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I tried to calculate Spearman's rank coefficient by hand.

Data:

enter image description here

enter image description here

But when I use python, it returns a different value.

rawdata = pd.DataFrame(
    [
        [3,4],
        [5,4],
        [6,2],
        [6,4],
        [8,9],
        [11,7]
    ],
    columns=['Set of A','Set of B'])
print(rawdata)
correlation, pval = spearmanr(rawdata)
print(f'correlation={correlation:.6f}, p-value={pval:.6f}')

It returns:

correlation=0.585239, p-value=0.222365

What am I doing wrong here?

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1 Answer 1

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The formula you're using is a simplified version for when there are no ties. Try the full version of the formula:

$r_s=\frac{\Sigma_i(a_i-\bar{a})(b_i-\bar{b})}{\sqrt{\Sigma_i(a_i-\bar{a})^{2}(b_i-\bar{b})^{2}}}$

Where $a_i$ and $b_i$ are the fractional ranks you've already found in your table

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  • $\begingroup$ You seem to be writing about the Pearson correlation coefficient. The calculations presented in the question clearly deal with ties (in a standard manner). $\endgroup$
    – whuber ♦
    Feb 14, 2020 at 18:09
  • $\begingroup$ I believe that Spearman correlation is equal to the Pearson correlation of the fractional ranks $\endgroup$
    – CFD
    Feb 14, 2020 at 18:11
  • 1
    $\begingroup$ Thank you for modifying and explaining your notation. Doing the calculation confirms your answer is a correct explanation of the discrepancy (+1). $\endgroup$
    – whuber ♦
    Feb 14, 2020 at 18:36

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