When trying to explain rank correlation, first thing that should come in front is the grading or rank given to a particular group of individuals.

For example, if we suppose that based on two characteristics n group of individuals are ranked then the equation will come as:

r = 1 – 6∑di2/ n(n2 – 1)

Here (x, y;), i= 1,2…, n is the ranked individuals

Here, di=XiYi

One thing must be clarified before using – 1≤ r ≤ 1that one rank will not appear more than once. When the ranks are repeated then another correlation factor needs to be added. If we take that ranks are repeated m number of times then the equation will be added asm(m2 – 1) with Σd2

Let us measure the correlation of coefficient of the data mentioned below:

X85748550657874607490
Y78917858607280556870

Here n = 10, so the solution is:

            X           YRank X (x)Rank Y (y)d = x – y          d2
85

74

85

50

65

78

74

60

74

90

78

91

78

58

60

72

80

55

68

70

2.5

6

2.5

10

8

4

6

9

6

1

3.5

1

3.5

9

8

5

2

10

7

6

-1

5

-1

1

0

-1

4

-1

-1

-5

1

25

1

1

0

1

16

1

1

25

072

In this solution, you can see that 85 are repeated two times for which the rank became 2.5 where it could be 2 and 3. Then 74 from the X series, is also repeated three times giving it the rank of 6 where it could be 5, 6, 7. Coming next to 78 from Y series which is repeated three times and getting the rank of 3.5 where it could have been 3 and 4.

As mentioned above, for the repeated ranks, a correlation factor will be added for each distribution, these are written below:

2(4 – 1)/12 = 1/2

3(9 -1)/12 = 2

2(4 – 1)/12 = 1/2

After measuring these three correlation factor we come to:

½ + 2 + ½ = 3

Hence we come to the rank correlation:

r = 1 – 6(72 + 3)/ 10 (100 – 1) = 0.545

 

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