# 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

for any *x* ∈ {0, 1, 2, ... *n*}. If *np* and *n*(1 − *p*) are large (sometimes taken to mean ≥ 5), 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(*Y*) = *np* and var(*Y*) = *np*(1 − *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 λ then the variance of *X* is also λ, and

if *Y* is normally distributed with expectation and variance both λ.

See also Yates' correction for continuity.

## References

- Devore, Jay L.,
*Probability and Statistics for Engineering and the Sciences*, Fourth Edition, Duxbury Press, 1995.