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Spearman Rank Correlation Coefficient
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\(r_{s} = 1-\frac{6\times \Sigma^{n}_{i=1}d^{2}_{i}}{n\times (n^{2}-1)}\)

  • \(r_{s}\) - Spearman rank correlation coefficient
  • \(d_{i}\) - difference between the ranks of each pair of observations on X and Y
  • \(n\) - number of observations

A test based on the Spearman rank correlation coefficient is an example of a nonparametric test.

It is one of the most frequently used tests to examine the correlation between two variables. We often employ it when we cannot use a t-test because random variables don't meet assumptions about the distribution.

To calculate the Spearman rank correlation coefficient, 1) first we have to give a number to each observation from a sample. 1 is for the largest observation, 2 is for the second largest one, 3 is for the third largest one, and so on. The process of giving each observation a number is carried out for both random variables. 2) Then we calculate the Spearman rank correlation coefficient as given above. 3) To decide whether to reject the null or not, we use a t-test (test statistic for tests concerning correlation coefficient) if the sample is large enough (more than 30). In other cases, we use special tables to find critical values.

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