9-3-2016 Update:

One of the best known technique for transforming correlation coefficient (r) values into weighted additive quantities is the r-to-Z transformation due to Fisher.

Fisher’s r-to-Z transformation is an elementary transcendental function called the inverse hyperbolic tangent function. The reverse, a Z-to-r transformation, is therefore a hyperbolic tangent function.

In Windows computers, these functions are built-in in their scientific calculator program which is accessible by navigating to Start > All Programs > Accessories > Calculator. Microsoft Excel also has these built-in as the ATANH and TANH functions.

Fisher’s r-to-Z transformation is applicable only to bivariate normal distributions; i.e. if the (x, y) paired variables both describe bell-shaped curves. Non trivial errors arise if one of the variables is not normally distributed.

5-22-2016 Update: We have developed a tool for easily computing these transformations and explaining the bivariate normality restriction.

The tool is available at

http://www.minerazzi.com/tools/fisher/transformations.php

Beware of Sloppy Calculations

Correlation coefficient arithmetic averages are not computable directly from individual values.

Indeed, it is not possible to add, subtract, average or take standard deviations out of raw r values.

Unfortunately some researchers with a limited knowledge on Statistics have published papers containing such gross errors. What is worse, reviewers of those papers are either not statisticians or have been lazy enough to overlook at the concept, leading graduate students and post docs into error.

Search marketers are also buying into the error. An example of this are the SEOs from SeoMOZ (now MOZ) promoting quack “science” and sloppy “statistical studies”. If you are an SEO and still want to believe their snakeoil marketing, that’s up to you.