Which formula calculates Spearman's Rank correlation coefficient?

The formula for Spearman's Rank correlation coefficient is indeed given by

[

\rho = \frac{1 - 6 \sum d^2}{n(n^2 - 1)}

]

where (d) represents the difference in ranks for each pair of observations, and (n) is the number of observations.

Spearman's Rank correlation is a non-parametric measure that assesses how well the relationship between two variables can be described by a monotonic function. Instead of using raw data, Spearman's method ranks the data first and then applies this formula to quantify the strength and direction of the association between the ranks.

In this formula, the term (6 \sum d^2) penalizes larger differences in ranks, emphasizing that larger discrepancies between ranks yield a lower correlation. The denominator, (n(n^2 - 1)), ensures the coefficient is normalized between -1 and 1, making it interpretable as a correlation coefficient. This characteristic makes Spearman's Rank correlation useful in situations where the relationship might not be linear, or where data do not satisfy the assumptions of parametric tests.

Choosing this formula reflects an understanding of how Spearman's method works, focusing on ranks rather than