Fisher's Z Transformation

Fisher's Z transformation is a procedure that rescales the product-moment correlation coefficient into an interval scale that is not bounded by + 1.00. It may be used to test a null hypothesis that an obtained correlation is significantly different from some hypothesized value (usually a nonzero value, because a t test is available to test whether ρ = 0), to test the significance of the difference between two independent correlations, to find the average of several correlations, or to form a confidence interval (CI) for a correlation coefficient.

Like all statistics, the correlation coefficient is subject to sampling variation. For a given population, the sample correlation coefficient (r) has a sampling distribution around its population parameter, ρ (Greek lowercase letter rho), and this distribution has a standard error, the standard error of the correlation coefficient, σr. However, the sampling distribution of r has a different shape, depending on the value of ρ. The possible values of r are limited to the range between +1.0 and −1.0. If the value of ρ is about zero, sample deviations can occur equally in either direction, and the distribution is symmetric. However, as ρ departs from zero, this symmetry is lost because one end of the sampling distribution is more restricted than the other due to the limiting values of + 1.0. There is more space on one side of the parameter than on the other. For extreme values of ρ, the sampling distribution of r becomes markedly skewed. The skew becomes noticeable at about ρ= .40.

A solution to this problem was developed by R. A. Fisher, who proposed a transformation for r that would normalize the distribution. The resulting index is called Fisher's Z or ZF. Note that this is not the same quantity as the standard score or the critical ratio, each of which is a deviation of an observed value from the [Page 361]distribution mean, divided by the standard deviation. The statistic Fisher developed,

has the advantage that its sampling distribution is almost exactly normal for any value of ρ and has a standard error of

This standard error does not depend on the value of ρ, unlike the standard error used in the t test for testing the hypothesis that ρ= 0. Also, σZF is calculated, not estimated, so the test statistic is a critical ratio Z and is more powerful than the t test.

Because r can have only a limited number of values (to 2 decimal places) and the distribution is symmetric around zero, some statistics books contain tables for transforming r to ZF and ZF back to r. However, the equations for transforming a correlation to the metric of ZF and back again are so simple that they are easily entered into a cell of a spreadsheet or programmed as a macro. The equation for the reverse transformation is

where the first form of the expression is in the notation used for computations by EXCEL and the second is in standard notation. However, many spreadsheets also include functions for converting to and from ZF in their function libraries. For example, EXCEL has a function called FISHER for the r-to-ZF transformation in Equation 1 and one called FISHERINV for the reverse transformation in Equation 3. Pasting FISHER into a cell produces the dialog box in Figure 1. Entering the observed r for X produces ZF. In this example, r = .80 produces ZF = 1.0986.