If X is a binomial variable with parameters n and p, then the transformed variable Y, given in radians by , has an approximate normal distribution with mean sin-1(√p) 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 with expectation μ, then, for μ>1, the transformed variable Z, given by , has 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.
Subjects: Probability and Statistics.