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Spearman's Rank Correlation

Calculating the correlation coefficient requires the two samples to be normally distributed. In the case of non-normal distributions, Pearson's correlation coefficient will lead to wrong results. A remedy to this situation may be the use of Spearman's rank correlation rs (). Basically, rs differs from Pearson's correlation only in that the values are converted to ranks before computing the coefficient (the numerical equivalence is only true for untied data, in the case of tied data Pearson's and Spearman's coefficient will be slightly different).

The equation for the correlation coefficient may be simplified by replacing the observation values with the rank numbers, resulting in the following formula:

with Di being the differences of the rank numbers. The equation is valid when n is greater than 4.

In the case of tied observations (observations which are numerically equal) one has to take the arithmetic average of the rank numbers associated with the ties.

Example

Two persons taste 10 red Italian wines, grading them by an ordinal scale between 1 and 5. The results are as follows:

                                 Grades     Grades
                    Wine No.    Person 1   Person 2
                       1           1          2
                       2           2          3
                       3           4          5
                       4           5          4
                       5           2          2
                       6           2          2
                       7           4          3
                       8           3          4
                       9           1          3
                      10           4          2
In order to calculate Spearman's rank correlation coefficient we first have to sort the grades for each person. The resulting rank numbers are averaged for tied observations:
                                Grades     Rank  Rank
                    Wine No.    Person 1        (ties)
                       1           1         1    1.5
                       9           1         2    1.5
                       6           2         3     4
                       5           2         4     4
                       2           2         5     4
                       8           3         6     6
                      10           4         7     8
                       7           4         8     8
                       3           4         9     8
                       4           5        10    10


                                 Grades    Rank  Rank
                    Wine No.    Person 2        (ties)
                       5           2         1    2.5
                       1           2         2    2.5
                      10           2         3    2.5
                       6           2         4    2.5
                       2           3         5     6
                       7           3         6     6
                       9           3         7     6
                       8           4         8    8.5
                       4           4         9    8.5
                       3           5        10    10

The final table of ranks includes the differences of the ranks as well as the squared differences:


Wine No.    Person 1  Rank       Person 2  Rank      Rank      squared
                                                   Difference   Diff.
   1           1       1.5         2       2.5        -1          1
   2           2        4          3        6         -2          4
   3           4        8          5       10         -2          4
   4           5       10          4       8.5        1.5        2.25
   5           2        4          2       2.5        1.5        2.25
   6           2        4          2       2.5        1.5        2.25
   7           4        8          3        6          2          4
   8           3        6          4       8.5       -2.5        6.25
   9           1       1.5         3        6        -4.5       20.25
  10           4        8          2       2.5        5.5       30.25
------------------------------------------------------------------------
                                               sum    0.0       76.50

The sum of the squared differences is used to calculate Spearman's rank correlation coefficient as follows:

rs = 1 - (6*76.5/(10*(100-1)) = 0.5364


Last Update: 2006-Jšn-17