UNIT 7 RANK CORRELATION

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1 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 Solutos / Aswe 7. INTRODUCTION I seco ut of ths block, we have scusse the correlato wth ts propertes a also the calculato of correlato coeffcet. I correlato coeffcet or prouct momet correlato coeffcet, t s assume that both charactetcs are measurable. Sometmes charactetcs are ot measurable but raks ma be gve to vuals accorg to ther qualtes. I such stuatos rak correlato s use to kow the assocato betwee two charactetcs. I ths ut, we wll scuss the rak correlato a calculato of rak correlato coeffcet wth ts merts a emerts. We wll also stu the metho of cocurret evato. I Secto 7., ou wll kow the cocept of rak correlato whle Secto 7.3 gves the ervato of Spearma s rak correlato coeffcet formula. Merts a emerts of the rak correlato coeffcet are scusse Sub-secto There mght be a stuato whe two tems get same rak. Ths stuato s calle te or repeate rak whch s escrbe Secto 7.4. You wll lear the metho of cocurret evato Secto 7.5. Objectves After reag ths ut, ou woul be able to epla the cocept of rak correlato; erve the Spearma s rak correlato coeffcet formula; escrbe the merts a emerts of rak correlato coeffcet; calculate the rak correlato coeffcet case of te or repeate raks; a escrbe the metho of cocurret evato. 7. CONCEPT OF RANK CORRELATION For the calculato of prouct momet correlato coeffcet characte must be measurable. I ma practcal stuatos, characte are ot measurable. The are qualtatve charactetcs a vuals or tems ca be rake 45

2 Correlato for Bvarate Data orer of ther merts. Ths tpe of stuato occu whe we eal wth the qualtatve stu such as hoest, beaut, voce, etc. For eample, cotestats of a sgg competto ma be rake b juge accorg to ther performace. I aother eample, stuets ma be rake fferet subjects accorg to ther performace tests. Arragemet of vuals or tems orer of mert or profcec the possesso of a certa charactetc s calle rakg a the umber catg the posto of vuals or tems s kow as rak. If raks of vuals or tems are avalable for two charactetcs the correlato betwee raks of these two charactetcs s kow as rak correlato. Wth the help of rak correlato, we f the assocato betwee two qualtatve charactetcs. As we kow that the Karl Peao s correlato coeffcet gves the test of lear relatoshp betwee two varables a Spearma s rak correlato coeffcet gves the cocetrato of assocato betwee two qualtatve charactetcs. I fact Spearma s rak correlato coeffcet measures the stregth of assocato betwee two rake varables. Dervato of the Spearma s rak correlato coeffcet formula s scusse the followg secto. 7.3 DERIVATION OF RANK CORRELATION COEFFICIENT FORMULA Suppose we have a group of vuals a let,,..., a,,..., be the raks of vuals charactetcs A a B respectvel. It s assume that o two or more vuals have the same rak ether charactetcs A or B. Suppose both charactetcs X a Y are takg rak values,, 3,,. The sum of raks of charactetcs A s (sce X s takg values,,,) ( ) () From, the formula of sum of atural umbe. Mea of varable X s ( ) ( ) Sce both varables are takg same values,,, the ( )

3 47 Rak Correlato If varace of X s eote b the ) ( ) ( ) ( ) ( ) ( )... ( () Substtutg the value of equato (), we have )... ( (Sce X s takg values,,,) 6 ) )( ( (From the formula of sum of squares of atural umbe) 4 ) ( 6 ) )( ( 4 ) ( 6 ) ( ) 3( ) ( ) ( ) ( ) ( σ (from the formula (a - b)(a + b) = b a ) Sce both varables X a Y are takg same values, the wll have same varace, thus σ σ

4 Correlato for Bvarate Data Let be the fferece of the raks of the th vual two charactetcs, Sce ( ) ( ) Squarg a summg over = to, we have ( ) ( ) ( ) ( ) ( )( ) ( ) Dvg equato (3) b, we have ( ) ( ) ( ) ( )( ) (3) ( )( ) Cov(, ) Cov(, ) We kow that, r, whch mples that Cov(, ) r. (4) Substtutg Cov(, ) r equato (4), we have r Sce,, the r r ( r) ( r) r 48

5 6 (Sce ) ( ) r Rak Correlato We eote rak correlato coeffcet b r s, a hece 6 (5) ( ) Ths formula was gve b Spearma a hece t s kow as Spearma s rak correlato coeffcet formula. Let us scuss some problems o rak correlato coeffcet. Eample : Suppose we have raks of 8 stuets of B.Sc. Statstcs a Mathematcs. O the bass of rak we woul lke to kow that to what etet the kowlege of the stuet Statstcs a Mathematcs s relate. Rak Statstcs Rak Mathematcs Soluto: Spearma s rak correlato coeffcet formula s 6 ( ) Let us eote the rak of stuets Statstcs b R a rak Mathematcs br. For the calculato of rak correlato coeffcet we have to f whch s obtae through the followg table: Rak Statstcs R Rak Mathematcs R Dfferece of Raks R R Here, = umber of pare observatos = ( )

6 Correlato for Bvarate Data Thus there s a postve assocato betwee raks of Statstcs a Mathematcs. Eample : Suppose we have raks of 5 stuets three subjects Computer, Phscs a Statstcs a we wat to test whch two subjects have the same tre. Rak Computer Rak Phscs Rak Statstcs Soluto: I ths problem, we wat to see whch two subjects have same tre.e. whch two subjects have the postve rak correlato coeffcet. Here we have to calculate three rak correlato coeffcets Rak correlato coeffcet betwee the raks of Computer a Phscs r Rak correlato coeffcet betwee the raks of Phscs a Statstcs 3s r Rak correlato coeffcet betwee the raks of Computer a 3s Statstcs Let R, R a R3 be the raks of stuets Computer, Phscs a Statstcs respectvel. Rak Rak Rak = 3 = 3 3 = 3 Computer Phscs Statstcs R (R ) (R ) (R 3 ) R R R 3 R R Total Thus, Now =3, 3 =3 a 3 = ( ) ( ) r3s ( ) r3 s s egatve whch cates that Computer a Phscs have opposte tre. Smlarl, egatve rak correlato coeffcet r 3s shows the opposte

7 tre Phscs a Statstcs. r 3s 0. 3 cates that Computer a Statstcs have same tre. Sometmes we o ot have rak but actual values of varables are avalable. If we are tereste rak correlato coeffcet, we f raks from the gve values. Coserg ths case we are takg a problem a tr to solve t. Eample 3: Calculate rak correlato coeffcet from the followg ata: Rak Correlato Soluto: We have some calculato the followg table: Rak of (R ) Spearma s Rak correlato formula s 6 ( ) 6 7(49 ) Now, let us solve a lttle eercse. Rak of (R ) E) Calculate Spearma s rak correlato coeffcet from the followg ata: = R -R Merts a Demerts of Rak Correlato Coeffcet Merts of Rak Correlato Coeffcet. Spearma s rak correlato coeffcet ca be terprete the same wa as the Karl Peao s correlato coeffcet;. It s eas to ueta a eas to calculate; 5

8 Correlato for Bvarate Data 3. If we wat to see the assocato betwee qualtatve charactetcs, rak correlato coeffcet s the ol formula; 4. Rak correlato coeffcet s the o-parametrc veo of the Karl Peao s prouct momet correlato coeffcet; a 5. It oes ot requre the assumpto of the ormalt of the populato from whch the sample observatos are take. Demerts of Rak Correlato Coeffcet. Prouct momet correlato coeffcet ca be calculate for bvarate frequec strbuto but rak correlato coeffcet caot be calculate; a. If >30, ths formula s tme cosumg. 7.4 TIED OR REPEATED RANKS I Secto 7.3, t was assume that two or more vuals or uts o ot have same rak. But there mght be a stuato whe two or more vuals have same rak oe or both charactetcs, the ths stuato s sa to be te. If two or more vuals have same value, ths case commo raks are assge to the repeate tems. Ths commo rak s the average of raks the woul have receve f there were o repetto. For eample we have a seres 50, 70, 80, 80, 85, 90 the st rak s assge to 90 because t s the bggest value the to 85, ow there s a repetto of 80 twce. Sce both values are same so the same rak wll be assge whch woul be average of the raks that we woul have assge f there were o repetto. Thus, both 80 wll receve the average of 3 a 4.e. (Average of 3 & 4.e. (3 + 4) / = 3.5) 3.5 the 5 th rak s gve to 70 a 6 th rak to 50. Thus, the seres a raks of tems are Seres Raks I the above eample 80 was repeate twce. It ma also happe that two or more values are repeate twce or more tha that. For eample, the followg seres there s a repetto of 80 a 0. You observe the values, assg raks a check wth followg. Seres Raks m(m ) Whe there s a repetto of raks, a correcto facto ae to the Spearma s rak correlato coeffcet formula, where m s the umber of tmes a rak s repeate. It s ver mportat to kow that ths correcto facto ae for ever repetto of rak both characte.

9 I the ft eample correcto facto ae oce whch s (4-)/ = 0.5, whle the seco eample correcto facto are (4-)/ = 0.5 a Rak Correlato 3 (9-)/ = whch are ae to. Thus, case of te or repeate rak Spearma s rak correlato coeffcet formula s m(m ) 6... ( ) Eample 4: Calculate rak correlato coeffcet from the followg ata: Epeture o avertsemet Proft Soluto: Let us eote the epeture o avertsemet b a proft b Rak of (R ) Rak of ( R ) = R -R m(m )... ( ) Here rak 6 s repeate three tmes rak of a rak.5 s repeate twce rak of, so the correcto facto 3(3 ) ( ) Hece rak correlato coeffcet s 3(3 ) ( ) (64 ) 53

10 Correlato for Bvarate Data X 63 6( ) There s a egatve assocato betwee epeture o avertsemet a proft. Now, let us solve the followg eercses. E) Calculate rak correlato coeffcet from the followg ata: E3) Calculate rak correlato coeffcet from the followg ata: CONCURRENT DEVIATION 54 Sometmes we are ot tereste the actual amout of correlato coeffcet but we wat to kow the recto of chage.e. whether correlato s postve or egatve, coeffcet of cocurret evato serves our purpose. I ths metho correlato s calculate betwee the recto of evatos a ot ther magtues. Coeffcet of cocurret evato s eote b r c a gve b r c (c k) (6) k where, c s the umber of cocurret evato or the umber of + sg the prouct of two evatos, k.e. total umber of pare observato mus oe. Ths s also calle coeffcet of correlato b cocurret evato metho. Steps for the calculato of cocurret evato (see the Eample 5 smultaeousl) are:. The ft value of seres s take as a base a t s compare wth et value.e. seco value of seres. If seco value s greater tha ft value, + sg s assge the Colum ttle D. If seco value s less tha the ft value the - sg s assge the colum D.. If ft a seco values are equal the = sg s assge. 3. Now seco value s take as base a t s compare wth the thr value of the seres. If thr value s less tha seco - s assge agast the

11 thr value. If the thr value s greater tha the seco value + s assge. If seco a thr values or equal tha = sg s assge. 4. Ths proceure s repeate upto the last value of the seres. 5. Smlarl, we obta colum D foeres. Rak Correlato 6. We multpl the colum D a D a obta colum D D. Multplcato of same sg results + sg a that of fferet sg s - sg. 7. Fall umber of + sg are coute the colum D D, t s calle c a we get coeffcet cocurret evato b the formula (6). 8. I the formula, se a outse the square root, sg + a - epes o the value of ( c k ). If ths value s postve tha + sg s take at both places f ( c k ) s egatve - sg s cosere at both the places. Let us scuss the followg problem. Eample 5: We have ata of come a epeture of worke of a orgazato the followg table: Icome Epeture F whether correlato s postve or egatve b coeffcet of cocurret evato. Soluto: Coeffcet of cocurret evato s gve b r c (c k) k Let us eote the come b a epeture b a we calculate c b the followg table: Chage of recto sg for (D ) Chage of recto sg for (D ) D D c = 9 Here, c = 9 a k = = 0 the we have 55

12 Correlato for Bvarate Data r = (Both sgs are + because c k s postve) Thus correlato s postve. Now, let us solve the followg eercses. 8 0 E 4) F the coeffcet of correlato betwee suppl a prce b cocurret evato metho for the followg ata: Year Suppl Prce E5) Calculate coeffcet of cocurret evato for the followg ata: SUMMARY I ths ut, we have scusse:. The rak correlato whch s use to see the assocato betwee two qualtatve charactetcs;. Dervato of the Spearma s rak correlato coeffcet formula; 3. Calculato of rak correlato coeffcet fferet stuatos- () whe values of varables are gve, () whe raks of vuals fferet charactetcs are gve a () whe repeate raks are gve; 4. Propertes of rak correlato coeffcet; a 5. Cocurret evato whch proves the recto of correlato. 7.7 SOLUTIONS /ANSWERS 56 E) We have some calculatos the followg table: Rak of (R ) Rak of (R ) = R -R =6

13 6 6 6(36 ) E) We have some calculatos the followg table: Rak of (R ) Rak of (R ) = R -R m(m )... ( ).5 Here, rak 4.5 s repeate twce rak of a rak s repeate thrce rak of so the correcto facto ( ) 3(3 ) a therefore, rak correlato coeffcet s ( ) 3(3 ) 6.5 7(49 ) Rak Correlato 6(.5.5) E3) We have some calculatos the followg table: 57

14 Correlato for Bvarate Data 58 Rak of Rak of = R -R (R ) (R ) Rak correlato coeffcet s m(m ) 6... ( ) Here, rak 4 a 6.5 s repeate thrce a twce respectvel rak of a rak s repeate thrce rak of, so the correcto facto 3(3 ) ( ) 3(3 ) a therefore, rak correlato coeffcet s 3(3 ) ( ) 3(3 ) (49 ) (0) E4) Coeffcet of cocurret evato s gve (c k) r c k Let us eote the suppl b a prce b a we calculate c b the followg table: Chage of Drecto Chage of Drecto D D sg for (D ) sg for (D ) c =

15 Now c = a k = - = 5 r = (Both sgs are - because c k s egatve) = 0.45 Thus, correlato s egatve. E5) Coeffcet of cocurret evato s gve (c k) r c k Let us eote the suppl b a prce b a we calculate c b the followg table: Rak Correlato Chage of Drecto Chage of Drecto sg for (D ) sg for (D ) D D c = 3 Now c = 3 a k = = 5 r = (Both sgs are + because c k s postve) r = 0.45 Thus, correlato s postve. 59

UNIT 6 CORRELATION COEFFICIENT

UNIT 6 CORRELATION COEFFICIENT 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.