Continuity correction

In probability theory, if a random variable X has a binomial distribution with parameters n and p, i.e., X is distributed as the number of "successes" in n independent Bernoulli trials with probability p of success on each trial, then


 * $$P(X\leq x) = P(X<x+1)$$

for any x &isin; {0, 1, 2, ... n}. If np and n(1 &minus; p) are large (sometimes taken to mean &ge; 5), then the probability above is fairly well approximated by


 * $$P(Y\leq x+1/2)$$

where Y is a normally distributed random variable with the same expected value and the same variance as X, i.e., E(Y) = np and var(Y) = np(1 &minus; p). This addition of 1/2 to x is a continuity correction.

A continuity correction can also be applied when other discrete distributions supported on the integers are approximated by the normal distribution. For example, if X has a Poisson distribution with expected value &lambda; then the variance of X is also &lambda;, and


 * $$P(X\leq x)=P(X<x+1)\approx P(Y\leq x+1/2)$$

if Y is normally distributed with expectation and variance both &lambda;.

See also Yates' correction for continuity.