,Xir 1t.176 Monf;"

.":'!'7, The ~ta:ndard deviation of a symmetrical diS1ribution is .3. What inUit tJe$ value of the distribution about'ih~ mean in order that the distributiorl~ mesokurtic?

8. Given in a frequency distribution,

N = IOO,tjd =50,T.jd 2 =1967.2,lId3 = 292S.8,Ljd 4 =86650,2. '~l and ~2' F~i "\~ 9.. Thr first four moments about the value 4aro;: -1.5,1.5,17 and lO8.Find'.~?

first four moments (i) about the mean (ii) abo~t the origin. io. The first four moments about the the mean are 0, 3,2, 3.6 and 120 . inean is 11. Find the moment about the origin.

. . ,jlJ. The first four mome~1S ofa distribution about the value 4 are l.3.s, 10,5,4~ . ~ ~nd 282, Find the four central moments. , . . .~

12. The first three moments ofa distribution about the value 3 are 2,10 and ~ . . . ':~

Show that the first three moments about the originare 5;31 and 141. Also s') that the mean is 5, variance is 6 and fJ3 =-74.

1~ ':~~

11, If about the origin the first three moments moments are 3,24 and 76, shi that about the value 2, the first three moments are 1, 16 and-40. - .

' ; , , 1 14. If the first four moments about the value 4are 1,6,16 and 72 show thatJt ~Tst four central moments are 0,5,0 and 41. Comment on the nature ofdi':>'

tion. ,r~' ",

~lR~: 15. Given that the first moments ,abouUts moments of a distribut{!lr 0,120,0,36000, find PI and ~2' , '/

~\/j/ ~ Chapter 8

CorrelatibD.. ..

Measures of location, dispersion and skewness aie\cllP'aet~riStics of~a single variable concerning' a'statistitaJ' data. In, tbiSchapter, we jntroduceJhe concept of correlation ,wl.tich is :oneof the me~odsof sludying>thci relationship between tWo variables;. In statistiealanaJYsisWe come' across,the st\ld-y:'oftwo:variables wherein the 'change' ill the: Value 'of one variable'produ~s'a change in ,thevalue botbervanable..:lnthat .case we'; say tnat'the

Correlation

variable. Forexample, we should expect"zero correlation between wei~~ -of a person and ,the colour o(his hair or the height of a persollll.n4iti, colour ofhis hair. . Slmplt:C.orrelatlon,,;

( . Th~' C()~relation between'two variables is ~led si~ple corre1a~iqi; !J'he correlatIOn in the case of more than two vanables 15 called mulhpl.: correlation. .~\: Scatter diagram . . Let us ~o~ider~';'~et of~air~dY~~~ of the variables xandY"11 'example,...r rejllesentsthe heightsof:persons andy.their weights.A1o~, ;lhe horizontal axis we iepresent theheight ank4100g the'vertical~4'

(' ,~weight Plottllevalues(.r,y) ODa graphpaper;Weget a conectionofd~ Thelfiguresoiobtainedis;called'i scatter diagram., Fr.omthe scat_, diagram we. can:,obtain-a~rough idea of th~correlation' bttweeD,the".~

variables x andy. Ifall these dots' cluster around II ~e thecotrelatio~ called a linear correlation. If the dots closter.. around a curve,.,Ui

'correlatiou is called a non-linear or curve linear correlation. We can at' get an idea of wllether the correlation is positive Of ~gative fioti1 i scatter diagram~ They arc illustrated in thefonowing diagrams :

y

...

~: ~ ~. .' .., .../..' " .. . . . . .

:. ; .. .\. ".', .\:.~:;~.. ~~~.,~( , j.(

POllt~ilJMtrrClJ;,..tlolf .. . Neg'atlNLlMtIt'CfI"t1dt1l Fl.' 1 :,PiJoy 2 i, '.l' i.",>

I" \ . . -,\ .

. ... \ ...: ,. ~ ~' .

. I'tMltIN Ntllt-/_f;! r;O,I1',14J Fi

. &'... . ,'-',

Statiltic:iI7;_ ~ ""..-.

19 en f, .

Fi.' . /I'I"tl" PJDlt-llnltIt coiN.".,.:,

'I"y

!

, , o ' o '

:.,_ . -. '~~~~r~ ",". .. 7Fl. 6 ......~, FtB .

Ptrfett PolltiN .Perftet NelGtivt

U"eQT.Corrtllltlon Urrear Corrtl"tlon ..... ," .. ," . '" " . '.. it':

.. Scattetd1~amsi~~:.~o~~:~~~~iC)f.~e ~elationbetw~cm~ variables. It gwcs no iiif'ormatton abOut the degree of relabonsbip between the variables. For ~A.uantitative~e~urement of the degree of relationship between two variables ~lrCiliOQ has given the formula..

. .. -,..... , .. ~ ....",.,.

:. (' . ,\.1' ( . r = -..l!.

'.1", (1 (1' I Y

pis ca1led the product moment correlationbetwccnx andy and is defined by, '.,

_ 1: {X'~.x> (Y"~Yr . ( p - "N

and Ox and 0y .arc the standard deviations.of Xand Y respec:tively. p is also called covariance between X and Y. . .

Vf(Y-Yj~" (:=Vi(X~Xi ' .. = r' (1x 1\1' . , 0y N

.".

Corr~latioD

The formula for the correlation coefficien~ , can also be expressed. in the form, '

~ (X - X) (Y - 5), = V!:(x - X)2 ..; ~..{~ ~"~2

". ,':t

_ It is conventionally takena~x';;';X""'Xandy = . 1:xy,

wewnte, =;..; ... 2.:i:X . ~:~"':':~'\~'" ,:~c'AY:~'~ \;.}t,~,:(!:;\-,,~.,:,".~: 1.;.'~ ": ..:,:.~. ': ....

The above formula is eXpressed in terms of deviations ,of the variables from their means. Instead, if the actual values hf the

'obserVations are taken then the formala can be wrltten as. . . "

,.,,:! xY _ l:X.l:Y ; N ,

, = V"IX2 _-cj2 ' V}:~ _mt . , N .". N

NUY - :EX .l:Y =

. '"N IX2 - (L\')2 Vii1:~ '7,~n2 " htstelld of the-;deViations from their:lJieans,"We"deviations arc'

measured from the value A, and ,B for X and Y ',!('= XLA; dJ ~Y,~Bi,t~e ~t:r:~t~~if~~ffi~~ilU\~',~n'

". - .~ .. ~~ ... ~." ,'".' L: '-':~':~~-,:..:-,~:~ ~ ~, ..::..\ ", .:N~ .. i~.! '. ) ... ", ", ,".

r = Vw-~ VIdl-~N . N = .,. ...~~d)'-~ Ltr ~F.

"NW - (Idt)2. "NIdi _

Thus. we have the foUowiPg (Qrplulae for calculating the correlatiori;

coefficient between two vaiiablesx'lU1dy;

(1) ',_ .';9'. ' ( l:;~'

I

This fonil~l~ is used.when devialio.s 'are .me'alured from th~:~' mean. ';"

.l8f

"

I

Y - Yand hence

. .

. ~"

~l>les~y t!lking bj(, .,

.

. ";.' ,

(Idy)z:;: ..

.

s~tjca

_ N Uy - ~'-~.Y" r - " Nr.; _(Ex)l . VNri.- (1:.y)i

" .,. '.,\ .

ifno ,assuD1C?d aver~e ist.~e~ f9r x~qy~~,nes. .. . . " ,- .- ..... -' . !.-..... ;".

_ N Idxdy - La Idy r - .'VNId1- _ (~)1 ~.VN~._(Ljy)2

This formula is applied when deviationSfor,~and Yseries are taken ;from somcuslimedvatues. " ;';"

1~:"~iY.~~ fr~qUl:ricY~t~bU,ti~ofv~~plesxaD4YC9~elati9n 'coeffige~fisgiven bydi~'f~rintila;" .. ,'. ';"'" :,' ;', '. , _ N~fdxdy - Ifdx .l:fdy

, - , .. " .VN IliJXl ~ (I fdx)2 ....-JN 1: fdi - (1:. fdy)2 Numerical value of the c:~~..tlOD c:oem~eDt

.The ~ff1pc':lt ofcoITclatj9n r liesbetween ~ 1 an4 +1 inc1~ive. oftho~e valuCs. . coefficient caliedrabkeorrelation cOefficient.We rank' the obserVations . in ascending or descending order using the numbets 1.;2j 3, ;~; .....nand measure the degree bfrelationshipbetween the ranks' instead of actual numericalvalues. The rank correlation coefficientwhen there aren ranks in each variable is given by the formula (due toSpearman),

.;j~~~'~k~~ji:;'i').i':""-""

Correlation

fiEiil': .c;=L~" . . '. II (II~ -1),

,..f:'~:

':W ," ".I~}~,,~,

)~ '::5'

'"

':1;~.~~

d

,r l~,

\i'

where d =x - y is the"differericebetweeitranks oftOiTespOnding pairs ,.~~ ob, andy.. , . . . . '.,

. . .Ii = number of obselVatioDS.;~ .' Note: TIe nnks: When the values ofvariables x andy are Biven J

we caa'rank the values iii each of the variables,anddetermme: the:W(~=::zrc~~~~~~T~~1 correction (actor in the formula forp:'TheformUlil forpisgiVeli by, '. '.~

. 6['4'- + Ln

9 18

(M.U. B.Com. Sept. 1985)

x

1 2 3 4 5 6 7 8 9

45

x

ni 193..,,: ,::':1;;," "\ 4:',:: "'89:';;:"230'" 187

:f'.:' y:':, x-A~~ y-iJ.-F-~- ''''':2 :.i:'2 l#4y.~':::f .... - " .. ': ~~" , '. :'. "~i

IX)' 56 '; ". .', Ilf>

~~~ lI

Correlation

-306 - 2.08 = v'255-2.0~ .ff047~:"

- 308.08 - 308.08 ." = v'232;92 viOl =, ..... = ....,0.732.A .. '

Example 7: In order to find the coefficient of,correlation between

\-

=:. .,j~

1.. variables.t andy'from 12 pairs'ofobscn:@ons, the foJ)owi~ calculations:Were made: '.. ,:,1

/ .Lt == .~ ' Iy *'), ". ~'.::!:'670 , " '. I y7..::: '285, . 1:ty =334~jf On subsequent verifications, it. was.founrl th~(tht pair (t

y = 4)was copicd:'wrongly, the codect \llllue being. (x =1{),y =:-14). Fj' Ihe correct vaJucofthe correlation. coefficient (M.~. B.Com., May 19,' Solution: u =30 U =670 lXy = 334 .

Iy =.'5 Ii =285,.. .' N "';.12 Wrong pair = (x = 11,y = 4) Correct pair = ~ = 10,y == 14)

elm'eet values:

lX = 30 - 11 + 10 = 29 Iy = 5 - 4 + 14 ~. 15 U .= 670 - 121 + 100 ::::: 649 .Ii = 285 - 16 +,J96 = 465 Uy = 334 - 44 +,'140 = 430

'"'r ...f!. N = 12

r = --- ..' .....

. 2 '. 2VIx2 _ (lX) 'V Ii _ (Iy),,... "'i.. 'Y.:., N

.. 29 )( 15":; . ;.430 :-. ". ... 12";';":~" "

= v 649 _'292 / ~~\~ '152 .' Of .. '. 1~; '~'" 12

430 ....;.56.25 .'. ,;' = V649 - !'l[08~:::\.1'.~5~"t8:'5 - .:~393. 75, ;\ = ':393.75 d,O 775 - . -- 508.27'

d!Y

-1068

Statistics

-518 -180 -64

o 2'

,.-20 ':"'32 -72

-lQO -84

~ample,8 :fiQd the~fficie~tofcot:tel~fipnbet\veenoutputaJ,ld,~ost .. of an,.alitoIDobil~ fa~tory fJ:omthe folJ().wmg dilta :

Outp,utQf (;elI'S . '. '" ',: . (in Iholls~~~t: 3.5 4.2 5,;,~ 6:5' io 8.2 8;8 9.0 9.7 10.0 Cost of cars

(thous,lln~ Rs.): 9.8 9.0 8.8',8.4 8.3 8.2 8.2 '8.0 8.0 8.1 The correlation coefficient is unaffected by the change oforigin and

the scale. . Multiply outputs by 10 and then subtract 35. MUltiply the cost (in\ thousands of. Rs;) by Hi and s~btract 80. (M.V. B,Com. 1978)'

SeIU.~o~:' . '" .' . .

hJ89

x dx dy dy2, y .0.;

o 18 -37 14 . 1369' 196 7 . 10 -30 6 900 36

21 8 -1'6 4 256 16 30 4 ~7 . 0 '~49 o

.:. 4'35 3 -2 -1 1 47 2 10 .'. r2. 100 4 53 2 16- ''';fo'J 256 4 55 ,0 18 -4 324 16 62 ... o ',25 -4 625 16 65 1 28. .-3 ~. 9

375 48 58 4667 298 dx =x-37 dy=y-4

Idxdy _ Idx. Idy . N

r = y 'l:itr 2 .. r~I 2 . ,' ..IcW- "- ('j.il V '. IdY~';"~ ", .,'.. N

1(l68'':'' .5 x 8

V -10

=

52 y.... 82,: 4667 - - 298 ;,... 10 ., ..', 10

1068~A = v4667 =TI v298 - 6.4

-1072 -1072. = v4664.5 V292Jj = 11~.2. =0.91.

I

Correlation (1)1!'(.(' ExaDiple: 9 The following ,table 19ivesthc age~distributioQ i of' tltl

'JlOPufatioa 8.Ild,thenWhberofWlemJjloyed in a town. 'v

r--' Age . I Numberofperso;;;-1N~,fJet'(j/:;"';" .' ill '000 I unemployed

20-30 40 ~/400 30~40 '55 .1;0 . . 1,100' ~"'50,' ~2:1~ '" 96lJ' 50.;.60 20 s:o 1,600 60':';70 ',' :,~~: ",.. ' ;,1,600,' .

j; .. " .." ::.~:' ':>'"";1 :.'('.:!d:-': :..": '.';,~"':"'''' ":~";

Fmd the cOefficient of correlation r between the midvalues of thcj age groups and percentages ofunemployed m. different age-eonstitit~

. ." '" , .: '.~ Solutton : Let x represent the' 'age (mid~value of classes) and ;Y''

represent petcentage~eniplo~. x y) .dx dy dXJ dy2

25 l' ':';2 -4 4 '. 16 :1 35 2 -1 -3 1 9

45 3 -,-2. 40 ,~ 55 8 1 3 '1 9 65 20 2 15 '4 225

" c "

Z25 34 .0 9 10 ,263 ~.;" ~x.:.. 45

Letdx = 10 ,dy=y-S

." ,rdi':'i~\, -; .U:il!:dY","-'\ r , = V~_@r ';;~al-~:

N ","" ,." N

44 ... Ox'9\, = . ~S'

-J 10 - V263 _fl0 .. ,~S

:. 44 ' . v'IO ~. =-0.88

; ~-

"%J!' . ~~tiCs ii8umtl1e to: ThefQJlowiiJ.g;are the~ ranks obtained by 10 students in

i', '.' " SlatisticsandMatbeJ118lio;: Statistics: '12. 3:' 4' 56 7 8' 9' 10 Mathematics: . :.

1;4 .

2 5 '3 . .

9 7 10 6. 8

~ ... To what extent is the knowledge of students in the two subjects.related? .! .....'.' .

:;

, ;. I

X" ;Y x-y~d d2 ~: 2

1 4

0 ....2

0 4 I .

3 4 ""I,"

-1'5,

1 -1

" "" "~" " 1 1 I

5 3 2 4 6 7 ."

'. 9: 7'

-3 o ' ~..;..",

9 0

8 9

10., :. '6: ..'.

-2 .. 3

4 9

iO .. :, ,,~';;;/:.;:;., . 2" . 4 ~ ; : " '36

f~

~; ~: "\ ~""i1ie.~:~lmkC{;rielaHoniSgiveh~Y' ,." ..... I "ie;

,J': .,. p ':;;"u '1" ..'.:' .ni6';d~o'!' .. . "i, . , ." W(N~'''''1) :~ ~i;iij

6)(36 . , = ,1 ,-, iO x 99 = 1 - 0.219 = 0.781.

Example1.l: :.TJJi 'c.pmp~tiiciis' i~\~ beariiy cOntest are ranked 'by thre.e judges in th~ folloWing oIder : .... . ., '. ;lfrstjQdge:; 1 4 6 '3 2 9 /. '. 8 10: 5' Second-judge: ' 2 6 5 4 7 10 9' 3 8' l'

Thi~dJildge; .' ~. 7 4 5' Hi 8 9 2 6, 1 ,t,rs'~ the method of rank CQrrelation cOefficient to &eterminewhich

p.ir ofjildges have the nearest approach to cOmmon taste in beauty? Solution:

.,t:" ,"

Let,; y. t denote the ranks by ~st,2nd, and3rdjudges respectively

Comlalion-

Statistics x - y z dxy 'dyz dZi

1 2 ~3 ,,1 "':'1 6[T.d2 +m

":,'; . Ii,' q ... II

-~r I

1Il1!... )( ate

... 0

......

......

,;:!,

'j"~ . . : .

f

~.r

It .5

~. I;:

C"l

oJ.,o

o

v .....

o N

MI,;.', -I. . , .

-s ~ "

~ .......

~ ".; ~ ., S;, to-S 0\ ~ f' -0 " -N N "" ;.~ .M 01... '. . ' ... ~

--

I"' I I co ... '11"' N ..,. 00 M'0 .~ '"

00 .... M 'D 'It

Correlation

" ~ fi 0 ~ "/ "- t""I N N '4) ,''It1 ~. ...... ..... co M 0 ,-0 N I00' ~ ..... N N M ...'.:; I I

I"' 0 0 0 0 01'" 0 0 o '0." N ClCl N 0 t'l "., ,... N H ... 0 1 l! ::;t ..... M N tf'l." V 00 l"' I N N M

. I .. I...

-.;"

-, .. ., I I 'lI"'

N '"

co 0t- M .". I N N.."...... , I H

~~ I~ ,{- ..... ..M I"' t"l ~ S ~ ~ ~'0 'D I"' ~ I~ " '-, ~

)95.Business Statistics

'Example 15 : Find the CoeffiCient of Correlation for the follGwingdatli

I; "g ;~ ~;c,' z,,, ;i ;;J . (I . ;E Nov 94)

Solution: .X Y dx dy dx1 dy2 dxdy 33 17 40 28 60 30 19 32 83 38 95 49

392 194 .dx =x-65

r" i, ..,

,- JNr.dx1. -Cf.''I!:....J~dy1. _ (Idy)1. .' ', ~_.~~;:;... ",

'6xl178-'lx2 . . ", ' .....

= ~6x2970-22 J6x570-22 . = 7064 . . = '7068 = 0.9075

./17756 J3414 , 7788.1 EsaQlPle' 16 : I:tnd the coefficient ofCorrelation between r and y for the following set ofobServations: .

x I:. IS 20 25 . 30 35 ,40 . .9.Q.,~ _ WO.s, 120.7 140.9 _160.1 170.8

!. or

dy .'., .tJx2..x y di ill dX.iIx'3,. 15 90.5 -3 9.5 9 90.25 -28.5 20 1008 -2 0.8 ' 4 0.64 '-1.60 25 120.7 -1 20.7 .1 428.49

-1.68 i 30 1409 o '40.9 0 167.2&1 0

-,~ 160.1 ' 1 - ' . .'60.1 .. 3612.0135 1 . 60.1 40 110.8 2 70.8. 4 5012.64 . 141.6

165 783.8 -:-3 20Z.8 19' ." 10816.84 150.9

...,.30 . -:-15 900. 225 450 ,'-:25 .. -4 .... 625 16 100

-5 -2 25 4 10 14 0 0 0. 19~ ,.

. -,

18" ~

6 324 - '-36 108 30 17 900 289 510 2- 2- 2970" .'570 1178

dy =y-32 . "

~::-~J;~,':' ....

.J~

:'I~, ilY

197

::: 8x24-0xO ::: 24. :::0.603 .Ja x36 -o.Ja x 44 ..136 x 44

dx:::: x-63, dy.=,Y-69 ,.. ':mihiy- tdxUiy

r::: JNIdx2_(~)2 ~NIdy2 _(UJy)2

._ ..._.. ...

x y dx dy 'fix' dY dxdy 65 61 -3 -2 9' 4 6 66 68 -2 -1 4 1 2 61', 65 -1 .. .,..4. '. 1 " 16 ,4 67 ., .

.68 -1 -1 1 1 1 . 68 72 0 3 0 9 0 69 72 2 0 4 0 0

. 70 69 2 0 4 0 o .. n 71 4 2 16 4 . 8

344 552 0 0 36 . 44 24 . "'.

.. . ..~ .

xample 18 : Find the coefficient of Correlation for the following data : .I x I 65 66' 67 67 69 10 12 I Y 61 68.65' .68 72 69 11

(B.EBarathiy~r Univ. Nov. '95Kamaraj Unh'. Apr '96)

, EXIlJ.Dp1t: .9:and export using the follO\\iing data and comment on the result. Product (in crore tons): 27 Exorts(incrorelons):l7.

dy=y-l00

905.4 +623. 4 c 'J;:=11=4=_9=.Jr64=90=1.0=4=-=4=11=27=.S=4

_ 6x J.50.9 -(-3(207.8 - J6'x 19-(-3)2~~6x1081684-(202.8)2

= 1528.8 = 1528.8 ::::0.9246' JIlSJ23773,2" 1653.46' .'

x-30dx=- 5

NIdxdy-UixUiyr= .JNIdx2 _(Idr)2 ~~dy2 _(Itly)2

Let

x.

I 27 ..28 29

x y dx dy tJx2 elf dxdy . .- 1 9 -3 ... -2 9 4 6

2 8 ,;",2 -3 cr~ 9 6 3 10 -J . -1 0\ 1 0 4. 12 0 1 10 0

-0 :,'.

. '0" .,.

'4 . ., 9

. :;26,.:,::',

1%

Ex~ple 17 : Compute thecoefficicmtofcorrelation beiWeenx andy from the;; following data : ..

I 5 11 l' 0 tv '0 find tile cQ~fficient of -1 0 1 0 . .NI.dxdy - IdrIdy .

30 19 0 . 0 O' 0ji'.i; I,r::: JNIdx2_(r.dx)2~NIdldIdy)2 ;.~;

32 21 2 2 4 4 32 20 2 1 4 1

7x...26-0xJt 26' . 21 3 .2. 9 4::: =~=0.9286 I 33~h x 28';"0..17 x28-0 28 211 135 1 2 31 14

".:{

'\1:l":?: ..

" i:i'

,199.:$jkess Statistics198

;.J'J__ ' - ...." ....

d:t'=x-30dy=y-19. x y dx

35 32 ~3 34 30 -4 40 31 2 43 32 5 56 53 18 . 20 20 -18 38 33 0

266 . 231 0

dJil d.l dXdy'"~ -1 . 9 1 3'NI.dxdy.- r.dxUly.r ;: . -3, 16 9 12.~N!(lx2_('dx>"Z ~NLdi-(r.dy~2 -8 4 64 -16 -I 25 1 -51x 20-]x 2 138 138 '

= . ;:;:--;: 0.968$, 20 324 400 360 J7xjl'-'F~7;d4":'22 J216.jg4-142.49' '~ 20 324 400 3~O \~ r~: 0 0 0 0

Examllle2f) : Calculate Karl PearsOn's CoeffiCient of Correlation'frorn th~ 0 702 584 600 foilowing data, using 20 as the working mean for the pricemd 70 as the work~.

Let d'C =x- 38 dy:: y-33'ing.' mean for demand. " .. . ,~' .j"

. Price I 14 16 17 18 19 . 20 21 22 Nr.dxdv - r.dx l.dyr ;: ., Demand I 84 78 70 75' 66 . 67 62 58 .. ~ N~,fx2 ..., (r.dx) 2J t.,rr.dy2 ~ (r.c-M 2

Solution :. ~ 7)( 60 .

.r y di(x-20) dy(y-70) dfl dY .. dxdy ;: " ;: 0,937 ..: .....

.J? x702 ..17 x 884 14 84 -6 14 36 196 -84 , Example 22 : Calculate Karl pearson's coefficient of Correlation for the fol16 78 --4 8 16 64 -32

lowing data of prices and deriiand~~J~il05), (54.98), (85,53). (9U9). (59.84).17 70 -3 0 9 0 0 (95,40), (68,73), (29,59); (73,6.3):~72,52)18 75-2 .5 4 25 -10

Solution' I. 19 66 -1 -4 1 16 4 20 67 0 -J 0 9 0

..

21 62 1 -8 1 64 -8 2 '. -12 ..

_-2422 58 4 144 23.; 69 3 . .,...10" 9 .100 . , -30.

- ~-.

m ~ E .;"~le1.3 ; Find the Coefficient of Correlation betwee;X ~n~ y fro.~.]f ollowmg data : ,:~~ . :\~ n:: 10, Ex :: 60,: ~v:: 60,. Uy =305, :u-2 = 400, Ey 2 == 580 Of Solution:

NEdxdv - EdxEdyr= .

J,vIdx2 _(Ldx)2 ~lVr.dy2 _(ttry)2 ~

lOx 305 -60 x 60

-JH)X4.00;;'T60J2~10 ,

202 C9rrcl;t " ," .,;~

Example 28 : Find the coefficient ofCorrelation between age and playing h~'" from the following data :

_,liS Statistics ,_': PJ~ "t~:,_.ll.l.: - -. ,.. " . , . . ' '. _ . . ~.,~ . jjmpJe 29 : Calculate Karl Pearson's coefficient of Correlation}?~t,~~~11x It y from the bh'ariate sample of 140 pairs of x, andy as distributed below:

f~ ~" I .10-20 20-30' 30-40 40-50

'-20 20 26 -30 . 8 d4 37

,0-40 - 4 18 3 I-50

- - 4 6 -

Age(years) No. ofStudcIlts Regular Players 15-16 20C' 15016-17 270 16217-:-18 340 17018-19 360 18019-20 400 ISO20-21 200 120'. .' .... . (C.A Nov. tion :'

'Solution : Let x denote the age mid-v~ue and y denote the percentage of 15 25 35 45 f dy dents whopJav-' j:" i'_ ,\~ ~ 26 .... ... ; 120

- -46 -1

x

15,5 16.5 ,

17.5 .

18.5 ,.

19.5 .20.S

x y -2 15 ..:..1 12 .0 10 1 10 2 9 3 12 3 68

y 75

-60 -'

. ,~O

50 45 60

x'4 1 , 0

.

4 ; 9 -' ..

19.....

1

yz 225 144, 100 100 ,81 144 794

.

xy .. :-~p, : .. -12 :

-12 10 18 '

.; 36 .. 22

r = /ff.dxdy.,: 'Idxtdy.,. ; /vr.dx2 _(~)2 JNIdy~~~(}4y)2

IlS ~ 14 g37 59' 0 Let 24 lIS 43 ' 25 1

..'y Z4 i6 10 2 .y=-

. 5 28 -1

44 0

59 -; . 1

9 2

140 ",. 2

-28 0 59 18' 49 28 0 '59 '~.1> '~123 20. 0 26. '. 30 '76 ~, .

fdy .. -46

0 25 20 :-1

fdY 46 0 25 ' .

40 HI

ftdxdy 20

,0 . 24

32 96

J.. ...

I . ,_ ~dxdy - Ldxtdy . '"

r =-JNIdx2 - (Iih)2 JNIdy2.-,- (tdy)2 ',.. 140x76 .... 49-xl .

...::' .. ' ., . '.' =0705 .. .':... J140x 123-492 J140x 111-(-1)2 . .. .'

xample 30 :'From the following table ofbivariate frequency distribution caJ. ulatc the coefficient of Correlation between heights and weights ofchildi'en.

Height (in inches) ;'.' 'eight in pounds 40-44; 4~8 48-52 52-56 56-60 60264' Total

35-55 4 - 40 60' - - - 104

\:::1

t;~ !'O '}1! If

II,filt'" I: r-~ : ; ,

r'.

~f f f ! \ 1

6x22-,3x68 192,..204

=J6x 19- 9~6x79.-682.=.fU4-9 ..}4764-4624 55-75 75-95 24 88 8 12 32 -8 124 48 '

-396 ,-72-.,-=-'-=-,5939 .

95-115 '. 4 8 12 .J105lc 140 121.~4 115-135 4 4 8

P5-155 4 4 Total 4 40 84 100 52 20

A 9 11 12 ,8, 1 6 2 -54 B 10 14 J4 7 ,4' 5 2, -1 4

'..".'" . - '.' i'-, ~ .

C 15 12 16 ' 3, 6.5 3 ' 0-' ,3.5,0' D 14 " 13 ,15 .. ,A ,5 4 ,0 1 0, f E 16 10 17 2 8 2 0 6 0 36 F 11 15 ,10 6 3 8 -2 -5 4, 25 G 12 12 11 5 6.S ,7 -2 ' -0.5 4 0.25 H 17 16 18 1 2 1 0 1 0 1

i

16 '101.25 ;

6'f."i2 6x16 .; ~ ... p =1 =1---=0,81

.1)' ti(,,! -I) , 8 x63

206 (i) Rank each set of the data (ii) Calculate 2 appropriate rank correlation coefficients (iii) Write abrief report on your ::indings. Solution : Let x denote the score 'in edu.,Tblm ~r50ns :wij(.' . :~ , high edllcationaltest scoremay be preferred forj()b~.'However,: thereisa ne~ live rank correlation in :apptitude test:~~d a~s~srn~nHest ~c~~'~il~' h~~CI

'. \. , ... '-'" -'. ". p. ~ ...

apptitude test is not a good indicator ofjob performance.

Example 34 : The rnarksotnained by the students in physics and mathematic~y are as follows, ' Marks in phYsics: 35 23 47 17 10 43 9 (i 28. Marks in Maths : 3Q 33 45 23 8 4912 4 11 ~

Compute the ,ranks for t~e two subj:cts and the coefficient of Correlation of,~ ranks, '

"~ ~f".

,2fj7

',~~>: ,~ i1:~ ';~

!1~h!:'l 1ft 11 "

'1"'1I:",I

i:il ,~

'.

AJK IStal1stlcs"\~ ~h Calculate coefficient of correlation by the metho~QC.CONCURRENT DEVIATION

. concurrent deviation from the following data. . ". , , Method orcalculltting correlatloDcoefficient: Earlier we have seen 85 ,62 48 84 95 103 100 85 1152 different methods of.calculating correlation co-efficient$'. (i) Pearson's '~~ 23 19, 21 25 25 28 ,,27 26 30t,nethod (ii) Spearman's method, In Pearson's method the deviation~ ofx,~

and y values from their. means are considered for calculatil1g th '~i.~ correlation cOefficient. In spearman's method we consider only the ranks 1~ ill the variables x andy 'and the correlation between the ranks is Ii. calculated.. Now we consider another method of calculating theri correlation coefficient. Even thougli this, method may not measure the J~

~egree ofcoefficient as good as in the "ther two methods it gives in a weryJ simple Way the coefficient of correlation which will enable us to,note~' whether the variables are dependent or not. '

7 2

+ +

+ +

+

+

+

J)xDy

+

+

+

+

.+ '

Change in, direction .0/ variable Y

Dy

, ~ti';'cl1arige

20

y

25

23

25

26.

~,

21 19

-#} ..

-+ I', 30'

+

+,

+

,+

, , Disagreement =

"Q1.ange in direction of variable X

Dx

No. of concurrent deviation ==

r c

x

48

85

84

95

)~ 84

85t~ 62

115

100 103

Note:

Example 2:

vi Y

....'....:.... ~ / '''x 84 (y . Zo ::'S..olution :

..:v< - ~ V~(2C-N) , f;~Te - - - N , where Te is the coefficient of concurrent deviiltions, C is the number of concurrent deviations and N is the number of pairs of deviations compared. The sign of re is-determined as fol1ow~ If 2C - N is negative then' -' sign is taken both inside and outside the square root. In this case Te is negative.1f2C - N is positive then "+' sign is taken both inside and ou,tside the square root)rhe value of'Te 'will always lie between -1 and +, l.i.e., -1 :s rc :s 1.

= :to {i1iC-:::-N)Note: 'N' is not the number of pairs of observation but only the " N l~ .!i.number of pairs of deviations. i.e., N = n - 1where ',,' is the number of --.- " ---

Pitirs of observations. (2x7-9) = .1'... , '9~rits ' , , ,,'

. .+5 .. ,"(~) It is the simplest formula for calculating 'T'. V7}= .. +..,", ':'- == + .f.tlf.33' = 0.74I), " (2) It is easily determined by this method whether the variables are .". . dependent or not. 'UtheJ'~"isn9change in the direction for a pair of values o(x or

Demerits '~seriesthen'itwiD not be count.ed for ~ncurreritdeviation. (1) This method does not differentiate small and bigchanges in the values . fi~~tbe

210 ~

Solution:

X IChangeindirection of Y

Chimgeln direction of I DxDy

varillbleX variable Y Dx

60 ..... , 23 59 __ , ;':0 36 ,+ 72

,,'."":.

+' -',

10 , ., ~

51 -

, ~' ' + 55 .+ 3~ 54

-44

1 + 65 + 33

concurrent deviation Ie = 0 Disagreements '= I 6

r = ... h (2C-N) c . V=: N

= ,"';:i; (2 xO.- 6) , ', '6

_

/y -6 ,'.",

-t) - -, = .;..Vf.=' -1 .-

6 '"" ,,',

PROBABLE ERROR In a Dumber of situations webavec~cfulated. the correl

coefficient between two variables x and Y,for.example between t of husband and,t,hea~e ofwife, qte heighro,!fater~ctJ1eigh~:~f~'!l~~t calculate the correlation coe~:Ient we hafe t~kenas,a~R~~:~f;'~2:mJ ofvalues fro~ a lar~e populatton" Th~.~lT~!~!~gq~{fj .....'fljfi.d.. fromtht s~Wple maYQQt beth~ s!ll11~ a~tbe,,~~m:.ija;~~Q", population. It is possi~le .todeterminet~e\:,~ts' b~ coefficient ofcorrilanon oftbe p~put~(' .,fsample correlation coefficierlt'Prci~~l the con:elation.~~~~&(j" ,\> ,0'

Correlation , .. 2l1' :':' .. , .~..

whereS.E. is the standard error ofcorrelation coefficient and is given by l-i~'=7N'

The Iimlts for the population correlation coefficient are given by p ,= r:t P.E.

, where 'p' denotes the corielat1oneoefficient in thepopu1ation~The following facts fr!>m Probable ~rroi' are significant.

. (i) If the valu~or ,is less thaD the probable error, then there is no evidence of correlation. .'

,(ii) If the ~:Of"i~e ~a~ ~ix time~ the probable error, then the presence of co~elatlOn coeffiCIent IS certain. .

(iii)' Since 'T' lies'between.:";1 an~ +1 (-1 :$ .,. :s 1) the probable error is never negative. ' .. Note: The formula for P.E. is v~ued'only if (1) the ~pIechosen

dr is a simple random sample and (2) the popolation. is'noma!. :umple1: Compute the prob~le' error assuming the correlation

coefficient of0.9 from a sample of 25,pairs of items.. f'r$~luUoD: I' ... '0.9; n'" 25 . ' .. :'.~~ ": ., 2

., P.E. = 0.6745 x 1 - (0.9)

=0.6745 x 0.D38 = 0.0256. :~"ple~: 'If r}'= 0.8 and n ~ 64 find 'out the probable error ofthe

,L~::;,:;h\::,.-~fficient of correlation and determine the limits for the }t~(i~';lif)~~=::::' \,'....\ P;~, = 0.6745 X -""':!:=~

,;,;:"O:~745)(:04S: ...0.0304 "

."" .' ' .. , !~ftiqtf1:~are P.E; ~:''':.'- ".~~.;"::.':.' .~'I:.... :;:~/,::::>:;, ..:: .... ", . .