If X is a binomial variable (see binomial distribution) with parameters n and p, then the transformed variable Y, given in radians byhas an approximate normal distribution with mean and variance 1/(4n + 2). The approximation improves as np increases and should not be used if np<1.

If X is a Poisson variable (see Poisson distribution) with expected value μ, then, for μ>1, the transformed variable Z, given byhas an approximate standard normal distribution. Observed values of Z are referred to as Freeman–Tukey deviates. Both transformations were proposed in a 1950 paper by M. F. Freeman and Tukey.