# Continuity Correction

A continuity correction may be applied when a continuous distribution is used to approximate a discrete distribution.   It is used in order to improve the approximation.

Probably the most straightforward example involves approximating the binomial distribution with the normal distribution.

Consider a binomial random variable X with probability of “success” equal to p and n number of trials. A simple example of a binomial random variable is the number of heads obtained when one flips a “fair” coin 10 times. Here p = 0.5 and n = 10. For a given p, as the number of trials increases, the normal distribution can be used to approximate the binomial distribution. The normal approximation is best when p is “close” to 0.5 since the binomial distribution is symmetric at that value. The approximation improves as n gets larger and larger.

For example, the probability of obtaining at most 13 heads out of 30 flips of fair coin is about .292332. Using the normal approximation without the continuity correction is:

We can improve the approximation by adding the continuity correction.

In this case, we need to add the area between 13.5 and 13 to this calculation. Doing so gives the probability:

This is a much better approximation.

The continuity correction can be applied to any discrete distribution approximated by a continuous distribution.  This is of practical importance whenever the normal distribution is used in approximating a p-value for a test statistic that has a discrete distribution. In this situation, the use of the correction avoids inflation of the type 1 error rate. See Sheskin (1997).