According to the central limit theorem ( CLT), the distribution function F n of a normalized sum n −1∕2( X 1 + … + X n ) of n independent random variables X 1, …, X n , having a common distribution with mean zero and variance σ 2 > 0, converges to the distribution function Φ σ of the normal distribution with mean zero and variance σ 2, as \(n\rightarrow \infty \) . We will write Φ for Φ 1 for the case σ = 1. The densities of Φ σ and Φ are denoted by ϕ σ and ϕ, respectively. In the case \(X_j^{\prime }s\) are discrete, F n has jumps and the normal approximation is not very good when n is not sufficiently large. This is a problem which most commonly occurs in statistical tests and estimation involving the normal approximation to the binomial and, in its multi-dimensional version, in Pearson’s frequency chi-square tests, or in tests for association in categorical data.