Page 126 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_HINDI

vFkZ'kkL=k esa lkaf[;dh; fof/;k¡

uksV 8-4 i`Fkq'kh"kZRo vFkok doqQnrk vFkok f'k[kjh;rk (Kurtosis)

i`Fkq'kh"kZRo dk vFkZ (Meaning of Kurtosis)—xzhd Hkk"kk esa Kurtosis 'kCn dk vFkZ gS mHkjk gqvk mHkjkiu
(bulge or bulginess)A blh rjg lkaf[;dh eas Hkh i`Fkq'kh"kZRo izlkekU;rk osQ lUnHkZ esa vko`fÙk oØ dh f'k[kjh;rk
dk |ksrd gSA vr% izlkekU; oØ dh rqyuk esa fdlh vko`fÙk oØ osQ uqdhysiu ;k 'kh"kZRo (Peakedness) vFkok
piVsiu (Flatness) osQ eki dks i`Fkq'kh"kZRo dgrs gSaA

ØkWDlVsu ,oa dkmMsu osQ vuqlkj] ¶i`Fkq'kh"kZRo dk eki ml ek=kk dks O;Dr djrk gS ftlesa ,d vko`fÙk caVu
dk oØ uqdhyk ;k piVs 'kh"kZ okyk gksrk gSA¸ “(A measure of Kurtosis indicates the degree to which a
curve of a frequency distribution is peaked or flat topped.” Croxton & Cowden)
Lihxy osQ vuqlkj] ¶i`Fkq'kh"kZRo izk;% izlkekU; oØ dh rqyuk esa fdlh vko`fÙk oØ osQ uqdhysiu (peakedness)
dh ek=kk dk eki gSA¸ (Kurtosis is the degree of peakedness of a
distribution usually taken to a normal distribution.”—Spiegel) B

i`Fkq'kh"kZRo ls ;g irk pyrk gS fd ;fn fdlh Js.kh dks fcUnqjs[kk
A
(graph) ij vafdr fd;k tk, rks D;k og izlkekU; oØ gksxk vFkok
izlkekU; oØ ls vf/d piVk oØ gksxk vFkok izlkekU; oØ ls vf/ C
d uqdhyk oØ gksxkA vr% ;g crkrk gS fd Js.kh osQ eè; Hkkx esa
vko`fÙk;ksa dk teko (Concentration of frequencies in the middle (A) eè;e 'kh"kZ okyk
of the distribution) oSQlk gSA ;fn osQUnz esa vko`fÙk;ksa dk teko lkekU; (B) uqdhys 'kh"kZ okyk
gS rks og vko`fÙk oØ eè;e 'kh"kZ okyk ;k izlkekU; (Mesokurtic or (C) piVs 'kh"kZ okyk
Normal) dgykrk gSA ;fn vko`fÙk;ksa dk teko lkekU; oØ dh rqyuk
fp=k 8-2
esa Js.kh osQ eè; Hkkx esa cgqr vf/d l?ku :i ls osQfUnzr gS vFkkZr~ in
ewY; cgqyd osQ vkl&ikl l?kurk osQ lkFk xqfPNr gksrs gSa rks og oØ yEcs ;k uqdhys 'kh"kZ okyk (Lepto-
kurtic) dgykrk gSA osQUnz esa vko`fÙk teko cgqr de gksus ij og piVs 'kh"kZ okyk (Platy-kurtic or Flat) oØ
dgykrk gSA
fp=k 8.2 ls Li"V gS fd mÙkyrk (Convexity) dh n`f"V ls rhuksa oØksa esa vR;f/d vUrj gSA Curve A lkekU;
'kh"kZ] Curve B uqdhys 'kh"kZ ,oa Curve C piVs 'kh"kZ okyk oØ gSA
LVqMsUV (fofy;e ,l- xkSlsV) us bu oØksa dh piVs 'kh"kZ okys oØ dh rqyuk NksVh iw¡N vkSj piVh ihB okys
tkuoj IySfVil ls vkSj uksadnkj oØ dh rqyuk Å¡ps 'kh"kZ o yEch iw¡N okys daxk: ls dh gSA

i`Fkq'kh"kZRo dk eki (Measurement of Kurtosis)—i`Fkq'kh"kZRo dh eki prqFkZ ,oa f}rh; ifj?kkrksa osQ vk/kj
ij ifj?kkr vuqikr (moment ratio) }kjk dh tkrh gSA blosQ fy, dkyZ fi;lZu ls fuEu lw=k dk iz;ksx fd;k
gSμ
μ prqFkZ ifj?kkr Fourth moment
i`Fkq'kh"kZRo dh eki (β ) = 4 = =
2
μ 2 2 (f}rh; ifj?kkr) 2 (Second moment) 2
fuoZpu (Interpretation)—β dk ewY; 3 osQ cjkcj gksrk gS rks oØ lkekU; gksrk gSA β dk ewY; 3 ls vf/d
2
2
rFkk de gksus ij oØ lkekU; ugha gksrk gSA
;fn β = 3 rks oØ eè;e 'kh"kZ okyk (Meso-kurtic) gSA
2
;fn β > 3 rks oØ yEcs ;k uqdhys 'kh"kZ okyk (Lepto-kurtic) gSA
2
;fn β < 3 rks oØ piVs 'kh"kZ okyk (platy-kurtic) gSA
2
i`Fkq'kh"kZRo osQ eki osQ fy, γ dk Hkh iz;ksx fd;k tk ldrk gSμ
2

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