalansinte iruvashathumulla vaalukal mothatthil balance cheiyunnu ennaanu ardhamaakkunnathu; ithu oru samamiti vitharanathinte kaaryamaan, ennaal oru wall neelavum nertthathum mattonn cheruthum ennaal kozhuppullathumaya asamamaya vitharanathinum ithu baadhakamaan.
aamukham.
chuvadeyulla chithratthile rund vitharanangalum pariganikkuka. oro graphilum, vitharanathinte valathuvashathulla moolyangal idathuvashathulla moolyangalil ninnu vyathyasthamayi kurayunnu. eee tapering vashangale "vaalukal" ennu vilikkunnu, koodaathe oru vitharanathinu rund tharatthilulla vakrata eathaanennu nirnnayikkaan avaa oru drishya maarggam nalkunnu
oru dattaa seereesile vakrata chilappol graafikkaayi mathramalla, moolyangalude lalithamaaya parishodhanayiloodeyum nireekshikkappedaam. udaaharanathinu, 50 kendra moolyathil thulyamaayi vitharanam cheyyappedunna samkhyaa kramam (49,50,51) pariganikkuka. sharaasharikk valare thaazheyulla oru moolyam cherkkunnathiloode namukk eee sreniye negatteevu scude vitharanamaakki maattaan kazhiyum, ithu orupakshe negatteevu outlayar aayirikkam, udha.jii. (40,49,50,51). athinaal, shreniyude sharashari 47.5 aayi maarunnu, median 49.5 aanu. nonparametric scuvinte soothravaakyathe adisthaanamaakki, formula _ 1 aayi nirvachichirikkunna scue negatteevu aanu. athupole, sharaasharikk valare mukalilulla oru moolyam cherkkunnathiloode namukk sequence positteev aayi valachodikkaam, ithu orupakshe positteev outlayar aayirikkum, e.jii. (49,50,51,60), ivide sharashari 52.5 umm sharashari 50.5 umm aanu.
neratthe soochippichathupole, vakrathayude poojya moolyamulla oru unimodal vitharanam eee vitharanam anivaryamaayum samamithiyaanennu ardhamaakkunnilla. ennirunnaalum, oru cemetric unimodal allengil multimodal vitharanathinu ellaayppozhum poojya vakrathayundu.
sharaashariyudeyum madhyathinteyum bandham.
vakrata sharashariyum median thammilulla bandhavumaayi nerittu bandhappettirikkunnil negatteevu vakramulla oru vitharanathinu medianekkaal kooduthalo kuravo aayirikkam, athupole positteev vakrathinum.
formula 3 sharashari, formula 4 median, formula 5 staanderd deviation ennee formulakalaayi nirvachikkapetta nonparametric scue enna pazhaya aashayathil, eee bandhatthinte adisthaanathil scueness nirvachikkunnu positteev/raitt nonparametric scue ennaal medianekkaal (valathuvashath) valuthaanu, athesamayam negatteevu/lephtt nonparametric scue ennaal medianekkaal (idathuvashathu) kuravaanu. ennirunnaalum, vakrathayude aadhunika nirvachanathinum parambaraagatha nonparametric nirvachanathinum ellaayppozhum ore adayaalam illah vitharanagalude chila kudumbangalodu avar yojikkunnundenkilum, avaa chila kesukalil vyathyaasappettirikkunnu, avaye samyojippikkunnath thettidharippikkunnathu.
vitharanam samamithiyaanengil, sharashari medianu thulyamaanu, vitharanathinu poojya vakrathayundu. vitharanam samamitiyum unimodal aanenkil, sharashari = median = mooad. ithu oru nanayam toss cheiyunnathinteyo allengil 1,2,3,4 enna shreniyudeyo kaaryamaan.. ennirunnaalum, neremarichu pothuve sariyallennu shradhikkuka, i.iii. poojya vakrata (chuvade nirvachichirikkunnathu) sharashari medianu thulyamaanennu ardhamaakkunnilla.
2005leeii oru jenal lekhanam choondikkaanikkunnu pala paatapusthakangalum oru perumaattachattam padippikkunnu, athil sharashari valath vakrathinu keezhilulla meediyante valathum idathuvashathulla vakrathinu keezhilulla meediyante idathumaanu. eee niyamam athisayakaramaaya aavruthiyumaayi paraajayappedunnu. multimodal vitharanangalilo oru wall neelamullathum ennaal mattonn bharamullathumaaya vitharanangalilo ithu paraajayappedaam. ennirunnaalum, saadhaaranayaayi, meediyante idathuvashathum valathuvashathumulla pradeshangal thulyamallaatha vyathiriktha vitharanagalil niyamam paraajayappedunnu. atharam vitharanangal sharashari, median, scue enniva thammilulla paatapusthaka bandhathinu virudhamaanennu mathramalla, meediyante paatapusthaka vyaakhyaanathinum virudhamaanu.
udaaharanathinu, njangalude veedukaliludaneelam praayapoorthiyaaya thaamasakkaarude vitharanathil, valathuvashathaan vakrata. ennirunnaalum, bhooribhaagam kesukalum modinekkaal kuravo thulyamo aayathinal, sharashari idathuvashathu irikkunnu. thalphalamaayi, valath vakrathinu keezhilulla meediyante sharashari valath enna tamb niyamam parajayappettu.
nirvachanam.
mathsyathozhilaalikalude vakrathayude moment coffificient.
oru random variable "x"-nte scueness formula _ 6 moonnaamathe standardised moment formula _ 7 aanu, ithu ingane nirvachichirikkunnu
ivide "m" sharashari, "σ" staanderd deviation, e prathekshikkunna oopparettar, "m" 3 moonnaamathe kendra nimishavum "κ" t "" t "-th cumulantukalumaanu. ithine chilappol piyarsans moment coffificient oof scueness allengil kevalam moment coffificient oof scueness ennu vilikkunnu, ennaal piyarsante mattu scueness staattistikkukali thettidharikkaruthu (chuvade kaanuka). avasaanathe samathvam moonnaamathe kyumulant 'κ' 3-nteyum randaamathe kyumulant 'κ' 2-nte 1.5th powerinteyum anupaathathil vakrata prakadippikkunnu. ithu randaamathe cumulantinte chathuram normalise cheytha naalaamathe kyumulant enna nilayil kurtosisinte nirvachanathinu samaanamaan. 
vakrathaye chilappol vakrata ["x"] ennum soochippikkunnu.
σ" parimithamaanengil, "μ" parimithavum aanu, muun soothravaakyam vipuleekarikkunnathiloode vakramathvam kendramallatha moment e ["x" 3] yude adisthaanathil prakadippikkaan kazhiyum,
udaaharanagal.
vakrata ananthamaayirikkaam, eppol ennathupole
moonnaamathe kyumulantukal ananthamaayirikkumbozho allengil eppozhaanenno
moonnaamathe kyumulant nirvachikkappedaathidath.
parimithamaaya vakrathayodukoodiya vitharanagalude udaaharanangalil inipparayunnava ulppedunnu.
sample vakrata.
n" moolyangalude oru saambilinu, janasamkhyayude vakrathayude rund swaabhaavika esttimettarukal ivayaanu
oppam
ivide formula _ 14 sample sharashari, "s" sample staanderd deviation, "m" 2 (pakshapathaparamaaya) sample randaamathe kendra nimisham, "m" 3 sample moonnaamathe kendra nimisham. formula _ 15 ennathu moments esttimettarinte oru reethiyaanu. 
sample scueness" ennathinte mattoru pothuvaaya nirvachanam ithaanu
ivide formula _ 17 moonnaamathe kyumulantinte advitheeya samamiti pakshapatharahithamaaya esttimettarum formula _ 18 randaamathe kyumulantinte (i.iii. sample vyathyasam). eksalilum minitab, sas, spss ennivayulppede niravadhi stattistikkal paakkejukalilum kaanapedunna pathippaanu eee krameekaricha fisher-piyarsan standardised moment coffificient formula _ 19.
adisthaana kramarahithamaya variable formula _ 20 saadhaaranayaayi vitharanam cheyyappedunnu enna anumaanathinu keezhil, formula _ 21, formula _ 15, formula _ 23 ennee moonnu anupaathangalum formula _ 25, i ulla janasamkhyaa vakrata formula _ 24-nte pakshapaatharahithavum sthirathayullathumaaya esttimettarmaaraanenna kaanikkan kazhiyum.iii.avayude vitharanangal sharashari 0, vyathiyaanam 6 (fisher, 1930) ennivayulla oru saadhaarana vitharanathilekku othucherunnu. athinaal sample vakrathayude vyathyasam mathiyaya valia saambilukalkku ekadesham formula _ 26 aanu. kooduthal krithyamaayi paranjaal, oru saadhaarana vitharanathil ninnu "n" valuppathinte kramarahithamaya saambilil,
saadhaarana saambilukalil, formula _ 21 nuu moonnu esttimettarukalude cheria vyathyaasamund,
normal allatha vitharanagalkk, formula _ 21, formula _ 15, formula _ 23 enniva saadhaaranayaayi poppuleshan scueness formula _ 6-nte pakshapathaparamaaya estimettarmaaraanu; avayude prathekshikkunna moolyangalkku yadhaartha scunasil ninnu vipareeta adayaalam polum undakam. udaaharanathinu, 0.01,066,0.33 bhaarangalulla-99,0,02 enniva kendreekarichu valare nerttha gausiyans adangunna oru mishravitharanathinu ekadesham-9.77 nte vakrata soothravaakyam undu, ennaal 3 soothravaakyangalude oru saambilil _ 23 nuu ekadesham 0.32 prathekshikkunna moolyamundu, kaaranam saadhaaranayaayi moonnu saambilukalum vitharanathinte positteev moolyamulla bhagathaanu, athu mattoru reethiyil vakramaanu.
aaplikkeshanukal.
daattayudeyo vitharanathinteyo swabhaavasavisheshathakal vyakthamakkaan histogram saadhaarana kwandyle plot samyojippichu upayogikkaan kazhiyunna oru vivaranathmaka sthithivivarakkanakkaanu vakrata.
saadhaarana vitharanathil ninnulla vitharanathinte vyathiyaanathinte dishayeyum aapekshika vyapthiyeyum vakrata soochippikkunnu.
vyakthamaaya vakrathayodoppam, oru sharaashariyude aathmavishwaasa edavela polulla staanderd stattistikkal anumana nadapadikramangal thettayirikkumennu mathramalla, yadhaartha kavarej level naamamaathratthil ninnu vyathyasthamayirikkum (udha.jii.95 sathamaanam) level aanenkilum avaa oro vashathum asamamaya pishak saadhyathakalkk kaaranamakum.
cornish-fisher vipuleekaranathiloode vitharanagalude ekadesha saadhyathakalum alavukalum (dhanakaaryathil apakadasaadyathayulla moolyam polullava) nedaan vakrata upayogikkaam.
pala modalukalum saadhaarana vitharanam oohikkunnu; i.iii.daatta sharaashariyekkurich samamithiyaan. saadhaarana vitharanathinu poojyatthinte vakrathayundu. ennaal vaasthavathil, dattaa pooyintukal poornnamaayum samamithiyaayirikkilla. athinaal, daattasettinte vakrathayekkurichulla dhaarana soochippikkunnathu sharaashariyile vyathiyanangal positteev aano negatteevu aano ennaanu.
sample scunus, sample curtosis ennivaye adisthaanamaakkiyulla oru nanmayude phitt normality testtaanu dii 'augostinoyude keey-squared testt.
vakrathayude mattu alavukal.
kaal piyarsan nirddeshicha lalithamaaya kanakkukoottalukal ulppede vakrathayude mattu alavukal upayogichittundu (piyarsante moment coffificientaaya vakrathayumaayi thettidharikkaruthu, mukalil kaanuka). eee mattu nadapadikal ivayaanu
piyarsante aadyathe vakrata gunakam (mooad vakrata).
piyarsan mooad scueness allengil aadyathe scueness coffificient nirvachichirikkunnathu ingane
piyarsante randaamathe vakrata gunakam (median vakrata).
piyarsan median scueness allengil second scueness coffificient nirvachichirikkunnathu ingane
ithu nonparametric scuvinte lalithamaaya gunithamaan.
alavu adisthaanamaakkiyulla nadapadikal.
yoolinte gunakam (1912 muthal) ennum vilikkappedunna bouliyude vakrathayude alavu (1901 muthal) ingane nirvachikkappedunnu 

