Spearman Correlation formula

## What is the definition of the Spearman Correlation test?

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

• p : Spearman correlation
• rs : Spearman correlation
• di : rg(Xi) - rg(Yi): difference between the two ranks of each observation (for example, one can have the second best score on variable X, but the ninth on variable Y.)
• n : number of scores

Note: if there are tied ranks, the following method can be used to calculate the Spearman Correlation:

$\rho = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt {\sum_i (x_i - \bar{x})^2 \sum_i (y_i - \bar{y})^2}}$

• x : value of x
• x̄ : mean of x
• y : value of y
• ȳ : mean of y
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