saankhyiki

ek graaf jismein saamaanya vitran (Normal distribution) pradarshit hai.

saankhyiki ek ganiteeya vigyaan hai jismein kisi vastu/avayav/tantr/samudaaya se sambandhit aakadon ka sangrah, vishleshan, vyaakhya ya spashteekaran aur prastuti ki jaati hai. yeh vibhinn kshetron mein laagoo hai - academic anushaasan (academic disciplines), is se praakrutik vigyaan, saamaajik vigyaan, maanaviki, sarkaar aur vyaapaar aadi.

saankhyikeeya tareekon ko deta ke sangrah ke sangrahan athva varnan ke liye istemaal kiya ja sakta hai. ise varnanaatmak saankhyiki (descriptive statistics) kaha jaata hai. iske atirikt, deta mein paitarn ko is tarah se model kiya ja sakta hai ki vah nishkarshon ki yaadruchhikta aur anishchitta ka kaaran bane aur fir is prakriya ko us vidhi, ya jis janasankhya ka adhyayan kiya ja raha ho, uske baare mein anumaan lagaane ke liye kiya jaata hai. ise anumaanit saankhyiki (inferential statistics) kaha jaata hai. varnanaatmak tatha anumaanit saankhyiki, donon mein vyaavahaarik saankhyiki sammilit hai. ek aur vidya hai - ganiteeya saankhyiki (mathematical statistics), jo vishay ke saiddhaantik aadhaar se sambandh rakhati hai.

anukram

parichay

saankhyiki (Statistics) sabhyata ki gati mein ankon ka yogadaan bada hi mahatvapoorn raha hai aur ank paddhati ke vikaas ka bahut bada shreya Bhaarat ko praapt hai. manushya ke gyaan ki pratyek shaakha ankon ki rini hai.

saankhyiki ka vigyaan bhi bahut kuchh kaam ankon se leta hai, jinhein "aaainkade" kehte hain, parantu in ankon ke kuchh vishisht lakshan hote hain. staitistiks shabd ki vyutpatti ka pata lagaate samay iske naam mein aaj tak hue anek kraantikaari parivrtanon ko jaanakar aashcharya hota hai. praacheen kaal mein raajyon ke tulanaatmak varnan ke liye staitistiks shabd ka prayog hota tha, jismein ankon ya aaainkadon ka koi sthaan hi naheen hota tha. staitistiks shabd ka mool laitin shabd staitas (itaalavi bhaasha "staiti", jarman "staitistiks"") hai, jiska arth hai 'raajaneetik raajya'. 18veen shati tak is shabd ka arth kisi raajya ki visheshataaon ka vivran tha. ataev kuchh praacheen lekhakon ne staitistiks ko raajya vigyaan ke naam se niroopit kiya hai. kramash: is shabd ko maatraatmak saarthakata praapt hui aur do vibhinn arthon mein iska prayog chalta raha. ek or yeh ankon se niroopit "janm aur mrutyu aaainkade" jaise tathyon se aur doosari or ankaatmak aaainkadon se upayogi nishkarsh nikaalne ke vidhi nikaaya, arthaat vigyaan se sambandhit tha. 19veen shati ke antim kaal se hamein "ujjval, saamaanya, mad" aadi sheershakon mein bachchon ki saankhyiki jaise vivran milte hain, jinse is gyaan shaakha ki parimaanonmukhata (quantitative direction) spasht hoti hai.

is prakaar ham dekhte hain ki vaigyaanik paddhati ki vishisht shaakha ke roop mein saankhyiki ka siddhaant apekshaakrut Abhinav upaj hai. iska mool roop laaplaas aur gaaus ki krutiyon mein dhooaindha ja sakta hai, lekin iska adhyayan 19veen shati ke chauthe charan mein jaakar samruddh hua. gaaltan aur kaarl piyrsan ke prabhaav se is vigyaan mein vilkshan pragati hui aur aagaami teen dashakon mein is vigyaan ki aadhaar shilaaeain sadrudh ho gain. yeh kah dena uchit hai ki din-din nae nae kshetron mein prayukt hone vaale is vishay ki imaarat abhi teji se banan ki sthiti mein hai. shodh kaarya, vah bhi visheshat: saankhyiki ke ganiteeya siddhaant mein, aisi teji se ho raha hai aur nae tathya aisi teevr gati se saamane aa rahe hain ki un sabki jaankaari rakhana bhi kathin ho raha hai. maanav gyaan aur kriya ke vividh kshetron mein is vishay ki prayukti din-din badh rahi hai aur badi upayogi siddh ho rahi hai.

baahya vishv ki ulajhi hui jatiltaaon se niyamon ke parichaalan ka gyaan praapt karna vigyaan ke pramukh uddeshyon mein se hai, jisse kuchh maulik siddhaanton ke aadhaar par vividh praakrutik ghatnaaon ki vyaakhya ki ja sake. in niyamon ke parichaalan ke gyaan se hamein "kaaran" aur "prabhaav" ke sambandh mein jaankaari hoti hai. kisi suniyojit prayog mein ham praaya: kaaranon ki jatil paddhati ke sthaan par saral paddhati ki sthaapana kar sakte hain, jismein ek baar mein ek hi kaaran se paristhiti ka vichran karaaya jaata hai. yeh sambhavat: aadarsh sthiti hai aur bahut se kshetron mein is prakaar ka prayog sambhav naheen hai. udaaharan ke liye, prekshak saamaajik tathyon ka prayog naheen kar sakta aur use un paristhitiyon ko, jo uske vash mein naheen hain, jyon ka tyon lekar chalana padta hai.

saankhyiki anek kaaranon se prabhaavit aaainkadon se sambandhit hai. kaaranon ke janjaal se ek ke atirikt baaki sabhi kaaranon ko chhaaaintakar sulajhaana prayogon ka uddeshya hai. yeh sabhi sthitiyon mein sambhav na hone ke kaaran vishleshan ke liye saankhyiki mein kaaran samooh ke prabhaavaadheen aaainkadon ko sveekaar kiya jaata hai aur aaainkadon se hi yeh bhi jaanane ki koshish ki jaati hai ki kaun-kaun se kaaran mahatva ke hain aur inmein se pratyek kaaran ke parichaalan se prekshit prabhaav par kiska kitna asar pada hai. isi mein hamaare gyaan ki is shaakha ki vilkshan aur vishisht shakti hai, jisse iski samruddhi hui hai aur yeh praaya: sarvavyaapak ho gayi hai.

udaaharanaarth, maan lein ki gehooain ki upaj par vibhinn khaadon ka prabhaav hamein gyaat karna hai. iske liye yeh paryaapt naheen hai ki khaadon ki sankhya ke baraabar bhookhand chunakar, pratyek bhookhand mein ek-ek khaad ke upachaar se fasal ugaai jaae aur upaj mein jo antar ho, use khaad ke prabhaav ka maapak maan liya jaae; kyonki yeh siddh kiya ja sakta hai ki ek hi khaad ke prabhaav se bhinn-bhinn bhookhandon mein upaj bhinn hoti hai. bhookhandon mein upaj ki bhinnata ke kaaran anek hote hain. vibhinn maatra mein khaad ke prabhaav ka adhyayan kiya jaae, arthaat vibhinn talon, vibhinn faarmon aur vibhinn varshon mein prayog kiye jaaeain, to adhyayan aur bhi jatil ho jaata hai. lekin "vichran ka vishleshan" (Analysis of Variance) naamak vishisht saankhyik vidhi ke dvaara, jiska mukhya shreya aar.A. fishr (R.A. Fisher) ko hai, ham samagr vichran ko khandit karke, bhinn-bhinn kaaranon se vichran nikaalkar, vaidh nishkarshon par pahuainch sakte hain. aajkal krushi ke atirikt kai doosare kshetron mein bhi is pravidhi ka prayog ho raha hai.

vyashti ka adhyayan na karke, samashti naam se abhihit samooh ya samudaaya ka adhyayan karna saankhyiki vigyaan aur maulik dhaarana hai. iski paribhaasha ham vaigyaanik paddhati ki us shaakha ke roop mein kar sakte hain jo ginkar ya maapakar praapt samashtigt gunon ka, jaise kisi manushya varg ki ooainchaai ya bhaar se, kisi khaas ghaan mein nirmit dhaatu dandon ki tanaav saamarthya jaisi praakrutik ghatnaaon ke aaainkadon se, ya sankshep mein aavrutti kriya (repetitive operation) se praapt kisi bhi prayogaatmak aaainkade ka adhyayan karti hai.

at: saankhyikeevid ka pehla karttavya aaainkadon ka sangrah karna hai. yeh vah swayam kar sakta hai, ya anya uddeshya se ekatrit doosare aaainkadon ka prayog kar sakta hai. pehle prakaar ke aaainkadon par pradhaan aur doosare prakaar ke aaainkadon ko gaun kehte hain. aaainkadon ke prayog kar kisi parinaam par pahuainchane ke poorv, unki vishvasaneeyata ki jaaainch kar leni chaahiye.

saankhyikeeya adhyayan ka doosra kadam ekatrit aaainkadon ka vargeekaran aur saaraneekaran hai. yadi prekshanon ki sankhya adhik hai, to aaainkadon ka vargeekaran abheesht hi naheen, aavashyak bhi hai. sanghanan karte samay kuchh maatra mein soochanaaon ka tyaag karna padta hai. kintu mastishk vruhad ank raashi ka arth samajhne mein asamarth hota hai. at: aaainkadon se niroopit tathya ka adhimoolyan karne ke liye sanghanan aavashyak hai. sanghanan ke baad aaainkadon ko baaranbaarata-bantan-saarani ke roop mein niroopit karte hain.

is saarani se niroopak sankhyaaon ko, jo ekal sankhyaaeain hoti hain, pehchaanana saral hai aur maadhya (mean), maadhyamik (median), bahulak (mode) aadi se aaainkadon ki keindreeya pravrutti tatha maanak vichlan (standard deviation) dvaara aaainkadon ke apakirn aur vichran aadi gunon ko niroopit karte hain.

aaainkadon ko vakr rekhaachitron, chitralekhon (pictograms) aadi dvaara bhi prastut kiya ja sakta hai aur is prakaar ke prastuteekaran se praaya: mastishk ko aaainkadon ki vyaakhya, bhavishyavaani, anumaan aur ant mein poorvaanumaan (forecasting). kuchh saankhyikeevid poorvaanumaan ko saankhyikeevid ka kartavya naheen maanate, lekin adhikaansh maanate hain.

kisi janasankhya ki samashti ke adhyayan mein, pratyek sadasya ka alag-alag adhyayan, sankhya ki vipulata aur shram tatha laagat ke apavyaya ke kaaran, vyaavahaarik naheen thaharata. at: jan samudaaya ke sambandh mein gyaan praapt karne ke liye, ham sadasyon ke chayan ka, jinhein pratidrsh kehte hain, adhyayan karte hain. pratidrsh mool samashti ki jaankaari pradaan karta hai. soochana nirpeksh nishchitta ke roop mein ho, aisi aasha naheen ki ja sakti. ise praaya: sanbhaavita ke roop mein hi prakat karte hain. saankhyiki ke is bhaagako aaganan (estimation), kehte hain.

saankhyikeevid ko kuchh praathamik kaaryon ke liye, jaise sanchayan, vargeekaran, saaraneekaran, lekhaachitreeya upasthaapan (presentation) aadi ke liye vishisht prashikshan ke saath hi praarambhik ganit ki bhi aavashyakta hoti hai aur baad mein aaganan, anumaan aur poorvaanumaan ke liye uchch ganit aur sanbhaavita ke siddhaant ki sahaayata leni padti hai.

arthashaastr, samaaj vigyaan aur vaanijya ke kshetron mein, berojgaari badh rahi hai ya ghat rahi hai, bhavanon ki kami hai aur yadi hai, to kis seema tak, kuposhan ho raha hai ya naheen, sharaababandi se aparaadhon mein kami hui hai ya naheen, aadi prashnon ka samaadhaan saankhyiki ke dvaara hota hai.

janan vigyaan, jeev vigyaan aur krushi mein saankhyikeeya vidhiyon ka prayog ab anivaarya ho chala hai. jeev vigyaan mein ek nai shaakha jeev saankhyiki nikli hai, jiske antargat jeev vijnyaaaneeya vichranon ka saankhyiki adhyayan kiya jaata hai.

kuchh praagaitihaasik narakhopadiyaaain kisi ek maanav vigyaan ke jaati ki hain ya do vibhinn jaatiyon ki, maanav vigyaan ke is du:saadhya prashn ka hal nikaalne mein kaarl piyrsan ne sarvapratham saankhyiki ka prayog kiya tha.

manovijnyaaan aur shiksha ke kshetr mein vyaavasaayik prashikshan ke liye, maanav mastishk ka adhyayan karte samay, buddhi, vishesh yogyata aur abhiruchi aadi ke sandarbh mein saankhyikeeya takaneeki ki sahaayata li jaati hai.

chikitsa ke kshetr mein saankhyikeeya aaainkade aur vidhiyaaain donon hi param upayogi hain. mahaamaari vigyaan (epidemiology) aur jan svaasthya mein aaainkadon ki aavashyakta padti hai aur kisi nai aushadhi ya teeke (inoculation) ki dakshata ka pata lagaane ke liye aayurvejnyaaanik anusandhaan mein saankhyikeeya vidhiyon ke gyaan ki aavashyakta hoti hai.

jyotish, beema aur mausam vigyaan, saankhyiki ki laabhaprad yuktiyon ke anya kshetr hain. saankhyiki ka prayog yadaakada saahitya mein bhi hua hai. kuchh samay poorv tak aisi dhaarana thi ki bhautiki, rasaayan aur injeeniyri mein saankhyiki ki koi aavashyakta naheen hai. in yathaarth vijnyaaanon mein saankhyikeeya siddhaanton ke prayog se sachmuch bahut badi kraanti hui hai. saankhyikeeya gun niyantran, jo utpaadan injeeniyri ke antargat saankhyikeeya vidhiyon ka anukoolan hai, isi kraanti ki den hai. baadh niyantran, sadak suraksha, telephone, yaataayaat aadi ki samasyaaon mein saankhyikeeya pranaaliyon ka prayog safal raha hai.

bhavishya mein saankhyiki ka aur bhi vyaapak prasaar sambhav hai. kuchh vishyon ke liye yeh maulik mahatva ke vichaar aur kuchh ke liye anusandhaan ki shaktishaali vidhiyaaain, pradaan karti hai. bina vishay khandan ki aashanka ke kaha ja sakta hai ki saankhyiki sarvavyaapi vishay banta ja raha hai.

saankhyikeeya paddhatiyaan

praayogik tatha avlokan hetu adhyayan

ek saankhyikeeya anusandhaan pariyojana ke liye chhaanabeen karne ka saajha dhyeya hai aapad (causality) aur visheshataya bhavishyavaktaaon ke moolyon mein parivartan, athva svatantr charon (independent variable) ka anukriya athva aashrit charon (dependent variable) par kya prabhaav padta hai, iske baare mein nishkarsh nikaalna.aniyt saankhyikeeya adhyayan ke do pramukh prakaar hain, prayogaatmak adhyayan aur avlokan adhyayan.adhyayan ke donon prakaar mein, ek svatantr char (ya charon) ke matabhedon ka, ek aashrit char ke vyavahaar par asar ka avlokan kiya jaata hai. donon prakaar ke beech ka antar is baat par nihit hai ki vaastavik adhyayan ko kaise aayojit kiya jaata hai. nihit hai. koi bhi bahut prabhaavi ho sakta hai.

ek prayogaatmak adhyayan mein sammilit hai, iske antargat ki pranaali ka maap, is pranaali se chhedchhaad aur usi prakriya ka prayog kar, atirikt maap lena, yeh nirdhaarit karne ke liye ki kya pranaali se chhedchhaad ne maap ke moolyon mein sanshodhan kiya haiisake vipreet, ek paryavekshaneeya adhyayan mein prayogaatmak herafer shaamil naheen hai. iske bajaay, deta ekatrit kiya jaata hai aur bhavishyavaktaaon aur pratikriya ke beech ke sah sambandh ki jaaainch ki jaati hai.

ek praayogik adhyayan ka ek udaaharan hai prasiddh haavathorn prabhaav (Hawthorne studies), jisne pashchimi ilektrik company ke haavathorn sanyantr mein kaarya parivesh ko badalne ka prayaas kiya. shodhakarta yeh jaanana chaahate the ki kya roshani mein vruddhi se shramikon dvaara assembly line (assembly line) par ki gayi utpaadakata mein vruddhi hogi .shodhakartaaon ne sabse pehle to is plant ki utpaadakata ko naapakar, vahaan ke ek hisse ki roshani mein sanshodhan kiya aur yeh jaaainch ki, ki kya roshani mein parivartan ka utpaadakata par koi prabhaav pada.yeh ullekhaneeya hai ki vaastav mein, utpaadakata (prayogaatmak sharton ke tahat) behtar nikli.(dekho haavathorn prabhaav (Hawthorne effect).) haalaanki, is adhyayan ki aaj, prayogaatmak prakriyaaon mein trutiyon ke liye aalochna ki jaati hai, vishesh roop se, niyantran samooh (control group) ki kami aur andhepan (blindness) ke kaaran.

paryavekshaneeya adhyayan ka ek udaaharan hai, jo dhoomrapaan aur fefade ke Cancer ke beech sahasanbandh ka anveshan karta hai. is prakaar ke adhyayan mein aam taur par, ek sarvekshan ki madad se, jis kshetr mein dilchaspi ho, uske aankadon ko ikaththa kar, unka saankhyikeeya vishleshan kiya jaata hai. is maamale mein, shodhakartaaon ne dhoomrapaan karne vaalon aur na karne vaale, donon ke aankadon ko ikattha kiya hoga, shaayad case-control adhyayan (case-control study), ke maadhyam se aur fir pratyek samooh mein fefadon ke Cancer ke maamalon ki sankhya dhoondhi hogi.

ek prayog ke buniyaadi kadam hain;

  1. anusandhaan ki yojana, jismein shaamil hain, soochana strot tay karna, anusandhaan ke vishay ka chayan, tatha prastaavit anusandhaan ki vidhi aur naitik vichaar.
  2. prayogon ke design (Design of experiments) keindrit hoga system model par aur svatantr aur aashrit charon ki ant:kriya par.
  3. vistrut jaankaari ko chhupaate hue, tippaniyon ke sangrah ka saar nikaalna (Summarizing a collection of observations), jismein unki samaanataaon ko chinhit kiya gaya ho.(varnanaatmak aaainkade (Descriptive statistics))
  4. duniya ke baare mein avlokan par tippaniyon (the observations tell) ke baare mein aam sahamati tak pahunchana .(saankhyiki nishkarsh (Statistical inference))
  5. dastaavejeekaran / adhyayan ke parinaam prastut karna.

saankhyikeeya takaneek

anusandhaan (research)avlokan (observation) ke liye kuchh gyaat saankhyikeeya pareekshan (test) aur prakriyaaen (procedure) hain:

(Regression analysis)

vishisht vishay

jaaainch ke kuchh kshetron mein vyavahaarik saankhyiki ka upayog itna vyaapak hai ki unki ek vishisht shabdaavali hai. in vishyon mein shaamil hain:

  • saankhyiki vibhinn khelon mein, vishesh roop se besabaul (baseball) aur cricket (cricket)

aaainkadon ke saath vyaapaar mein ek mahatvapoorn aadhaar upakaran aur vinirmaan firm ka.iska upayog maap system parivrtaneeyata ko samajhne, niyantran prakriyaaon mein (jaise saankhyikeeya prakriya niyantran (statistical process control) athva S pi si), data ka saaraansh dikhne, tatha data-sanchaalit nirnaya lene ke liye kiya jaata hai. in bhoomikaaon mein yeh ek pramukh yantr hai aur shaayad akela vishvasaneeya upakaran bhi hai.

durupayog

yeh ek aam dhaarana hai ki prastutakarta saankhyikeeya gyaan ke durupyog (misused) ke liye, jyaadaatar aur jaanaboojh kar, aise raaste dhoondhate hain, jinse keval un aankadon ki vyaakhya karein, jo unke laabh ke anuroop hai beinjaamin disraili dvaara di ek prasiddh kahaavat hai, " jhooth teen prakaar ke hote hain: jhooth, shaapit, jhooth aur aaainkade (There are three kinds of lies: lies, damned lies, and statistics)"; aur Harvard ke pramukh Lawrence lovel (Lawrence Lowell) ne san 1909 mein likha tha ki aaainkade, veel paais ki tarah tab tak theek hain jab tak aap us vyakti ko jaante hon jisne unhein banaaya tha, tatha unke avayavon ke baare mein santusht hain..

yadi vibhinn adhyayan ek doosare ka khandan karte hon, tab aam janta ka aise adhyayanon par se vishvaas uth sakta hai. udaaharan ke liye, ek adhyayan se yeh pata chale ki koi aahaar ya gatividhi raktachaap ko badhaate hain, jabki koi doosra adhyayan yeh kahe ki vah raktachaap ko kam karte hain. aisi visangatiyaan prayogaatmak design mein maamooli her-fer dvaara uth sakti hain, jaise rogi samoohon, ya anusandhaan protokaul ke vah matabhed, jo aasaani se gair visheshagyon dvaara samajhe na ja sakein.(kabhi kabhi media reports aisi mahatvapoorn praasangik jaankaari ko poori tarah se najaraandaaj kar deti hain.)

ek nishchit namoone ko chunakar, (ya, khaarij ya sanshodhit kar), parinaam mein fer-badal kiya ja sakta hai. jaroori naheen ki is tarah ka jodtod durbhaavanaapoorn athva kutiltaapoorn ho; yeh shodhakarta ke anabhipret pakshapaat se utpann ho sakta hai. deta ka saaraansh dene vaale graaf bhi gumaraah karne vaale ho sakte hain.

is tathya ki gahan aalochna hui hai ki parikalpana pareekshan ka vah drushtikon, jiska vyaapak roop se prayog kiya jaata hai aur jo kai maamalon mein kaanoon ya viniym ke dvaara aavashyak hai, ek parikalpana (shoonya parikalpana (null hypothesis)) ki "tarafadaari" karta hai aur yeh ek bade adhyayan mein, kisi maamooli antar ke mahattv ko bhi badha-chadha kar dikha sakta hai. ek aisa fark jo uchch saankhyikeeya mahattv ka ho, tab bhi ho sakta hai ki vah bina kisi vyaavahaarik mahatva ka ho.(dekhein parikalpana pareekshan ki aalochna (criticism of hypothesis testing) aur shoonya parikalpana par vivaad (controversy over the null hypothesis).)

yeh aam report, ki ek parikalpana ko, mahatva ke diye gaye star par nakaar diya gaya, isse achhi pratikriya vah hogi, jismein pi-moolya (p-value) par jyaada mahattv diya gaya ho.yeh pi-moolya, baharahaal, is prabhaav ke aakaar ko naheen darshaata.ek doosra aam drushtikon hai vishvaas ke antaraal (confidence interval) ka varnan dena.haalaanki yeh bhi unhi gananaaon se praapt hote hain jaise parikalpana-testas athva pi-moolya, ve is prabhaav ke aakaar aur uske aas-paas ki anishchitta, donon ka varnan bhi karte hain.

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