NCERT पाठ्यपुस्तक पाठ 14 - सांख्यिकी, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 14 - सांख्यिकी, कक्षा 10, गणित Class 10 Notes | EduRev

 Page 1


lkaf[;dh 285
14
14.1 Hkwfedk
d{kk IX esa] vki fn, gq, vk¡dM+ksa dks voxhZo`Qr ,oa oxhZo`Qr ckjackjrk caVuksa esa O;ofLFkr
djuk lh[k pqosQ gSaA vkius vk¡dM+ksa dks fp=kh; :i ls fofHkUu vkys[kksa] tSls naM vkys[k]
vk;r fp=k (buesa vleku pkSM+kbZ okys oxZ varjky Hkh lfEefyr Fks) vkSj ckjackjrk
cgqHkqtksa osQ :i esa fu:fir djuk Hkh lh[kk FkkA rF; rks ;g gS fd vki voxhZo`Qr
vk¡dM+ksa osQ oqQN la[;kRed izfrfuf/ (numerical representives) Kkr djosQ ,d dne
vkxs c<+ x, FksA bu la[;kRed izfrfuf/;ksa dks osaQnzh; izo`fÙk osQ ekid (measures of
central tendency) dgrs gSaA geus ,sls rhu ekidksa vFkkZr~ ekè; (mean), ekè;d
(median) vkSj cgqyd (mode) dk vè;;u fd;k FkkA bl vè;k; esa] ge bu rhuksa
ekidksa] vFkkZr~ ekè;] ekè;d vkSj cgqyd] dk vè;;u voxhZo`Qr vk¡dM+ksa ls oxhZo`Qr
vk¡dM+ksa osQ fy, vkxs c<+k,¡xsA ge lap;h ckjackjrk (cumulative frequency) vkSj lap;h
ckjackjrk lkj.kh dh vo/kj.kkvksa dh ppkZ Hkh djsaxs rFkk ;g Hkh lh[ksaxs fd lap;h
ckjackjrk oØksa (cumulative frequency curves), tks rksj.k (ogives) dgykrh gSa] dks fdl
izdkj [khapk tkrk gSA
14.2 oxhZo`Qr vk¡dM+ksa dk ekè;
tSlkfd ge igys ls tkurs gSa] fn, gq, izs{k.kksa dk ekè; (;k vkSlr) lHkh izs{k.kksa osQ ekuksa
osQ ;ksx dks izs{k.kksa dh oqQy la[;k ls Hkkx nsdj izkIr fd;k tkrk gSA d{kk  IX ls] ;kn
dhft, fd ;fn izs{k.kksa  x
1
, x
2
,. . ., x
n
 dh ckjackjrk,¡ Øe'k%  f
1
, f
2
, . . ., f
n  
gksa] rks bldk
vFkZ gS fd izs{k.k  x
1
,  f
1
 ckj vkrk gS_ izs{k.k  x
2
,  f
2
 ckj vkrk gS] bR;kfnA
lkaf[;dh
Page 2


lkaf[;dh 285
14
14.1 Hkwfedk
d{kk IX esa] vki fn, gq, vk¡dM+ksa dks voxhZo`Qr ,oa oxhZo`Qr ckjackjrk caVuksa esa O;ofLFkr
djuk lh[k pqosQ gSaA vkius vk¡dM+ksa dks fp=kh; :i ls fofHkUu vkys[kksa] tSls naM vkys[k]
vk;r fp=k (buesa vleku pkSM+kbZ okys oxZ varjky Hkh lfEefyr Fks) vkSj ckjackjrk
cgqHkqtksa osQ :i esa fu:fir djuk Hkh lh[kk FkkA rF; rks ;g gS fd vki voxhZo`Qr
vk¡dM+ksa osQ oqQN la[;kRed izfrfuf/ (numerical representives) Kkr djosQ ,d dne
vkxs c<+ x, FksA bu la[;kRed izfrfuf/;ksa dks osaQnzh; izo`fÙk osQ ekid (measures of
central tendency) dgrs gSaA geus ,sls rhu ekidksa vFkkZr~ ekè; (mean), ekè;d
(median) vkSj cgqyd (mode) dk vè;;u fd;k FkkA bl vè;k; esa] ge bu rhuksa
ekidksa] vFkkZr~ ekè;] ekè;d vkSj cgqyd] dk vè;;u voxhZo`Qr vk¡dM+ksa ls oxhZo`Qr
vk¡dM+ksa osQ fy, vkxs c<+k,¡xsA ge lap;h ckjackjrk (cumulative frequency) vkSj lap;h
ckjackjrk lkj.kh dh vo/kj.kkvksa dh ppkZ Hkh djsaxs rFkk ;g Hkh lh[ksaxs fd lap;h
ckjackjrk oØksa (cumulative frequency curves), tks rksj.k (ogives) dgykrh gSa] dks fdl
izdkj [khapk tkrk gSA
14.2 oxhZo`Qr vk¡dM+ksa dk ekè;
tSlkfd ge igys ls tkurs gSa] fn, gq, izs{k.kksa dk ekè; (;k vkSlr) lHkh izs{k.kksa osQ ekuksa
osQ ;ksx dks izs{k.kksa dh oqQy la[;k ls Hkkx nsdj izkIr fd;k tkrk gSA d{kk  IX ls] ;kn
dhft, fd ;fn izs{k.kksa  x
1
, x
2
,. . ., x
n
 dh ckjackjrk,¡ Øe'k%  f
1
, f
2
, . . ., f
n  
gksa] rks bldk
vFkZ gS fd izs{k.k  x
1
,  f
1
 ckj vkrk gS_ izs{k.k  x
2
,  f
2
 ckj vkrk gS] bR;kfnA
lkaf[;dh
286 xf.kr
vc] lHkh izs{k.kksa osQ ekuksa dk ;ksx = f
1
x
1
 + f
2
x
2
 + . . . + f
n
x
n 
 gS rFkk izs{k.kksa dh la[;k
f
1
 + f
2
 + . . . + f
n  
gSA
vr%] budk ekè; x fuEufyf[kr }kjk izkIr gksxk %
x
 =
1 1 2 2
1 2
+ + +
+ + +
n n
n
f x f x fx
f f f
;kn dhft, fd mijksDr dks laf{kIr :i esa ,d ;wukuh v{kj S ¹cM+k flxek
(capital sigma)º ls O;Dr djrs gSaA bl v{kj dk vFkZ gS tksM+uk (summation) vFkkZr~
x
 =
1
1
n
i i
i
n
i
i
f x
f
=
=
?
?
bls vkSj vf/d laf{kIr :i esa] x = 
S
S
i i
i
f x
f
fy[krs gSa] ;g le>rs gq, fd i dk eku 1
ls n rd fopj.k djrk gSA
vkb, bl lw=k dk fuEufyf[kr mnkgj.k esa ekè; Kkr djus osQ fy, mi;ksx djsaA
mnkgj.k 1 : fdlh LowQy dh d{kk X osQ 30 fo|k£Fk;ksa }kjk xf.kr osQ ,d isij esa] 100
esa ls izkIr fd, x, vad] uhps ,d lkj.kh esa fn, x, gSaA bu fo|k£Fk;ksa }kjk izkIr vadksa
dk ekè; Kkr dhft,A
izkIrkad (x
i
) 10 20 36 40 50 56 60 70 72 80 88 92 95
fo|k£Fk;ksa dh la[;k ( f
i
) 1 1 3 4 3 2 4 4 1 1 2 3 1
gy : ;kn dhft, fd ekè; Kkr djus osQ fy,] gesa izR;sd x
i
 ls mldh laxr ckjackjrk
f
i 
}kjk xq.kuiQy dh vko';drk gSA vr%] vkb, bu xq.kuiQyksa dks lkj.kh 14-1 esa n'kkZ,
vuqlkj ,d LraHk esa j[ksaA
Page 3


lkaf[;dh 285
14
14.1 Hkwfedk
d{kk IX esa] vki fn, gq, vk¡dM+ksa dks voxhZo`Qr ,oa oxhZo`Qr ckjackjrk caVuksa esa O;ofLFkr
djuk lh[k pqosQ gSaA vkius vk¡dM+ksa dks fp=kh; :i ls fofHkUu vkys[kksa] tSls naM vkys[k]
vk;r fp=k (buesa vleku pkSM+kbZ okys oxZ varjky Hkh lfEefyr Fks) vkSj ckjackjrk
cgqHkqtksa osQ :i esa fu:fir djuk Hkh lh[kk FkkA rF; rks ;g gS fd vki voxhZo`Qr
vk¡dM+ksa osQ oqQN la[;kRed izfrfuf/ (numerical representives) Kkr djosQ ,d dne
vkxs c<+ x, FksA bu la[;kRed izfrfuf/;ksa dks osaQnzh; izo`fÙk osQ ekid (measures of
central tendency) dgrs gSaA geus ,sls rhu ekidksa vFkkZr~ ekè; (mean), ekè;d
(median) vkSj cgqyd (mode) dk vè;;u fd;k FkkA bl vè;k; esa] ge bu rhuksa
ekidksa] vFkkZr~ ekè;] ekè;d vkSj cgqyd] dk vè;;u voxhZo`Qr vk¡dM+ksa ls oxhZo`Qr
vk¡dM+ksa osQ fy, vkxs c<+k,¡xsA ge lap;h ckjackjrk (cumulative frequency) vkSj lap;h
ckjackjrk lkj.kh dh vo/kj.kkvksa dh ppkZ Hkh djsaxs rFkk ;g Hkh lh[ksaxs fd lap;h
ckjackjrk oØksa (cumulative frequency curves), tks rksj.k (ogives) dgykrh gSa] dks fdl
izdkj [khapk tkrk gSA
14.2 oxhZo`Qr vk¡dM+ksa dk ekè;
tSlkfd ge igys ls tkurs gSa] fn, gq, izs{k.kksa dk ekè; (;k vkSlr) lHkh izs{k.kksa osQ ekuksa
osQ ;ksx dks izs{k.kksa dh oqQy la[;k ls Hkkx nsdj izkIr fd;k tkrk gSA d{kk  IX ls] ;kn
dhft, fd ;fn izs{k.kksa  x
1
, x
2
,. . ., x
n
 dh ckjackjrk,¡ Øe'k%  f
1
, f
2
, . . ., f
n  
gksa] rks bldk
vFkZ gS fd izs{k.k  x
1
,  f
1
 ckj vkrk gS_ izs{k.k  x
2
,  f
2
 ckj vkrk gS] bR;kfnA
lkaf[;dh
286 xf.kr
vc] lHkh izs{k.kksa osQ ekuksa dk ;ksx = f
1
x
1
 + f
2
x
2
 + . . . + f
n
x
n 
 gS rFkk izs{k.kksa dh la[;k
f
1
 + f
2
 + . . . + f
n  
gSA
vr%] budk ekè; x fuEufyf[kr }kjk izkIr gksxk %
x
 =
1 1 2 2
1 2
+ + +
+ + +
n n
n
f x f x fx
f f f
;kn dhft, fd mijksDr dks laf{kIr :i esa ,d ;wukuh v{kj S ¹cM+k flxek
(capital sigma)º ls O;Dr djrs gSaA bl v{kj dk vFkZ gS tksM+uk (summation) vFkkZr~
x
 =
1
1
n
i i
i
n
i
i
f x
f
=
=
?
?
bls vkSj vf/d laf{kIr :i esa] x = 
S
S
i i
i
f x
f
fy[krs gSa] ;g le>rs gq, fd i dk eku 1
ls n rd fopj.k djrk gSA
vkb, bl lw=k dk fuEufyf[kr mnkgj.k esa ekè; Kkr djus osQ fy, mi;ksx djsaA
mnkgj.k 1 : fdlh LowQy dh d{kk X osQ 30 fo|k£Fk;ksa }kjk xf.kr osQ ,d isij esa] 100
esa ls izkIr fd, x, vad] uhps ,d lkj.kh esa fn, x, gSaA bu fo|k£Fk;ksa }kjk izkIr vadksa
dk ekè; Kkr dhft,A
izkIrkad (x
i
) 10 20 36 40 50 56 60 70 72 80 88 92 95
fo|k£Fk;ksa dh la[;k ( f
i
) 1 1 3 4 3 2 4 4 1 1 2 3 1
gy : ;kn dhft, fd ekè; Kkr djus osQ fy,] gesa izR;sd x
i
 ls mldh laxr ckjackjrk
f
i 
}kjk xq.kuiQy dh vko';drk gSA vr%] vkb, bu xq.kuiQyksa dks lkj.kh 14-1 esa n'kkZ,
vuqlkj ,d LraHk esa j[ksaA
lkaf[;dh 287
lkj.kh 14.1
izkIrkad (x
i
) fo|k£Fk;ksa dh la[;k ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
;ksx Sf
i
 = 30 Sf
i
x
i
 = 1779
vc
S
=
S
i i
i
f x
x
f
 = 
1779
30
 = 59.3
vr%] izkIr fd;k x;k ekè; vad 59-3 gSA
gekjs nSfud thou dh vf/dka'k fLFkfr;ksa esa] vk¡dM+s brus cM+s gksrs gSa fd mudk
,d vFkZiw.kZ vè;;u djus osQ fy, mUgsa lewgksa esa ck¡V dj (oxhZo`Qr djosQ) NksVk fd;k
tkrk gSA vr%] gesa fn, gq, voxhZo`Qr vk¡dM+ksa dks] oxhZo`Qr vk¡dM+ksa esa cnyus dh
vko';drk gksrh gS rFkk bu vk¡dM+ksa osQ ekè; Kkr djus dh fof/ fudkyus dh
vko';drk gksrh gSA
vkb, mnkgj.k 1 osQ voxhZo`Qr vk¡dM+ksa dks pkSM+kbZ] eku yhft,] 15 osQ oxZ
varjky cukdj oxhZo`Qr vk¡dM+ksa esa cnysaA ;kn jf[k, fd oxZ varjkyksa dh ckjackjrk,¡
fu£n"V djrs le;] fdlh mifj oxZ lhek (upper class limit) esa vkus okys izs{k.k vxys
oxZ varjky esa fy, tkrs gSaA mnkgj.kkFkZ] vad 40 izkIr djus okys 4 fo|k£Fk;ksa dks oxZ
varjky 25&40 esa u ysdj varjky 40&55 esa fy;k tkrk gSA bl ijaijk dks è;ku esa
j[krs gq,] vkb, budh ,d oxhZo`Qr ckjackjrk lkj.kh cuk,¡ (nsf[k, lkj.kh 14-2)A
Page 4


lkaf[;dh 285
14
14.1 Hkwfedk
d{kk IX esa] vki fn, gq, vk¡dM+ksa dks voxhZo`Qr ,oa oxhZo`Qr ckjackjrk caVuksa esa O;ofLFkr
djuk lh[k pqosQ gSaA vkius vk¡dM+ksa dks fp=kh; :i ls fofHkUu vkys[kksa] tSls naM vkys[k]
vk;r fp=k (buesa vleku pkSM+kbZ okys oxZ varjky Hkh lfEefyr Fks) vkSj ckjackjrk
cgqHkqtksa osQ :i esa fu:fir djuk Hkh lh[kk FkkA rF; rks ;g gS fd vki voxhZo`Qr
vk¡dM+ksa osQ oqQN la[;kRed izfrfuf/ (numerical representives) Kkr djosQ ,d dne
vkxs c<+ x, FksA bu la[;kRed izfrfuf/;ksa dks osaQnzh; izo`fÙk osQ ekid (measures of
central tendency) dgrs gSaA geus ,sls rhu ekidksa vFkkZr~ ekè; (mean), ekè;d
(median) vkSj cgqyd (mode) dk vè;;u fd;k FkkA bl vè;k; esa] ge bu rhuksa
ekidksa] vFkkZr~ ekè;] ekè;d vkSj cgqyd] dk vè;;u voxhZo`Qr vk¡dM+ksa ls oxhZo`Qr
vk¡dM+ksa osQ fy, vkxs c<+k,¡xsA ge lap;h ckjackjrk (cumulative frequency) vkSj lap;h
ckjackjrk lkj.kh dh vo/kj.kkvksa dh ppkZ Hkh djsaxs rFkk ;g Hkh lh[ksaxs fd lap;h
ckjackjrk oØksa (cumulative frequency curves), tks rksj.k (ogives) dgykrh gSa] dks fdl
izdkj [khapk tkrk gSA
14.2 oxhZo`Qr vk¡dM+ksa dk ekè;
tSlkfd ge igys ls tkurs gSa] fn, gq, izs{k.kksa dk ekè; (;k vkSlr) lHkh izs{k.kksa osQ ekuksa
osQ ;ksx dks izs{k.kksa dh oqQy la[;k ls Hkkx nsdj izkIr fd;k tkrk gSA d{kk  IX ls] ;kn
dhft, fd ;fn izs{k.kksa  x
1
, x
2
,. . ., x
n
 dh ckjackjrk,¡ Øe'k%  f
1
, f
2
, . . ., f
n  
gksa] rks bldk
vFkZ gS fd izs{k.k  x
1
,  f
1
 ckj vkrk gS_ izs{k.k  x
2
,  f
2
 ckj vkrk gS] bR;kfnA
lkaf[;dh
286 xf.kr
vc] lHkh izs{k.kksa osQ ekuksa dk ;ksx = f
1
x
1
 + f
2
x
2
 + . . . + f
n
x
n 
 gS rFkk izs{k.kksa dh la[;k
f
1
 + f
2
 + . . . + f
n  
gSA
vr%] budk ekè; x fuEufyf[kr }kjk izkIr gksxk %
x
 =
1 1 2 2
1 2
+ + +
+ + +
n n
n
f x f x fx
f f f
;kn dhft, fd mijksDr dks laf{kIr :i esa ,d ;wukuh v{kj S ¹cM+k flxek
(capital sigma)º ls O;Dr djrs gSaA bl v{kj dk vFkZ gS tksM+uk (summation) vFkkZr~
x
 =
1
1
n
i i
i
n
i
i
f x
f
=
=
?
?
bls vkSj vf/d laf{kIr :i esa] x = 
S
S
i i
i
f x
f
fy[krs gSa] ;g le>rs gq, fd i dk eku 1
ls n rd fopj.k djrk gSA
vkb, bl lw=k dk fuEufyf[kr mnkgj.k esa ekè; Kkr djus osQ fy, mi;ksx djsaA
mnkgj.k 1 : fdlh LowQy dh d{kk X osQ 30 fo|k£Fk;ksa }kjk xf.kr osQ ,d isij esa] 100
esa ls izkIr fd, x, vad] uhps ,d lkj.kh esa fn, x, gSaA bu fo|k£Fk;ksa }kjk izkIr vadksa
dk ekè; Kkr dhft,A
izkIrkad (x
i
) 10 20 36 40 50 56 60 70 72 80 88 92 95
fo|k£Fk;ksa dh la[;k ( f
i
) 1 1 3 4 3 2 4 4 1 1 2 3 1
gy : ;kn dhft, fd ekè; Kkr djus osQ fy,] gesa izR;sd x
i
 ls mldh laxr ckjackjrk
f
i 
}kjk xq.kuiQy dh vko';drk gSA vr%] vkb, bu xq.kuiQyksa dks lkj.kh 14-1 esa n'kkZ,
vuqlkj ,d LraHk esa j[ksaA
lkaf[;dh 287
lkj.kh 14.1
izkIrkad (x
i
) fo|k£Fk;ksa dh la[;k ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
;ksx Sf
i
 = 30 Sf
i
x
i
 = 1779
vc
S
=
S
i i
i
f x
x
f
 = 
1779
30
 = 59.3
vr%] izkIr fd;k x;k ekè; vad 59-3 gSA
gekjs nSfud thou dh vf/dka'k fLFkfr;ksa esa] vk¡dM+s brus cM+s gksrs gSa fd mudk
,d vFkZiw.kZ vè;;u djus osQ fy, mUgsa lewgksa esa ck¡V dj (oxhZo`Qr djosQ) NksVk fd;k
tkrk gSA vr%] gesa fn, gq, voxhZo`Qr vk¡dM+ksa dks] oxhZo`Qr vk¡dM+ksa esa cnyus dh
vko';drk gksrh gS rFkk bu vk¡dM+ksa osQ ekè; Kkr djus dh fof/ fudkyus dh
vko';drk gksrh gSA
vkb, mnkgj.k 1 osQ voxhZo`Qr vk¡dM+ksa dks pkSM+kbZ] eku yhft,] 15 osQ oxZ
varjky cukdj oxhZo`Qr vk¡dM+ksa esa cnysaA ;kn jf[k, fd oxZ varjkyksa dh ckjackjrk,¡
fu£n"V djrs le;] fdlh mifj oxZ lhek (upper class limit) esa vkus okys izs{k.k vxys
oxZ varjky esa fy, tkrs gSaA mnkgj.kkFkZ] vad 40 izkIr djus okys 4 fo|k£Fk;ksa dks oxZ
varjky 25&40 esa u ysdj varjky 40&55 esa fy;k tkrk gSA bl ijaijk dks è;ku esa
j[krs gq,] vkb, budh ,d oxhZo`Qr ckjackjrk lkj.kh cuk,¡ (nsf[k, lkj.kh 14-2)A
288 xf.kr
lkj.kh 14.2
oxZ varjky 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100
fo|k£Fk;ksa dh la[;k 2 3 7 6 6 6
vc] izR;sd oxZ varjky osQ fy,] gesa ,d ,sls ¯cnq (eku) dh vko';drk gS] tks
iwjs varjky dk izfrfuf/Ro djsA ;g eku fy;k tkrk gS fd izR;sd oxZ varjky dh
ckjackjrk mlosQ eè;&¯cnq osQ pkjksa vksj osQafnzr gksrh gSA vr%] izR;sd oxZ osQ eè;&fcanq
(mid-point) ¹;k oxZ fpÉ (class mark)º dks ml oxZ esa vkus okys lHkh izs{k.kksa dk
izfrfufèk (representative) ekuk tk ldrk gSA ;kn dhft, fd ge ,d oxZ varjky dk
eè; ¯cnq (;k oxZ fpÉ) mldh mifj vkSj fupyh lhekvksa dk vkSlr fudkydj Kkr
djrs gSaA vFkkZr~
oxZ fpÉ =
+
2
mifj ox Z lhek fupyh ox Z lhek
lkj.kh 14-2 osQ lanHkZ esa] oxZ 10&25 dk oxZ fpÉ 
10 25
2
+
 ,  vFkkZr~ 17.5 gSA blh
izdkj] ge vU; oxZ varjkyksa osQ oxZ fpÉ Kkr dj ldrs gSaA ge bu oxZ fpÉksa dks lkj.kh
14-3 esa j[krs gSaA ;s oxZ fpÉ x
i
’s dk dke djrs gSaA O;kid :i esa oxZ varjky osQ oxZ
fpÉ x
i
 osQ laxr ckjackjrk  f
i 
 fy[kh tkrh gSA vc ge mnkgj.k 1 dh gh rjg] ekè;
ifjdfyr djus dh izfØ;k dh vksj vkxs c<+ ldrs gSaA
lkj.kh 14.3
oxZ varjky fo|k£Fk;ksa dh la[;k (f
i
) oxZ fpÉ (x
i
) f
i
x
i
10 - 25 2 17.5 35.0
25 - 40 3 32.5 97.5
40 - 55 7 47.5 332.5
55 - 70 6 62.5 375.0
70 - 85 6 77.5 465.0
85 – 100 6 92.5 555.0
;ksx S f
i
 = 30 S f
i
x
i
 = 1860.0
Page 5


lkaf[;dh 285
14
14.1 Hkwfedk
d{kk IX esa] vki fn, gq, vk¡dM+ksa dks voxhZo`Qr ,oa oxhZo`Qr ckjackjrk caVuksa esa O;ofLFkr
djuk lh[k pqosQ gSaA vkius vk¡dM+ksa dks fp=kh; :i ls fofHkUu vkys[kksa] tSls naM vkys[k]
vk;r fp=k (buesa vleku pkSM+kbZ okys oxZ varjky Hkh lfEefyr Fks) vkSj ckjackjrk
cgqHkqtksa osQ :i esa fu:fir djuk Hkh lh[kk FkkA rF; rks ;g gS fd vki voxhZo`Qr
vk¡dM+ksa osQ oqQN la[;kRed izfrfuf/ (numerical representives) Kkr djosQ ,d dne
vkxs c<+ x, FksA bu la[;kRed izfrfuf/;ksa dks osaQnzh; izo`fÙk osQ ekid (measures of
central tendency) dgrs gSaA geus ,sls rhu ekidksa vFkkZr~ ekè; (mean), ekè;d
(median) vkSj cgqyd (mode) dk vè;;u fd;k FkkA bl vè;k; esa] ge bu rhuksa
ekidksa] vFkkZr~ ekè;] ekè;d vkSj cgqyd] dk vè;;u voxhZo`Qr vk¡dM+ksa ls oxhZo`Qr
vk¡dM+ksa osQ fy, vkxs c<+k,¡xsA ge lap;h ckjackjrk (cumulative frequency) vkSj lap;h
ckjackjrk lkj.kh dh vo/kj.kkvksa dh ppkZ Hkh djsaxs rFkk ;g Hkh lh[ksaxs fd lap;h
ckjackjrk oØksa (cumulative frequency curves), tks rksj.k (ogives) dgykrh gSa] dks fdl
izdkj [khapk tkrk gSA
14.2 oxhZo`Qr vk¡dM+ksa dk ekè;
tSlkfd ge igys ls tkurs gSa] fn, gq, izs{k.kksa dk ekè; (;k vkSlr) lHkh izs{k.kksa osQ ekuksa
osQ ;ksx dks izs{k.kksa dh oqQy la[;k ls Hkkx nsdj izkIr fd;k tkrk gSA d{kk  IX ls] ;kn
dhft, fd ;fn izs{k.kksa  x
1
, x
2
,. . ., x
n
 dh ckjackjrk,¡ Øe'k%  f
1
, f
2
, . . ., f
n  
gksa] rks bldk
vFkZ gS fd izs{k.k  x
1
,  f
1
 ckj vkrk gS_ izs{k.k  x
2
,  f
2
 ckj vkrk gS] bR;kfnA
lkaf[;dh
286 xf.kr
vc] lHkh izs{k.kksa osQ ekuksa dk ;ksx = f
1
x
1
 + f
2
x
2
 + . . . + f
n
x
n 
 gS rFkk izs{k.kksa dh la[;k
f
1
 + f
2
 + . . . + f
n  
gSA
vr%] budk ekè; x fuEufyf[kr }kjk izkIr gksxk %
x
 =
1 1 2 2
1 2
+ + +
+ + +
n n
n
f x f x fx
f f f
;kn dhft, fd mijksDr dks laf{kIr :i esa ,d ;wukuh v{kj S ¹cM+k flxek
(capital sigma)º ls O;Dr djrs gSaA bl v{kj dk vFkZ gS tksM+uk (summation) vFkkZr~
x
 =
1
1
n
i i
i
n
i
i
f x
f
=
=
?
?
bls vkSj vf/d laf{kIr :i esa] x = 
S
S
i i
i
f x
f
fy[krs gSa] ;g le>rs gq, fd i dk eku 1
ls n rd fopj.k djrk gSA
vkb, bl lw=k dk fuEufyf[kr mnkgj.k esa ekè; Kkr djus osQ fy, mi;ksx djsaA
mnkgj.k 1 : fdlh LowQy dh d{kk X osQ 30 fo|k£Fk;ksa }kjk xf.kr osQ ,d isij esa] 100
esa ls izkIr fd, x, vad] uhps ,d lkj.kh esa fn, x, gSaA bu fo|k£Fk;ksa }kjk izkIr vadksa
dk ekè; Kkr dhft,A
izkIrkad (x
i
) 10 20 36 40 50 56 60 70 72 80 88 92 95
fo|k£Fk;ksa dh la[;k ( f
i
) 1 1 3 4 3 2 4 4 1 1 2 3 1
gy : ;kn dhft, fd ekè; Kkr djus osQ fy,] gesa izR;sd x
i
 ls mldh laxr ckjackjrk
f
i 
}kjk xq.kuiQy dh vko';drk gSA vr%] vkb, bu xq.kuiQyksa dks lkj.kh 14-1 esa n'kkZ,
vuqlkj ,d LraHk esa j[ksaA
lkaf[;dh 287
lkj.kh 14.1
izkIrkad (x
i
) fo|k£Fk;ksa dh la[;k ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
;ksx Sf
i
 = 30 Sf
i
x
i
 = 1779
vc
S
=
S
i i
i
f x
x
f
 = 
1779
30
 = 59.3
vr%] izkIr fd;k x;k ekè; vad 59-3 gSA
gekjs nSfud thou dh vf/dka'k fLFkfr;ksa esa] vk¡dM+s brus cM+s gksrs gSa fd mudk
,d vFkZiw.kZ vè;;u djus osQ fy, mUgsa lewgksa esa ck¡V dj (oxhZo`Qr djosQ) NksVk fd;k
tkrk gSA vr%] gesa fn, gq, voxhZo`Qr vk¡dM+ksa dks] oxhZo`Qr vk¡dM+ksa esa cnyus dh
vko';drk gksrh gS rFkk bu vk¡dM+ksa osQ ekè; Kkr djus dh fof/ fudkyus dh
vko';drk gksrh gSA
vkb, mnkgj.k 1 osQ voxhZo`Qr vk¡dM+ksa dks pkSM+kbZ] eku yhft,] 15 osQ oxZ
varjky cukdj oxhZo`Qr vk¡dM+ksa esa cnysaA ;kn jf[k, fd oxZ varjkyksa dh ckjackjrk,¡
fu£n"V djrs le;] fdlh mifj oxZ lhek (upper class limit) esa vkus okys izs{k.k vxys
oxZ varjky esa fy, tkrs gSaA mnkgj.kkFkZ] vad 40 izkIr djus okys 4 fo|k£Fk;ksa dks oxZ
varjky 25&40 esa u ysdj varjky 40&55 esa fy;k tkrk gSA bl ijaijk dks è;ku esa
j[krs gq,] vkb, budh ,d oxhZo`Qr ckjackjrk lkj.kh cuk,¡ (nsf[k, lkj.kh 14-2)A
288 xf.kr
lkj.kh 14.2
oxZ varjky 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100
fo|k£Fk;ksa dh la[;k 2 3 7 6 6 6
vc] izR;sd oxZ varjky osQ fy,] gesa ,d ,sls ¯cnq (eku) dh vko';drk gS] tks
iwjs varjky dk izfrfuf/Ro djsA ;g eku fy;k tkrk gS fd izR;sd oxZ varjky dh
ckjackjrk mlosQ eè;&¯cnq osQ pkjksa vksj osQafnzr gksrh gSA vr%] izR;sd oxZ osQ eè;&fcanq
(mid-point) ¹;k oxZ fpÉ (class mark)º dks ml oxZ esa vkus okys lHkh izs{k.kksa dk
izfrfufèk (representative) ekuk tk ldrk gSA ;kn dhft, fd ge ,d oxZ varjky dk
eè; ¯cnq (;k oxZ fpÉ) mldh mifj vkSj fupyh lhekvksa dk vkSlr fudkydj Kkr
djrs gSaA vFkkZr~
oxZ fpÉ =
+
2
mifj ox Z lhek fupyh ox Z lhek
lkj.kh 14-2 osQ lanHkZ esa] oxZ 10&25 dk oxZ fpÉ 
10 25
2
+
 ,  vFkkZr~ 17.5 gSA blh
izdkj] ge vU; oxZ varjkyksa osQ oxZ fpÉ Kkr dj ldrs gSaA ge bu oxZ fpÉksa dks lkj.kh
14-3 esa j[krs gSaA ;s oxZ fpÉ x
i
’s dk dke djrs gSaA O;kid :i esa oxZ varjky osQ oxZ
fpÉ x
i
 osQ laxr ckjackjrk  f
i 
 fy[kh tkrh gSA vc ge mnkgj.k 1 dh gh rjg] ekè;
ifjdfyr djus dh izfØ;k dh vksj vkxs c<+ ldrs gSaA
lkj.kh 14.3
oxZ varjky fo|k£Fk;ksa dh la[;k (f
i
) oxZ fpÉ (x
i
) f
i
x
i
10 - 25 2 17.5 35.0
25 - 40 3 32.5 97.5
40 - 55 7 47.5 332.5
55 - 70 6 62.5 375.0
70 - 85 6 77.5 465.0
85 – 100 6 92.5 555.0
;ksx S f
i
 = 30 S f
i
x
i
 = 1860.0
lkaf[;dh 289
vafre LraHk esa fn, ekuksa osQ ;ksx ls gesa S f
i
x
i  
izkIr gksrk gSA vr%] fn, gq, vk¡dM+ksa
dk ekè; 
x
, uhps n'kkZ, vuqlkj izkIr gksrk gS%
x
 =
1860.0
62
30
ii
i
f x
f
S
= =
S
ekè; Kkr djus dh bl u;h fof/ dks izR;{k fof/ (direct method) dgk tk
ldrk gSA
ge ns[krs gSa fd lkjf.k;ksa 14-1 vkSj 14-3 esa] leku vk¡dM+ksa dk iz;ksx fd;k x;k
gS rFkk buesa ekè; ifjdfyr djus osQ fy, ,d gh lw=k dk iz;ksx fd;k x;k gSA ijarq
bu nksuksa esa gesa ifj.kke (ekè;) fHkUu&fHkUu izkIr gq, gSaA D;k vki lksp ldrs gSa fd
,slk D;ksa gqvk gS vkSj buesa ls dkSu&lk ekè; vf/d lgh gS\ nksuksa ekuksa osQ varj dk
dkj.k lkj.kh 14-3 esa dh xbZ eè;&¯cnq dYiuk gSA 59-3 lgh ekè; gS] tcfd 62 ,d
lfUudV ekè; gSA
dHkh&dHkh tc  x
i
 vkSj  f
i
  osQ eku cM+s gksrs gSa] rks x
i
 vkSj  f
i
  osQ xq.kuiQy Kkr djuk
tfVy gks tkrk gS rFkk blesa le; Hkh vf/d yxrk gSA vr%] ,slh fLFkfr;ksa osQ fy,]
vkb, bu ifjdyuksa dks ljy cukus dh fof/ lkspsaA
ge  f
i
  osQ lkFk oqQN ugha dj ldrs] ijarq ge izR;sd  x
i
 dks ,d NksVh la[;k esa
cny ldrs gSa] ftlls gekjs ifjdyu ljy gks tk,¡xsA ge ,slk oSQls djsaxs\ izR;sd  x
i
essa ls ,d fuf'pr la[;k ?kVkus osQ ckjs esa vkidk D;k fopkj gS\ vkb, ;g fof/ viukus
dk iz;Ru djsaA
blesa igyk pj.k ;g gks ldrk gS fd izkIr fd, x, lHkh x
i
  esa ls fdlh x
i 
 dks
dfYir ekè; (assumed mean) osQ :i esa pqu ysa rFkk bls ‘a’ ls O;Dr djsaA lkFk gh]
vius ifjdyu dk;Z dks vkSj vf/d de djus osQ fy,] ge  ‘a’ dks ,slk  x
i
  ys ldrs
gSa tks  x
1
, x
2
, . . ., x
n  
osQ eè; esa dgha vkrk gksA vr%] ge a = 47.5 ;k a = 62.5  pqu ldrs
gSaA vkb,  a = 47.5  pqusaA
vxyk pj.k gS fd  a vkSj izR;sd x
i
 osQ chp dk varj d
i
 Kkr fd;k tk,] vFkkZr~
izR;sd  x
i  
ls ‘a’ dk fopyu (deviation) Kkr fd;k tk,A
vFkkZr~ d
i
 = x
i
 – a
= x
i
 – 47.5
rhljk pj.k gS fd izR;sd d
i
 vkSj mlosQ laxr f
i  
dk xq.kuiQy Kkr djosQ lHkh  f
i 
d
i  
dk
;ksx Kkr fd;k tk,A ;s ifjdyu lkj.kh 14-4 esa n'kkZ, x, gSaA
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