In statistics, the Fisher transformation (or Fisher z-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficient r is near 1 or -1, its distribution is highly skewed, which makes it difficult to estimate confidence intervals and apply tests of significance for the population correlation coefficient ρ. The Fisher transformation solves this problem by yielding a variable whose distribution is approximately normally distributed, with a variance that is stable over different values of r. Definition Given a set of N bivariate sample pairs (Xi, Yi), i = 1, …, N, the sample correlation coefficient r is given by Here stands for the covariance between the variables and and stands for the standard deviation of the respective variable. Fisher's z-transformation of r is defined as where "ln" is the natural logarithm function and "artanh" is the inverse hyperbolic tangent function. If (X, Y) has a bivariate normal distribution with correlation ρ and the pairs (Xi, Yi) are independent and identically distributed, then z is approximately normally distributed with mean and standard error where N is the sample size, and ρ is the true correlation coefficient. This transformation, and its inverse can be used to construct a large-sample confidence interval for r using standard normal theory and derivations. See also application to partial correlation. Derivation Hotelling gives a concise derivation of the Fisher transformation. To derive the Fisher transformation, one starts by considering an arbitrary increasing, twice-differentiable function of , say . Finding the first term in the large- expansion of the corresponding skewness results in Setting and solving the corresponding differential equation for yields the inverse hyperbolic tangent function. Similarly expanding the mean m and variance v of , one gets m = and v = respectively. The extra terms are not part of the usual Fisher transformation. For large values of and small values of they represent a large improvement of accuracy at minimal cost, although they greatly complicate the computation of the inverse – a closed-form expression is not available. The near-constant variance of the transformation is the result of removing its skewness – the actual improvement is achieved by the latter, not by the extra terms. Including the extra terms, i.e., computing (z-m)/v1/2, yields: which has, to an excellent approximation, a standard normal distribution. Application The application of Fisher's transformation can be enhanced using a software calculator as shown in the