Continuity correction


In mathematics, a continuity correction is an adjustment made when a discrete object is approximated using a continuous object.

Examples

Binomial

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
for any x ∈. If np and np are large, then the probability above is fairly well approximated by
where Y is a normally distributed random variable with the same expected value and the same variance as X, i.e., E = np and var = np. This addition of 1/2 to x is a continuity correction.

Poisson

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 λ then the variance of X is also λ, and
if Y is normally distributed with expectation and variance both λ.

Applications

Before the ready availability of statistical software having the ability to evaluate probability distribution functions accurately, continuity corrections played an important role in the practical application of statistical tests in which the test statistic has a discrete distribution: it had a special importance for manual calculations. A particular example of this is the binomial test, involving the binomial distribution, as in checking whether a coin is fair. Where extreme accuracy is not necessary, computer calculations for some ranges of parameters may still rely on using continuity corrections to improve accuracy while retaining simplicity.