You are working with the text-only light edition of "H.Lohninger: Teach/Me Data Analysis, Springer-Verlag, Berlin-New York-Tokyo, 1999. ISBN 3-540-14743-8". Click here for further information. |
Table of Contents Bivariate Data Correlation Spearman's Rank Correlation | |
See also: correlation coefficient |
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 r_{s} (). Basically, r_{s} 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).
with D_{i} 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.
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 2In 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:
r_{s} = 1 - (6*76.5/(10*(100-1)) = 0.5364
Last Update: 2006-Jän-17