Fisher transformation

The Fisher transformation is a statistical technique, introduced by Ronald Fisher in 1915, that converts a sample Pearson correlation coefficient r into a variable whose distribution is approximately normal, using the formula z equals one half times the natural log of (1 + r) over (1 - r), which is the inverse hyperbolic tangent of r. It addresses a specific problem with correlation coefficients: as r approaches plus or minus one, its sampling distribution becomes highly skewed, which makes it difficult to build reliable confidence intervals or run standard significance tests directly on r. Because the Fisher-transformed variable z is approximately normally distributed with a variance that stays stable across different values of r, it is described as a variance-stabilizing transformation, and it lets researchers construct confidence intervals and hypothesis tests for correlation using ordinary normal-distribution methods.