The normal approximation to data is to approximate areas under the histogram of data, transformed into standard units, by the corresponding areas under the normal curve.

Many probability distributions can be approximated by a normal distribution, in the sense that the area under the probability histogram is close to the area under a corresponding part of the normal curve. To find the corresponding part of the normal curve, the range must be converted to standard units, by subtracting the expected value and dividing by the standard error. For example, the area under the binomial probability histogram for n = 50 and p = 30% between 9.5 and 17.5 is 74.2%. To use the normal approximation, we transform the endpoints to standard units, by subtracting the expected value (for the Binomial random variable, n×p = 15 for these values of n and p) and dividing the result by the standard error (for a Binomial, (n × p × (1−p))^1/2 = 3.24 for these values of n and p). The area normal approximation is the area under the normal curve between (9.5 − 15)/3.24 = −1.697 and (17.5 − 15)/3.24 = 0.772; that area is 73.5%, slightly smaller than the corresponding area under the binomial histogram. See also the continuity correction. The tool on this page illustrates the normal approximation to the binomial probability histogram. Note that the approximation gets worse when p gets close to 0 or 1, and that the approximation improves as n increases.