Statistics Assignment Help With Rank Correlation

Spearman’s Rank Correlation Coefficient

The spearman’s rank coefficient of correlation was developed by Charles Edward Spearman.

What is Spearman correlation coefficient?

Let us suppose that a group of n individuals is arranged in order of merit or proficiency in possession of two characteristics A and B. these ranks in two characteristics will, in general, be different. For example, if we consider the relation between intelligence and beauty, it is not necessary that a beautiful individual is intelligent also. Let (xi , yi ); i=1,2,………,n be the ranks of the i^th individual in two characteristics A and B respectively. Pearsonian coefficient of correlation between the ranks xi’s and yi ‘s is called the Spearman rank correlation coefficient between A and B for that group of individuals.

The formula for correlation of rank coefficient is given as:

rk = 1 – [6 ∑D² / N³ – N]

Where D = R₁ – R₂, between the paired items in the two rank series.

The value of rank correlation coefficient tells us about the degree of agreement between the 2 ranks.

See derivation for the rank coefficient of correlation by spearman below:

Assuming that no two individuals are bracketed equal in either classification, each of the variables X and Y takes the values 1,2,………..,n

Hence

Which is the spearman's formula for the rank correlation coefficient.

Properties of Spearman’s Rank correlation Coefficient

-1 ≤ r_k ≤ +1

Rank Correlation Coefficient Example

Calculate the Rank Correlation Coefficient in each of the following cases:

X	R₁	Y	R₂
10	1	5	1
20	2	6	2
30	3	7	3

To calculate the rank correlation coefficient, first we will determine the value of D = R₁ – R₂ in each of the entries:

X	R₁	Y	R₂
10	1	5	1
20	2	6	2
30	3	7	3

Then the Spearman’s rank correlation coefficient is calculated using the formula as:

rk = 1 – [6 ∑D² / N³ – N]

= 1- 6(0)

= +1

Thus the value of rank correlation coefficient equal to +1 implies that there is complete agreement in the order of ranks and the ranks are in the same direction.

Let us calculate the rank correlation coefficient in another example:

X	R₁	Y	R₂
10	1	7	3
20	2	6	2
30	3	5	1

To calculate the rank correlation coefficient, first we will determine the value of D = R1 – R2 in each of the entries:

X	R₁	Y	R₂	D	D²
10	1	7	3	-2	4
20	2	6	2	0	0
30	3	5	1	+2	4

Then the Spearman’s rank correlation coefficient is calculated using the formula as:

r_k = 1 – [6 ∑D² / N³ – N]

= 1 – (6*8)/9-3

= -1

Thus the value of rank correlation coefficient equal to -1 implies that there is complete agreement in the order of ranks and the ranks are in opposite direction

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