Day 7: Spearman's Rank Correlation Coefficient

Spearman's Rank Correlation Coefficient

We have two random variables, and :

If and denote the respective ranks of each data point, then the Spearman's rank correlation coefficient, , is the Pearson correlation coefficient of and .

Example

So,
Similarly,

equals the Pearson correlation coefficient of and , meaning that .

Special Case: and Don't Contain Duplicates

Here,

is the difference between the respective values of

and

Proof

Let's define be the rank of and be the rank of . Both and are permutations of set , because data sets and contain no duplicates in this special case.

Mean of and :

Standard Deviation of and :

So,

Calculating :

We know that:

So,

Covariance of and :

Spearman's Rank Correlation Coefficient:
We know that the Spearman's rank correlation coefficient of and is equal to the Pearson correlation coefficient of and . So,