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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Applying spearman's correlation

What does the Speaman Correlation measure?

  • The Spearman correlation (denoted as p (rho) or rs) 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
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