The Effects of an Arcsin Square Root Transform on a Binomial Distributed Quantity

This document provides proofs of the following: • The binomial distribution can be approximated with a Gaussian distribution at large values of N . • The arcsin square-root transform is the variance stabilising transform for the binomial distribution. • The Gaussian approximation for the binomial distribution is more accurate at smaller values of N after the variance stabilising transform has been applied. The conclusion contains some comments concerning the relationship between the variance stabilising transform and the improved accuracy of the Gaussian approximation, which holds for both the binomial and Poisson distributions. 1 Gaussian Approximation to the Binomial Distribution The binomial distribution P (n|N) = N ! n!(N − n)! nq(N−n) (1) gives the probability of obtaining n successes out of N Bernoulli trials, where p is the probability of success and q = 1− p is the probability of failure. The shape of the distribution approaches that of a Gaussian distribution at large N as a consequence of the central limit theorem. In order to demonstrate this, it is necessary to expand the distribution as a Taylor series around the maximum. First, let n′ be the position of the maximum of P (n), giving n = n′+η, and rather than expanding the distribution itself, expand the natural logarithm of the distribution. Expanding as a Taylor series f(a+ h) = f(a) + hf ′(a) + h 2! f ′′(a) + ...+ h(n−1) (n− 1)! (n−1)(a) + ... (2) gives ln[P (n′ + η)] = ln[P (n′)] + ηB1 + η 2! B2 + η 3! B3... (3) where Bk = ∣