# Effect size estimation for nonpar-test. Does anybody know the mathematical derivation of r=z/sqrt(N) ? (Rosenthal 1991)

In order to calculate an effect size estimator for a non-parametric test such as the Wilcoxon test, the following formula is applied: $r= \frac{Z}{\sqrt{N}}$

This formula is suggested by several forum contributions and is also used by SPSS for the purpose described above. Originally the formula was developed to my knowledge by Rosenthal (1991, p.19).

Beside just using the formula, I am interested at how to arrive at it as I do not see the connection between the correlation coefficient and the z value. If someone could provide an explanation of how to arrive at the formula or a derivation, I would be glad.

The only hint I have about the formula is in the following table equation 2.1 and 2.2. Rosenthal does not provide unfortunately an explanation about this relationship. Note: There has been already a discussion about this topic but rather in the form how to use and interpret it and not how to arrive at it. Effect size to Wilcoxon signed rank test?

Rosenthal, R. (1991). Applied Social Research Methods: Meta-analytic procedures for social research Thousand Oaks, CA: SAGE Publications Ltd doi: 10.4135/9781412984997

• Minimal references like Rosenthal (1991) can possibly be guessed but are in general a bad idea. Please give full details as expected in an academic thesis or paper. – Nick Cox Jan 16 '18 at 17:29
• You are absolutely right. Rosenthal, R. (1991). Applied Social Research Methods: Meta-analytic procedures for social research Thousand Oaks, CA: SAGE Publications Ltd doi: 10.4135/9781412984997 – Abar Jan 16 '18 at 17:39
• Please tell us what "$Z$" might be and how it is related to $r$. – whuber Jan 16 '18 at 18:48
• Z stands for the z value. How the z value like estimated by the Wilcoxon-Test and the correlation coefficient are related, is exactly my question. As for this formula there is no explanation in the stated paper. However, as I found out SPSS calc. for the Wilcoxon-test an effect size using this formula. – Abar Jan 16 '18 at 19:03
• @whuber : I hope this rephrasing helps to understand my question. – Abar Jan 16 '18 at 21:22