ixLHdkh

ixLHd;aul o;a; úYaf,aIKh lsÍfuka ;SrKj,gt<eöu i`oyd lrkq ,nk wOHhkh ixLHdkh f,iy`ÿkajkq ,efí'

o;a;

úp,H rdYshla yd iïnkaO w.hhka i`oyd o;a; hehslshkq ,efí'

ix.ykh

o;a; ckkh ù we;s iuia: moaO;sh ix.ykh f,iy`ÿkajkq ,efí'

Wod# ,xldfõ mdie,aj, isák 13 fY%aKsfha isiqkaf.aWi

ix.ykfhka o;a; ,nd .ekSu m%dfhda.slj wmyiqwjia:d j,oS ix.ykh ksfhdackh jk mßos f;dard .kq,nk ix.ykfha Wml=,lhlg ksheoshla hehs lshkq,efí'

ióËKhloS ,nd .;a o;a; wdfrdayK fyda wjfrdayKms<sfj<g ilia l< úg o;a; je, fyj;a o;a; wdj,shla,efí'

Wod# mdie,l 13 fYa%Ksfha isiqka 20 fofkl=g isákifydaor ifydaoßhka ixLHdj

1, 2, 1, 2, 3, 0, 4, 4, 0, 1, 2, 1, 3, 3, 2,3, 5, 2, 0, 1

ix.ykhla fyda ksheoshla ;=< tla tla o;a;h fhoS we;sjdr .Kk tys ixLHd;h f,i y`ÿkajhs ‘

tla tla m%Nskak o;a;h iy tajdfha ixLHd;h oelafjkmy; igyklg —wiuQys; ixLHd; jHdma;shla˜ hehs lshkq,efí'

by; WodyrKh ie,l+ úg"

ksÍËKh ixLHd;h

0 3

1 5

2 5

3 4

4 2

5 1

ksÍËK ixLHdj úYd, jk úg fuu wiuQys; ixLHd;jHdma;s ms<sfh< lsÍu wmyiq fõ' tu wjia:d j,oSo;a;hka iuQykh lrkq ,efí' fuu o;a; iuQyhkagmka;s m%dka;r hehs lshkq ,efí' mka;s m%dka;r weiqßkams<sfh< lrkq ,nk ixLHd; jHdma;s —iuQys; ixLHd;jHdma;shla˜ f,i ye`oska fõ'

Wod# my; oelafjkafka ridhk úoHdj úIhh i`oydmka;shl isiqka ,nd we;s ,l=Kq oelafjkixLHd; jHdma;shls‘‘

,l=Kq ixLHd;h

30 g wvq 2

30 - 39 8

40 - 49 20

50 - 59 6

60 - 69 2

70 - 79 2

mka;s iSud

m%dka;rhl by<u w.h tys by< iSudj f,io" my<uw.h my< iSudj f,io y`ÿkajhs‘

Wod# 30 - 39 m%dka;rfha by< iSudj } 39

my< iSudj } 30

mka;s udhsu

hï mka;s m%dka;rhl my< iSudj yd Bg wvqm%dka;rfha by< iSudj w;r yß ueo w.h tum%dka;rfha my< udhsu fõ'

Wod# 30 - 39 m%dka;rfha my< udhsu 𝑥 =29+30

2

mka;s ,l=K$ uOH w.h

uOH w.h =by< iSudj+ my< iSudj

2fyda

=by< udhsu+my< udhsu

2

o;a; yd f;dr;=re j, m%ia:dßl ksrEmKh

o;a; yd f;dr;=re m%ia:dßlj ksrEmKh l< yels l%ulsysmhls'

ia:ïN m%ia:dr

jg m%ia:dr

cd, f¾Lh

f¾Ld m%ia:dr

fldgq fl`os igyka

ixLHd; jHdma;shla ie,l+ úg tys o;a; jeä jYfhkatys uOH foig keUqre ù mj;S' fuh flakaøslm%jK;djh f,i y`ÿkajhs' flakaøsl m%jK;djh uekSui`oyd fhdod .kq ,nk ñkqï —flakaøsl m%jK;d ñkqï ˜f,i y`ÿkajhs'

flakaøsl m%jK;d ñkqï

iudka;r uOHkHh

Nß; uOHkHh

ud;h

iudka;r uOHkHh

wiuQys; o;a; i`oyd iudka;r uOHkHh

𝑥1, 𝑥2, 𝑥3, . . . 𝑥𝑛 f,i oS we;s o;a; iuQyfha uOHkHh"

𝑥 =𝑥1 + 𝑥2 + 𝑥3+ . . . + 𝑥𝑛

𝑛fuys 𝑛 hkq o;a; ixLHdj fõ'

Wod# 4, 6, 8, 10, 14, 24 ys uOHkHh fidhkak'

uOHkHh 𝑥 =4+6+8+10+14+24

6

= 11

wiuQys; ixLHd; jHdma;sh uOHkHh𝑥1, 𝑥2, 𝑥3, . . . 𝑥𝑛 hk w.hhka iys; o;a; iuQyhlixLHd; ms<sfj,ska 𝑓1, 𝑓2, 𝑓3, . . . , 𝑓𝑛 f,i oS we;s úg"

uOHkHh =𝑥1𝑓1+ 𝑥2𝑓2+ 𝑥3𝑓3+ ...+𝑥𝑛𝑓𝑛

𝑓1+ 𝑓2+ 𝑓3+ ...+ 𝑓𝑛= 𝑖=1𝑛 𝑥𝑖𝑓𝑖

𝑖=1𝑛 𝑓𝑖

Wod#𝒙𝒊 𝒇 fx

0 04 00

1 05 05

2 10 20

3 10 30

4 18 72

5 23 115

6 17 102

7 07 49

8 08 64

𝑓 = 100 𝑓𝑥 = 457

𝑥 = 𝑖=1𝑛 𝑥𝑖𝑓𝑖

𝑖=1𝑛 𝑓𝑖

=457

1000= 4.57

iuQys; ixLHd; jHdma;shl uOHkHh

fuysoS 𝑥 fjkqjg mka;s m%dka;rfha uOH w.h i,lhs'

Wod#

my; oelafjk ixLHd; jHdma;shg wod< o;a;hkaysuOHkHh fidhkak'

mka;sm%dka;rh

0 - 4 5 - 9 10 - 14 15 - 19 20 - 24 25 - 29 30 - 34

ixLHd;h 1 4 6 8 5 4 2

mka;s m%dka;rh ixLHd;h ( f ) uOH w.h (x) fx

-0.5 – 4.5 1 2 2

4.5 – 9.5 4 7 28

9.5 – 14.5 6 12 72

14.5 – 19.5 8 17 136

19.5 – 24.5 5 22 110

24.5 – 29.5 4 27 108

29.5 – 34.5 2 32 64

𝑓 = 30 𝑓𝑥 = 520

𝑥 = 𝑖=1𝑛 𝑥𝑖𝑓𝑖

𝑖=1𝑛 𝑓𝑖

=520

30= 17.33

o;a; ixLHdj yd tajd ixLHd;aulj úYd, jk úg fïwdldrhg uOHkHh .Kkh lsÍu wmyiq fõ' tjekswjia:dj,oS — fla; l%u˜ Ndú;d lrkq ,efí' fuysoSwod< úp,Hh fjk;a úp,Hhlg udre lrkq ,efí'

𝐴 hkq Wml,ams; uOHkHh o 𝑐 hkq ksh;hlao kï"

𝑑𝑖 =𝑥𝑖 −𝐴

𝑐f,i fla;khla Ndú;d lrkq ,efí'

𝑑 = 𝑑𝑖𝑛

𝑑 = 𝑥𝑖 − 𝐴𝑐𝑛

𝑑 =1

𝑐

𝑥𝑖 − 𝐴

𝑛

𝑑 =1

𝑐

𝑥𝑖 − 𝑛𝐴

𝑛

𝑑 = 𝑥 − 𝐴

𝑛

Wod# lv, úl=kkakl= osk 30 la ;=< lrk ,ofj<odfuka Tyqf.a ffoksl wdodhu oelafjk ixLHd;jHdma;shla my; oelafõ' 𝐴 Wml,ams; uOHkHh yd 𝑐

Ok ksh;hla jQ 𝑑 =𝑥 −𝐴

𝑐fla;kh wkqidrfhka

Tyqf.a ffoksl wdodhfï uOHkHh fidhkak'

ffokslwdodhuremsh,a

500 – 550 550 - 600 600 - 650 650 - 700 700 - 750 750 - 800

osk .Kk 2 5 10 7 4 2

ffokslwdodhu remsh,a

osk .Kk (f) 𝒙𝒊 𝒅𝒊 =𝒙𝒊 − 𝟔𝟐𝟓

𝟓𝟎

fd

500 – 550 2 525 -2 -4

550 – 600 5 575 -1 -5

600 – 650 10 625 0 0

650 – 700 7 675 1 7

700 – 750 4 725 2 8

750 - 800 2 775 3 6

𝑓 = 30 𝑓𝑑 =12

𝑑 = 𝑓𝑑

𝑓

𝑑 =12

30= 0.4

𝑥 = 625 + 50 𝑑

𝑥 = 625 + 50 × 0.4

𝑥 = 645

Nß; uOHkHh

o;a; j,g iudk jákdlula fkdmj;sk wjia:dj,oSo;a;fha jákdlu u; tajdg nrla mjrkq ,efí' fufiamjrk ,o Ndrhka weiqßka .Kkh lrkq ,nk uOHkHhNß; uOHkHh f,i y`ÿkajhs'

𝑥1, 𝑥2, 𝑥3, . . . 𝑥𝑛 hk o;a;j, jeo.;alu wkqj tajdgms<sfj,ska 𝑤1, 𝑤2, 𝑤3, . . . 𝑤𝑛 hk Ndrhka mejrE úg"

Nß; uOHkHh =𝑥1𝑤1+ 𝑥2𝑤2+ 𝑥3𝑤3+ . . .+𝑥𝑛𝑤𝑛

𝑤1+𝑤2+𝑤3+ . . .+ 𝑤𝑛

Wod#

mß.Kl l%shd lrjkakl=f.a ;k;=rla i`oyd mj;ajk ,o,sÅ; mÍËKhlg fmkS isá iqks,a" ir;a yd fmf¾rd kïwfmaËlhska ;sfofkl= tla tla úIhh i`oyd ,nd .;a,l=Kq my; oela fõ‘

whÿïlre bx.%Sis idudkH nqoaêh .Ks;h

iqks,a 35 75 90

ir;a 60 80 65

fmf¾rd 70 40 45

bx.%Sis" idudkH nqoaêh yd .Ks;h i`oyd ms<sfj,ska1, 2 yd 3 hk Ndrhka mejÍug kshñ;j we;akïfuu ;k;=r i`oyd iqÿiqu whÿïlre ks¾Kh lrkak'

iqks,af.a ,l=Kqj, Nß; uOHkHh =35 ×1+75 ×2+90 ×3

1+2+3= 75.83

ir;af.a ,l=Kqj, Nß; uOHkHh =60 ×1+80 ×2+65 ×3

1+2+3= 69.17

fmf¾rdf.a ,l=Kqj, Nß; uOHkHh =70×1+40 ×2+45 ×3

1+2+3= 47.5

jvd;au iqÿiq whÿïlre iqks,a fõ'

ud;h (𝑀0)

o;a; iuQyhl jeäu jdr .Kkla fhoS we;s o;a;hud;h f,i y÷kajhs¡

Wod( 2, 3, 3, 5, 5, 5, 5, 8

ud;h = 5

iuyr jHdma;s j,g ud; lsysmhla we;¡ ud; follawe;s jHdma;s —oaú ud;˜ jHdma;s f,io ud; follgjvd jeä .Kkla we;s jHdma;s —nyq ud; fyda wfkalud;˜ jHdma;s f,i y÷kajhs¡

we;eï jHdma;s i`oyd ud; ke;¡

wiuQys; ixLHd; jHdma;sh ud;h

wiuQys; ixLHd; jHdma;shl ud;h jkqfha WmßuixLHd;h iys; o;a;hhs¡

Wod(

ud;h = 2

𝒙 𝒇

1 10

2 14

3 11

4 08

5 05

iuQys; ixLHd; jHdma;sh ud;h

iuQys; ixLHd; jHdma;shl ud;h fiùu i`oyd my; iQ;%h fhdod .kS¡

= ud;h

𝑀0 = ud;h wh;a mka;s m%dka;rfha my< udhsu

∆1 = ud;h wh;a mka;s m%dka;rfha ixLHd;h;a Bg my< mka;s m%dka;rfha ^j.=fõ by<& ixLHd;h;a w;rfjki

∆2 = ud;h wh;a mka;s m%dka;rfha ixLHd;h;a Bg by< mka;s m%dka;rfha ^j.=fõ my<& ixLHd;h;a w;r fjki

𝑐 = mka;s m%dka;rfha ;ru$ m<,

𝑀0 = 𝐿 +∆1∆1 + ∆2

𝑐

Wod( <uqka iuQyhlf.a nr lsf,da.%Eï j,ska yd <uqkaixLHdj my; oela fõ¡

nrkg

45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75

<uqkaixLHdj

4 12 30 10 8 6

nrkg

<uqka ixLHdj(f)

45 – 50 4

50 – 55 12

55 – 60 30

60 – 65 10

65 – 70 8

70 – 75 6

ud; mka;sh

𝐿 = 55

∆1= 30 − 12 = 18

∆2= 30 − 10 = 20

𝑐 = 5

𝑀0 = 55 +18

18 + 20× 5

𝑀0 = 57.37