 Hello and welcome to the session. In this session, we are going to discuss Spearman's Strand Correlation Method. This method of determining correlation was given by Edward Spearman. It is used to determine correlation between qualitative data. For example, beauty on ST, character, etc., as their quantitative measurement is not possible. For example, a teacher can arrange the students in ascending or descending order of intelligence, but intelligence can't be measured quantitatively. Thus, Strand Correlation Method is used in such cases. Here, we assign the rank to the individual in some order, like the most intelligent individual may be given rank 1, next rank 2, and so on. Now, the formula for the computation of rank correlation is given by r is equal to 1 minus 6 into summation d square upon n into n square minus 1, where d is the difference between the corresponding ranks of the two series and n is the number of individuals in each series. Here, we should note that summation d is always 0, that is, the algebraic sum of the rank differences is always 0. Let us take an example. The marks obtained by students in economics and accountancy are as follows. Marks in economics are represented by x and marks in accountancy are represented by y. The x series is given as 42, 35, 54, 86, 25, 67, 70 and the y series is given as 60, 56, 45, 72, 35, 40, 65. We need to find out coefficient of rank correlation. Now, we shall assign ranks in x and y series denoted by r1 and r2 respectively. In the r1 series, we shall assign ranks 1, 2, 3 from the highest to the lowest. So, we assign rank 1 to 86, rank 2 to 70, rank 3 to 67, rank 4 to 54, rank 5 to 42, rank 6 to 35, rank 7 to 25. Similarly, in r2 series, we assign rank 1 to 72, rank 2 to 65, rank 3 to 60, rank 4 to 56, rank 5 to 45, rank 6 to 40 and rank 7 to 35. Now, we shall find out d, that is the difference between the corresponding ranks of the two series and d squared. Now, we have d is equal to r1 minus r2, that is 5 minus 3, that is 2, 6 minus 4, that is 2, 4 minus 5 is minus 1, 1 minus 1, 0, 7 minus 7 is 0, 3 minus 6 is minus 3, 2 minus 2 is 0. Here, we note that summation of d comes out to be 0. Now, we will calculate d squared, that is 2 squared which is equal to 4, 2 squared, that is 4, minus 1 squared is 1, 0 squared that is 0, again 0 squared that is 0, minus 3 squared is 9, 0 squared is 0. So, summation of d squared, that is sum of all the elements in d squared series is equal to 18. We know that rank correlation coefficient r is given by 1 minus 6 into summation d squared upon n into n squared minus 1, which is equal to 1 minus 6 into summation of d squared, that is 18 upon n into n squared minus 1 and we know that n is the number of individuals in each series. Here, n is equal to 7, so we have 7 into 7 square minus 1, 7 square is 49 minus 1, which is equal to 1 minus 6 into 18 is 108 upon 7 into 49 minus 1, that is 48. Which is equal to 1 minus 108 upon 7 into 48 is 336. On taking the LCM, we get 336 minus 108 upon 336, which is equal to 228 upon 336, that is 0.678. Therefore, r is equal to 0.678. Now, we shall discuss correlation for tied ranks when two or more individuals or items have the same score, that is there is a tie in their ranks. Then, the spearman's rank correlation coefficient formula r is equal to 1 minus 6 into summation d squared upon n into n squared minus 1 fails. Therefore, a correction or modification is necessary in the formula as this formula is based on the supposition that no rank is given to more than one item. Therefore, in the problems where the individuals are in a tie, we assign a common rank to each of the individuals and this common rank is the average of the ranks of these individuals. For example, let us consider the series where the X series is given as 95, 92, 91, 91, 91, 91, 81, 80, 78, 75, 75, 75 and 60. Here we can see that 91 is repeated 4 times and 75 is repeated 3 times. Here we assign rank 123 from highest to the lowest. So, we assign rank 1 to 95, rank 2 to 92, then the rank 3 plus 4 plus 5 plus 6 whole upon 4, that is equal to 4.5 will be assigned to each of 91. As 91 is repeated 4 times, next we assign rank 7 to 80, rank 8 to 78. Now again 75 is repeated 3 times. So, we rank 9 plus 10 plus 11 whole upon 3, that is 10 is assigned to each of 75 and then rank 12 is assigned to 60. We can also assign rank by starting from lowest, that is rank 1 to 60, then rank 2 plus 3 plus 4 upon 3, that is 3 to each of 75, then rank 5 to 78, rank 6 to 80, then the rank 7 plus 8 plus 9 plus 10 upon 4, that is 34 upon 4 which is equal to 8.5 to each of 91, then rank 11 to 92 and rank 12 to 95. Now we are going to discuss correlation factor. When the items with repeated values have been assigned common ranks, then some modification has to be made in the formula. A correction factor is added to this pair man's rank correlation formula and the modified formula is given by r is equal to 1 minus 6 into summation d square plus m1 cube minus m1 upon 12 plus m2 cube minus m2 upon 12 plus and so on will divided by m into m square minus 1 where m1 m2 m3 and so on are the number of times a value is repeated that is a correlation factor of m cube minus m upon 12 is added to summation d square for each repeating value in both the series. Let us take an example the data below is the series of demand and supply by the rank correlation between them the demand is given in 1000's unit and is denoted by x the x series is given as 80 75 64 64 64 20 20 35 67 42 in 1000's unit and supply is also given in 1000's unit denoted by y and the y series is given by 70 72 72 83 83 83 83 62 58 and 47 in 1000's unit now we shall calculate rank of x denoted by r1 and we assign rank 1 2 3 from the highest to the lowest so we assign rank 1 to 80 rank 2 to 75 now 64 is repeated 3 times so we assign the rank 3 plus 4 plus 5 upon 3 that is 12 by 3 which is equal to 4 to each of 64 next we assign rank 6 to 57 and rank 7 to 42 then rank 8 to 35 and now 20 is repeated 2 times so we assign the rank 9 plus 10 upon 2 that is 19 upon 2 which is equal to 9.5 to each of 20 similarly we assign ranks of y which is denoted by r2 here also we assign ranks 1 2 3 from the highest to the lowest and here 83 is the highest number which is repeated 4 times so we assign the rank of 1 plus 2 plus 3 plus 4 upon 4 that is 10 upon 4 which is equal to 2.5 to each of 83 next is 72 which is repeated 2 times so we assign the rank 5 plus 6 by 2 which is equal to 11 by 2 that is 5.5 to each of 72 then we assign the rank 7 to 78 to 62 9 to 58 and 10 to 47 now we shall calculate D and D square where D is equal to r1 minus r2 that is the difference between the corresponding ranks of the 2 series we have D is equal to r1 minus r2 therefore 1 minus 7 is equal to minus 6 2 minus 5.5 that is minus 3.5 4 minus 5.5 that is minus 1.5 4 minus 2.5 that is 1.5 4 minus 2.5 that is 1.5 9.5 minus 2.5 that is 7 9.5 minus 2.5 is 7 8 minus 8 is 0 6 minus 9 is minus 3 7 minus 10 is minus 3 and D square is given by minus 6 square that is 36 minus 3.5 square that is 12.25 minus 1.5 square that is 2.25 1.5 square is 2.25 again 1.5 square is 2.25 then 7 square that is 49 again 7 square that is 49 0 square is 0 minus 3 square is 9 minus 3 square is 9 we should note that summation of D is equal to 0 and summation of D square is equal to 171 and we know that rank correlation is given by the formula r is equal to 1 minus 6 into summation of D square plus m1 cube minus m1 by 12 plus m2 cube minus m2 by 12 plus and so on upon n into n square minus 1 where m1 m2 m3 are the number of times a value is repeated so here in x series we have m1 is equal to 3 as 64 is repeated 3 times m2 is equal to 2 as 20 is repeated 2 times now in y series we have m3 is equal to 2 as 72 is repeated 2 times and m4 is equal to 4 as 83 is repeated 4 times so we have r is equal to 1 minus 6 into summation of D square that is 171 plus 1 by 12 m1 cube minus m1 that is 3 cube minus 3 plus 1 by 12 m2 cube minus m2 that is 2 cube minus 2 plus 1 by 12 into m3 cube minus m3 that is 2 cube minus 2 plus 1 by 12 into m4 cube minus m4 that is 4 cube minus 4 whole upon n into n square minus 1 and m is 10 here so we have 10 into 10 square minus 1 so r is equal to 1 minus 6 into 171 plus 1 by 12 into 3 cube that is 27 minus 3 which is equal to 24 plus 1 by 12 into 2 cube minus 2 that is 8 minus 2 which is equal to 6 plus 1 by 12 into 4 cube that is 64 minus 4 which is equal to 60 plus 1 by 12 into 2 cube minus 2 that is 8 minus 2 which is equal to 6 upon 10 into 10 square minus 1 that is 100 minus 1 which is equal to 99 so we have 1 minus 6 into 171 plus 2 plus 1 upon 2 plus 5 plus 1 upon 2 upon 990 which is equal to 1 minus 6 into 171 plus 2 plus 1 by 2 plus 1 by 2 is equal to 1 plus 5 upon 990 which is equal to 1 minus 6 into 179 upon 990 that is 1 minus 6 into 179 which is equal to 1074 upon 990 on taking the LCM we get 990 minus 1074 upon 990 which is equal to minus of 84 upon 990 that is given by minus of 0.0848 therefore rank or relation coefficient r is given by minus of 0.0848 this completes our session hope you enjoyed this session