UNIT 6 CORRELATION COEFFICIENT

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1 UNIT CORRELATION COEFFICIENT Correlato Coeffcet Structure. Itroucto Objectves. Cocept a Defto of Correlato.3 Tpes of Correlato.4 Scatter Dagram.5 Coeffcet of Correlato Assumptos for Correlato Coeffcet. Propertes of Correlato Coeffcet.7 Short-cut Metho for the Calculato of Correlato Coeffcet.8 Correlato Coeffcet Case of Bvarate Frequec Dstrbuto.9 Summar. Solutos / Aswers. INTRODUCTION I Block, ou have stue the varous measures such as measures of cetral teec, measures of sperso, momets, skewess a kurtoss whch aalse varables separatel. But ma stuatos we are tereste aalsg two varables together to stu the relatoshp betwee them. I ths ut, ou wll lear about the correlato, whch stues the lear relatoshp betwee the two or more varables. You woul be able to calculate correlato coeffcet fferet stuatos wth ts propertes. Thus before startg ths ut ou are avse to go through the arthmetc mea a varace that woul be helpful uerstag the cocept of correlato. I Secto., the cocept of correlato s scusse wth eamples, that escrbes the stuatos, where there woul be ee of correlato stu. Secto.3 escrbes the tpes of correlato. Scatter agrams whch gve a ea about the estece of correlato betwee two varables s eplae Secto.4. Defto of correlato coeffcet a ts calculato proceure are scusse Secto.5. I ths ut, some problems are gve whch llustrate the computato of the correlato coeffcet fferet stuatos as well as b fferet methos. Some propertes of correlato coeffcet wth ther proof are also gve. I Secto. the propertes of the correlato coeffcet are escrbe whereas the shortcut metho for the calculato of the correlato coeffcet s eplae Secto.7. I Secto.8 the metho of calculato of correlato coeffcet case of bvarate frequec strbuto s eplore. Objectves After reag ths ut, ou woul be able to escrbe the cocept of correlato; eplore the tpes of correlato; escrbe the scatter agram; 5

2 Correlato for Bvarate Data terpret the correlato from scatter agram; efe correlato coeffcet; escrbe the propertes of correlato coeffcet; a calculate the correlato coeffcet.. CONCEPT AND DEFINITION OF CORRELATION I ma practcal applcatos, we mght come across the stuato where observatos are avalable o two or more varables. The followg eamples wll llustrate the stuatos clearl:. Heghts a weghts of persos of a certa group;. Sales reveue a avertsg epeture busess; a 3. Tme spet o stu a marks obtae b stuets eam. If ata are avalable for two varables, sa X a Y, t s calle bvarate strbuto. Let us coser the eample of sales reveue a epeture o avertsg busess. A atural questo arses m that s there a coecto betwee sales reveue a epeture o avertsg? Does sales reveue crease or ecrease as epeture o avertsg creases or ecreases? If we see the eample of tme spet o stu a marks obtae b stuets, a atural questo appears whether marks crease or ecrease as tme spet o stu crease or ecrease. I all these stuatos, we tr to f out relato betwee two varables a correlato aswers the questo, f there s a relatoshp betwee oe varable a aother. Whe two varables are relate such a wa that chage the value of oe varable affects the value of aother varable, the varables are sa to be correlate or there s correlato betwee these two varables. Now, let us solve a lttle eercse. E) What o ou mea b Correlato?.3 TYPES OF CORRELATION Accorg to the recto of chage varables there are two tpes of correlato. Postve Correlato. Negatve Correlato. Postve Correlato Correlato betwee two varables s sa to be postve f the values of the varables evate the same recto.e. f the values of oe varable crease (or ecrease) the the values of other varable also crease (or ecrease). Some eamples of postve correlato are correlato betwee

3 . Heghts a weghts of group of persos;. House hol come a epeture; 3. Amout of rafall a el of crops; a 4. Epeture o avertsg a sales reveue. Correlato Coeffcet I the last eample, t s observe that as the epeture o avertsg creases, sales reveue also creases. Thus, the chage s the same recto. Hece the correlato s postve. I remag three eamples, usuall value of the seco varable creases (or ecreases) as the value of the frst varable creases (or ecreases).. Negatve Correlato Correlato betwee two varables s sa to be egatve f the values of varables evate opposte recto.e. f the values of oe varable crease (or ecrease) the the values of other varable ecrease (or crease). Some eamples of egatve correlatos are correlato betwee. Volume a pressure of perfect gas;. Prce a ema of goos; 3. Lterac a povert a coutr; a 4. Tme spet o watchg TV a marks obtae b stuets eamato. I the frst eample pressure ecreases as the volume creases or pressure creases as the volume ecreases. Thus the chage s opposte recto. Therefore, the correlato betwee volume a pressure s egatve. I remag three eamples also, values of the seco varable chage the opposte recto of the chage the values of frst varable. Now, let us solve a lttle eercse. E) Eplore some eamples of postve a egatve correlatos..4 SCATTER DIAGRAM Scatter agram s a statstcal tool for etermg the potetalt of correlato betwee epeet varable a epeet varable. Scatter agram oes ot tell about eact relatoshp betwee two varables but t cates whether the are correlate or ot. Let (, ); (,,..., ) be the bvarate strbuto. If the values of the epeet varable Y are plotte agast correspog values of the epeet varable X the XY plae, such agram of ots s calle scatter agram or ot agram. It s to be ote that scatter agram s ot sutable for large umber of observatos..4. Iterpretato from Scatter Dagram I the scatter agram. If ots are the shape of a le a le rses from left bottom to the rght top (Fg.), the correlato s sa to be perfect postve. 7

4 Correlato for Bvarate Data Fg. : Scatter agram for perfect postve correlato. If ots the scatter agram are the shape of a le a le moves from left top to rght bottom (Fg. ), the correlato s perfect egatve. Fg. : Scatter agram for perfect egatve correlato 3. If ots show some tre a tre s upwar rsg from left bottom to rght top (Fg.3) correlato s postve. 8 Fg. 3: Scatter agram for postve correlato

5 4. If ots show some tre a tre s owwar from left top to the rght bottom (Fg.4) correlato s sa to be egatve. Correlato Coeffcet Fg. 4: Scatter agram for egatve correlato 5. If ots of scatter agram o ot show a tre (Fg. 5) there s o correlato betwee the varables. Fg. 5: Scatter agram for ucorrelate ata.5 COEFFICIENT OF CORRELATION Scatter agram tells us whether varables are correlate or ot. But t oes ot cate the etet of whch the are correlate. Coeffcet of correlato gves the eact ea of the etet of whch the are correlate. Coeffcet of correlato measures the test or egree of lear relatoshp betwee two varables. It was gve b Brtsh Bometrca Karl Pearso (87-93). 9

6 Correlato for Bvarate Data Note: Lear relatoshps ca be epresse such a wa that the epeet varable s multple b the slope coeffcet, ae b a costat, whch etermes the epeet varable. If Y s a epeet varable, X s a epeet varable, b s a slope coeffcet a a s costat the lear relatoshp s epresse as Y a bx. I fact lear relatoshp s the relatoshp betwee epeet a epeet varables of rect proportoalt. Whe these varables plotte o a graph gve a straght le. If X a Y are two raom varables the correlato coeffcet betwee X a Y s eote b r a efe as Cov(, ) r Corr(, ) () V() V() Corr(, ) Y. s cato of correlato coeffcet betwee two varables X a Where, Cov(, ) the covarace betwee X a Y whch s efe as: Cov (, ) ( )( ) a V () the varace of X, s efe as: V () ( ), Note: I the above epresso eotes the sum of the values for = to ; For eample meas sum of values of X for = to. If =.e. whch s equvalet to. If lmts are ot wrtte the summato epresso.e., whch cates the sum of all values of X. You ma f the scusse formulae wthout lmts ma books. We ca also wrte ( )( ) as ( )( ) N N a both have same meag. Smlarl, V () the varace of Y s efe b V () ( ) where, s umber of pare observatos. The, the correlato coeffcet r ma be efe as: 3

7 3 Correlato Coeffcet ) ( ) ( ) )( ( ) Corr(, r () Karl Pearso s correlato coeffcet r s also calle prouct momet correlato coeffcet. Epresso equato () ca be smplfe varous forms. Some of them are ) ( ) ( ) )( ( r (3) or r (4) or r (5) or r ().5. Assumptos for Correlato Coeffcet. Assumpto of Leart Varables beg use to kow correlato coeffcet must be learl relate. You ca see the leart of the varables through scatter agram.. Assumpto of Normalt Both varables uer stu shoul follow Normal strbuto. The shoul ot be skewe ether the postve or the egatve recto. 3. Assumpto of Cause a Effect Relatoshp There shoul be cause a effect relatoshp betwee both varables, for eample, Heghts a Weghts of chlre, Dema a Suppl of goos, etc. Whe there s o cause a effect relatoshp betwee varables the correlato coeffcet shoul be zero. If t s o zero the correlato s terme as chace correlato or spurous correlato. For eample, correlato coeffcet betwee:

8 Correlato for Bvarate Data 3. Weght a come of a perso over peros of tme; a. Rafall a lterac a state over peros of tme. Now, let us solve a lttle eercse. E3) Defe correlato coeffcet.. PROPERTIES OF CORRELATION COEFFICIENT Propert : Correlato coeffcet les betwee - a +. Descrpto: Wheever we calculate the correlato coeffcet b a oe of the formulae gve the Secto.5 ts value alwas les betwee - a +. Proof: Coser (Sce square quatt s alwas greater tha or equal to zero) ) Cov(, ) Cov(, a σ, Smlarl, σ X Varace of Sce Therefore, after puttg the values r r r (7) If we take postve sg equato (7) the r It shows that the correlato coeffcet wll alwas be greater tha or equal to-. If we take egatve sg equato (7) the r r It shows that correlato coeffcet wll alwas be less tha or equal to +. Thus r

9 If r, the correlato s perfect postve a f r correlato s perfect egatve. Propert : Correlato coeffcet s epeet of chage of org a scale. Descrpto: Correlato coeffcet s epeet of chage of org a scale, whch meas that f a quatt s subtracte a ve b aother X a quatt (greater tha zero) from orgal varables,.e. U a h Y b V the correlato coeffcet betwee ew varables U a V s k same as correlato coeffcet betwee X a Y,.e. Corr(, ) Corr(u, v). Correlato Coeffcet Proof: Suppose a a u a h b v the k a h u a a h u (8) a k v a a h v (9) where, a, b, h a k are costats such that a >, b >, h > a k >. We have to prove Corr(, ) Corr (u, v).e. there s o chage correlato whe org a scale are chage. a Smlarl, Cov (, ) Cov(, ) V () h V() h hk a h u a h ub k v b b v u uv v hkcov(u, v) V(u) V() k V(v) Corr(, ) a hu a h u u u Cov(, ) V()V(), Corr(, ) hkcov(u,v) h V(u)k V(v) 33

10 Correlato for Bvarate Data Corr(, ) Cov(u, v) V(u)V(v) Cor r(,) Corr(u, v).e. correlato coeffcet betwee X a Y s same as correlato coeffcet betwee U a V Thus, correlato coeffcet s epeet of chage of org a scale. Propert 3: If X a Y are two epeet varables the correlato coeffcet betwee X a Y s zero,.e. Corr(, ). Proof: Covarace betwee X a Y s efe b Cov(, ) (f varables are epeet the, ). Therefore, Thus, correlato s Cov(, ) Corr(, ) Cov(, ) V()V() V()V() As correlato measures the egree of lear relatoshp, fferet values of coeffcet of correlato ca be terprete as below: Value of correlato coeffcet Correlato s + Perfect Postve Correlato - Perfect Negatve Correlato There s o Correlato -.5 Weak Postve Correlato.75 - (+) Strog Postve Correlato.5 - Weak Negatve Correlato.75 - ( ) Strog Negatve Correlato 34 Let us scuss some problems of calculato of correlato coeffcet.

11 Eample : F the correlato coeffcet betwee avertsemet epeture a proft for the followg ata: Avertsemet epeture Correlato Coeffcet Proft Soluto: To f out the correlato coeffcet betwee avertsemet epeture a proft, we have Karl Pearso s formula ma forms [(), (3), (4), (5) a ()] a a of them ca be use. All these forms prove the same result. Let us take the form of equato (3) to solve our problem whch s r Corr(, ) Steps for calculato are as follow: ( )( ( ) ) ( ). I colums a, we take the values of varables X a Y respectvel.. F sum of the varables X a Y.e. 4 a 3 3. Calculate arthmetc meas of X a Y as a 4. I colum 3, we take evatos of each observatos of X from mea of X,.e. 3 4 =, 44 4 = 4 a so o other values of the colum ca be obtae. 5. Smlarl colum 5 s prepare for varable Y.e. a so o. 5 = 4, 55 = 5. Colum 4 s the square of colum 3 a colum s the square of colum Colum 7 s the prouct of colum 3 a colum Sum of each colum s obtae a wrtte at the e of colum. 35

12 Correlato for Bvarate Data To f out the correlato coeffcet b above formula, we requre the values of ( )( ), ( ) a ( ) whch are obtae b the followg table: ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 7 7 ( ) 3 ( )( ) Takg the values of( )( ), ( ) a ( ) from the table a substtutg the above formula we have the correlato coeffcet r Corr(, ) r Corr(, ) ( )( ( ) 3 7 ) 3 44 ( ) Hece, the correlato coeffcet betwee epeture o avertsemet a proft s.7. Ths cates that the correlato betwee epeture o avertsemet a proft s postve a we ca sa that as epeture o avertsemet creases (or ecreases) proft creases (or ecreases). Sce t les betwee.5 a.5 t ca be cosere as week postve correlato coeffcet. Eample : Calculate Karl Pearso s coeffcet of correlato betwee prce a ema for the followg ata. Prce Dema Soluto: I Eample, we use formula gve equato (3) whch evatos were take from mea. Whe meas of a are whole umber, evatos from mea makes calculato eas. Sce, Eample, meas a were whole umber we preferre formula gve equato (3). Whe meas are ot whole umbers calculato b formula gve equato (3) becomes cumbersome a we prefer a formula gve equato (4) or (5) or (). Sce here meas of a are ot whole umber, so we are preferrg formula ()

13 r Correlato Coeffcet Let us eote prce b the varable X a ema b varable Y. To f the correlato coeffcet betwee prce.e.x a ema Y usg formula gve equato (), we ee to calculate,,,, a whch are beg obtae the followg table: r r r r r r Corr(, ) (9 55) (4)(59) (9 4794) (4 4) (9 793) (59 59) Corr(, ) Corr(, ) Corr(, ) Corr(, ) (434 4) (77 78)

14 Correlato for Bvarate Data Note: We ca use stea of. Seco epresso cates sum over for = to. O the other ha frst epresso ( ) cates sum over all values of X. I the curret eample we are usg summato sg ( ) wthout lmt. Now, let us solve the followg eercses. E4) Calculate coeffcet of correlato betwee a for the followg ata: E5) F the coeffcet of correlato for the followg ages of husba a wfe: Husba s age Wfe s age SHORT-CUT METHOD FOR THE CALCULATION OF CORRELATION COEFFICIENT Whe values of varables are bg a actual meas of varables X a Y.e. a are ot whole umber ( Eample mea of X a Y.e. 4 a were whole umber ) the calculato of correlato coeffcet b the formula (), (3), (4), (5) a () s somewhat cumbersome a we have shortcut metho whch evatos are take from assume mea.e. stea of actual meas a, we use assume mea, hece ( ) a ( ) are replace b A a A where A a A are assume meas of (Assume mea ma be a value of gve varable of our choce) varables X a Y respectvel. Formula for correlato coeffcet b shortcut metho s Here, r Corr(, ) = No. of pars of observatos, A = Assume mea of X, A = Assume mea of Y, ( ) ( ) 38 =( A ): Sum of evato from assume mea A X-seres,

15 =( A ): Sum of evato from assume mea A Y-seres, ( A )( A) : Sum of prouct of evatos from assume meas A a A Y a seres respectvel, X Correlato Coeffcet = ( A ) : Sum of squares of the evatos from assume mea A, seres a = ( A ) : Sum of squares of the evatos from assume mea A seres. Note: Results from usual metho a short-cut metho are same. Eample 3: Calculate correlato coeffcet from the followg ata b shortcut metho: Soluto: B short-cut metho correlato coeffcet s obtae b r Corr(, ),, table. Let A = Assume mea of X =4 a ( ) ( ), a are beg obtae from the followg = -4 A = Assume mea of Y = 7 = = = = 4 7 = = 7 7 = = 4 7 = = 3 7 = Puttg the requre values above formula r r (5 5) (4 5) (5 7) (4 4) (5 39) (5 5)

16 Correlato for Bvarate Data r Thus, there s a ver hgh correlato betwee a. Now, let us solve a eercse. E) F correlato coeffcet betwee the values of X a Y from the followg ata b short -cut metho: Correlato Coeffcet Case of Bvarate Frequec Dstrbuto 4 I bvarate frequec strbuto oe varable s presete row a aother colum a correspog frequeces are gve cells (See Eample 4). If we coser two varables X a Y where, the strbuto of X s gve colums a the strbuto of Y s gve row. I ths case we aopt the followg proceure to calculate correlato coeffcet.. Other tha the gve bvarate frequec strbuto make three colums the rght of the table ( f, f a f ) two colums left of the table (m value for a class terval of varable () ), three rows the bottom of the table f, f a f a two rows the top of the table (m value for a for the gve value.e. f = ). Where f s the sum of all frequeces f a f s the sum of all frequeces for the gve values.e. f = f. F the m value a A.e. evato from assume mea A or step evato.e. A / h where h s such that A / h s a whole umber. 3. Appl step () for varable Y also. 4. F f b multplg b respectve frequec f a get 5. F f b multplg. F 7. F f b multplg f b multplg N N f. b respectve frequec f a get f. b respectve frequec f a get f. N b respectve frequec f a get f. 8. Multpl respectve a for each cell frequec a put the fgures left ha upper corer of each cell. N

17 9. F f b multplg f wth a put the fgures rght ha lower corer of each cell a we appl the followg formula: r f f ( f N where, N = f. ) f f N f Eample 4: Calculate the correlato coeffcet betwee ages of husbas a ages of wves for the followg bvarate frequec strbuto: Ages of Husbas ( Ages of Wves f N ) Total Total Correlato Coeffcet Soluto: Let, = ( 35)/, where assume mea A 35 a =. = ( 4) /, where assume mea A = 4 a h =. CI MV () CI MV () f f f f N = f f = - 8 = f = 98 f f f = -8 f f = f f v = 98 4

18 Correlato for Bvarate Data r ( 8 8) 98 ( 8) ( 8) r =.8.8 SUMMARY I ths ut, we have scusse:. Cocept of correlato;. Tpes of correlato; 3. The scatter agrams of fferet correlatos; 4. Calculato of Karl Pearso s coeffcet of correlato; 5. Short-cut metho of calculato of correlato coeffcet;. Propertes of correlato coeffcet; a 7. Calculato of correlato coeffcet for b-varate ata..9 SOLUTIONS / ANSWERS E) Whe two varables are relate such a wa that chage the value of oe varable affects the value of aother varable, the varables are sa to be correlate or there s correlato betwee these two varables. E) Postve correlato: Correlato betwee () Sales a proft () Faml Icome a ear of eucato Negatve correlato: Correlato betwee () No. of as stuets abset class a score eam () Tme spet offce a tme spet wth faml E3) Coeffcet of correlato measures the test or egree of lear relatoshp betwee two varables. It s eote b r. Formula for the calculato of correlato coeffcet s r Corr(, ) ( )( ( ) ) ( ) 4

19 E4) We have some calculato the followg table: ( ) ( ) ( ) ( ) Correlato Coeffcet ( )( ) Here 5 / 5 / 5 3 a 3 / 3 / 5 From the calculato table, we observe that ( ), ( ) 4 a ( )( ) Substtutg these values the formula ( )( ) r ( ) ( ) r Corr(, ) 4 Hece, there s perfect postve correlato betwee X a Y. E5) Let us eote the husba s age as X a wfe s age b Y Here, 8, 38, 4744,

20 Correlato for Bvarate Data We use the formula r Corr(, ) r r ( ( )( ) ( 394) (8)(38) ) ( ) ( 4744) (8 8) ( 34) (38 38) r = 4 4 Hece there s perfect postve correlato betwee X a Y. E) B short-cut metho correlato coeffcet s obtae b r Corr(, ),, followg table. Let ( ) ( ), a are beg obtae through the A = assume mea of X = 3 a = A = Assume mea of Y = 7 = Puttg the requre values above formula r r (5 5) ( ) (5 ) () (5 45) ( ) r

UNIT 7 RANK CORRELATION

UNIT 7 RANK CORRELATION Rak Correlato Structure 7. Itroucto Objectves 7. Cocept of Rak Correlato 7.3 Dervato of Rak Correlato Coeffcet Formula 7.4 Te or Repeate Raks 7.5 Cocurret Devato 7.6 Summar 7.7