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Rank Order Correlation

Dr. Anshul Singh Thapa

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Introduction

Spearman’s rank correlation was developed by the British psychologist C.E. Spearman. It is used when the variables cannot be measured meaningfully. Ranking may be more meaningful when the measurements of variables are suspect. Consider the situation where we are required to calculate the correlation between height and weight of students in a remote village. Neither measuring rods nor weighing scales are available. The students can be easily ranked in terms of height and weight without using measuring rods and weighing scales.

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There are also situations when you are required to quantify qualities such as fairness, honesty etc. Ranking may be a better alternative to quantification of qualities.
Moreover, sometimes the correlation coefficient between two variables with extreme values may be quite different from the coefficient without the extreme values. Under these circumstances rank correlation provides a better alternative to simple correlation.
Rank correlation coefficient and simple correlation coefficient have the same interpretation. Its formula has been derived from simple correlation coefficient where individual values have been replaced by ranks. These ranks are used for the calculation of correlation. This coefficient provides a measure of linear association between ranks assigned to these units, not their values.

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The calculation of rank correlation will be illustrated under three situations.

The ranks are given.
The ranks are not given. They have to be worked out from the data.
Ranks are repeated.

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The ranks are given

A	B
6	3
5	8
3	4
10	9
2	1
4	6
9	10
7	7
8	5
1	2

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Solution:

R¹	R²	(R¹ – R²) D
6	3	3
5	8	3
3	4	-1
10	9	1
2	1	1
4	6	-2
9	10	-1
7	7	0
8	5	3
1	2	-1

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Solution:

R¹	R²	(R¹ – R²) D	(R¹ – R²)² D²
6	3	3	9
5	8	3	9
3	4	-1	1
10	9	1	1
2	1	1	1
4	6	-2	4
9	10	-1	1
7	7	0	0
8	5	3	9
1	2	-1	1

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Solution:

R¹	R²	(R¹ – R²) D	(R¹ – R²)² D²
6	3	3	9
5	8	3	9
3	4	-1	1
10	9	1	1
2	1	1	1
4	6	-2	4
9	10	-1	1
7	7	0	0
8	5	3	9
1	2	-1	1
			ΣD² = 36

r_s = 0.782

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The ranks are not given. They have to be worked out from the data

Year	A	B
1	97.8	73.2
2	99.2	85.8
3	98.8	78.9
4	98.3	75.8
5	98.4	77.2
6	96.7	87.2
7	97.1	83.8

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Solution

Year	A	R₁
1	97.8	3
2	99.2	7
3	98.8	6
4	98.3	4
5	98.4	5
6	96.7	1
7	97.1	2

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Solution

Year	A	R₁	B	R₂
1	97.8	3	73.2	1
2	99.2	7	85.8	6
3	98.8	6	78.9	4
4	98.3	4	75.8	2
5	98.4	5	77.2	3
6	96.7	1	87.2	7
7	97.1	2	83.8	5

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Solution

Year	A	R₁	B	R₂	(R₁ – R₂) D
1	97.8	3	73.2	1	2
2	99.2	7	85.8	6	1
3	98.8	6	78.9	4	2
4	98.3	4	75.8	2	2
5	98.4	5	77.2	3	2
6	96.7	1	87.2	7	-6
7	97.1	2	83.8	5	-3

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Solution

Year	A	R₁	B	R₂	(R₁ – R₂) D	(R₁ – R₂)² D²
1	97.8	3	73.2	1	2	4
2	99.2	7	85.8	6	1	1
3	98.8	6	78.9	4	2	4
4	98.3	4	75.8	2	2	4
5	98.4	5	77.2	3	2	4
6	96.7	1	87.2	7	-6	36
7	97.1	2	83.8	5	-3	9

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Solution

Year	A	R₁	B	R₂	(R₁ – R₂) D	(R₁ – R₂)² D²
1	97.8	3	73.2	1	2	4
2	99.2	7	85.8	6	1	1
3	98.8	6	78.9	4	2	4
4	98.3	4	75.8	2	2	4
5	98.4	5	77.2	3	2	4
6	96.7	1	87.2	7	-6	36
7	97.1	2	83.8	5	-3	9
						ΣD² = 62

r_s = -0.107

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Ranks are repeated

A	B
50	110
55	110
65	115
50	125
55	140
60	115
50	130
65	120
70	115
75	160

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It may be noted that in series X, 50 has repeated thrice (m = 3), 55 has been repeated twice (m = 2), 65 has been repeated twice (m = 2).�In series Y, 110 has been repeated twice (m = 2), and 115 has been repeated thrice (m = 3)

A	B
50	110
55	110
65	115
50	125
55	140
60	115
50	130
65	120
70	115
75	160

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Solution

A	R₁
50	2
55	4.5
65	7.5
50	2
55	4.5
60	6
50	2
65	7.5
70	9
75	10

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Solution

A	R₁	B	R₂
50	2	110	1.5
55	4.5	110	1.5
65	7.5	115	4
50	2	125	7
55	4.5	140	9
60	6	115	4
50	2	130	8
65	7.5	120	6
70	9	115	4
75	10	160	10

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Solution

A	R₁	B	R₂	(R₁ – R₂) D
50	2	110	1.5	0.5
55	4.5	110	1.5	3
65	7.5	115	4	3.5
50	2	125	7	-5
55	4.5	140	9	-4.5
60	6	115	4	2
50	2	130	8	-6
65	7.5	120	6	1.5
70	9	115	4	5
75	10	160	10	0

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Solution

A	R₁	B	R₂	(R₁ – R₂) D	(R₁ – R₂)² D²
50	2	110	1.5	0.5	0.25
55	4.5	110	1.5	3	9.00
65	7.5	115	4	3.5	12.25
50	2	125	7	-5	25.00
55	4.5	140	9	-4.5	20.25
60	6	115	4	2	4.00
50	2	130	8	-6	36.00
65	7.5	120	6	1.5	2.25
70	9	115	4	5	25.00
75	10	160	10	0	00.00

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Solution

A	R₁	B	R₂	(R₁ – R₂) D	(R₁ – R₂)² D²
50	2	110	1.5	0.5	0.25
55	4.5	110	1.5	3	9.00
65	7.5	115	4	3.5	12.25
50	2	125	7	-5	25.00
55	4.5	140	9	-4.5	20.25
60	6	115	4	2	4.00
50	2	130	8	-6	36.00
65	7.5	120	6	1.5	2.25
70	9	115	4	5	25.00
75	10	160	10	0	00.00
					ΣD² = 134

r_s = 0.155

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Rank Order Correlation

R₁	R₂	R₃
1	3	6
6	5	4
5	8	9
10	4	8
3	7	1
2	10	2
4	2	3
9	1	10
7	6	5
8	9	7

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)
1	3	6	-2
6	5	4	1
5	8	9	-3
10	4	8	6
3	7	1	-4
2	10	2	-8
4	2	3	2
9	1	10	8
7	6	5	1
8	9	7	-1

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)	(R₁ – R₃)
1	3	6	-2	-5
6	5	4	1	2
5	8	9	-3	-4
10	4	8	6	2
3	7	1	-4	2
2	10	2	-8	0
4	2	3	2	1
9	1	10	8	1
7	6	5	1	2
8	9	7	-1	1

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)	(R₁ – R₃)	(R₂ – R₃)
1	3	6	-2	-5	-3
6	5	4	1	2	1
5	8	9	-3	-4	-1
10	4	8	6	2	-4
3	7	1	-4	2	6
2	10	2	-8	0	8
4	2	3	2	1	-1
9	1	10	8	1	-9
7	6	5	1	2	1
8	9	7	-1	1	2

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)	(R₁ – R₃)	(R₂ – R₃)	D₁₂²
1	3	6	-2	-5	-3	4
6	5	4	1	2	1	1
5	8	9	-3	-4	-1	9
10	4	8	6	2	-4	36
3	7	1	-4	2	6	16
2	10	2	-8	0	8	64
4	2	3	2	1	-1	4
9	1	10	8	1	-9	64
7	6	5	1	2	1	1
8	9	7	-1	1	2	1

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)	(R₁ – R₃)	(R₂ – R₃)	D₁₂²	D₁₃²
1	3	6	-2	-5	-3	4	25
6	5	4	1	2	1	1	4
5	8	9	-3	-4	-1	9	16
10	4	8	6	2	-4	36	4
3	7	1	-4	2	6	16	4
2	10	2	-8	0	8	64	0
4	2	3	2	1	-1	4	1
9	1	10	8	1	-9	64	1
7	6	5	1	2	1	1	4
8	9	7	-1	1	2	1	1

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)	(R₁ – R₃)	(R₂ – R₃)	D₁₂²	D₁₃²	D₂₃²
1	3	6	-2	-5	-3	4	25	9
6	5	4	1	2	1	1	4	1
5	8	9	-3	-4	-1	9	16	1
10	4	8	6	2	-4	36	4	16
3	7	1	-4	2	6	16	4	36
2	10	2	-8	0	8	64	0	64
4	2	3	2	1	-1	4	1	1
9	1	10	8	1	-9	64	1	81
7	6	5	1	2	1	1	4	1
8	9	7	-1	1	2	1	1	4

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Rank Order Correlation

R₁	R₂	R₃	(R₁ – R₂)	(R₁ – R₃)	(R₂ – R₃)	D₁₂²	D₁₃²	D₂₃²
1	3	6	-2	-5	-3	4	25	9
6	5	4	1	2	1	1	4	1
5	8	9	-3	-4	-1	9	16	1
10	4	8	6	2	-4	36	4	16
3	7	1	-4	2	6	16	4	36
2	10	2	-8	0	8	64	0	64
4	2	3	2	1	-1	4	1	1
9	1	10	8	1	-9	64	1	81
7	6	5	1	2	1	1	4	1
8	9	7	-1	1	2	1	1	4
						ΣD₁₂² = 200	ΣD₁₃² = 60	ΣD₂₃² = 214

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Solution