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
• r_s : Spearman correlation
• d_i : rg(X_i) - rg(Y_i): 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

What does the Speaman Correlation measure?

• The Spearman correlation (denoted as p (rho) or r_s) measures the strength and direction of association between two ranked variables.
• It is most commonly used to measure the degree and direction of a linear relation between two variables that are of the ordinal type.

What are the assumptions of the Spearman Correlation test?

• two variables that are either ordinal, interval or ratio (note: if the data are interval or ratio, normally you would use a Pearson correlation test)
• the relationship between the variables is monotonic, or (curvi)linear