The Normal Approximation to the Binomial

Computing binomial probabilities using the binomial PDF can be difficult for large n. If tables are used to compute binomial probabilities, calculations typically are only given for selected values of n <= 50 and for selected values of p. The Binomial Applet is much more flexible, since it allows any value of n <= 400 and any valid value of p. However, if n is quite large or if the binomial applet is not available, the normal distribution can be used to approximate the binomial distribution.

Binomial/Normal Distribution

The following Normal Approximation Applet can be used to experiment with computing approximate and exact binomial probabilities and to assess the conditions for which the normal approximation to the binomial distribution is good.

The normal distribution is a good approximation to the binomial when n is sufficiency large and p is not too close to 0 or 1. How large n needs to be depends on the value of p. If p is near 0.5, the approximation can be good for n much less than 20. However, it is better to be conservative and limit the use of the normal distribution as an approximation to the binomial when np > 5 and n(1 - p) > 5.

Examples

Example #1 computes binomial probabilities and quantiles for the distribution of hotel room occupancies and compares them to probabilities computed approximately using a normal approximation.

Exercises

Exercise #1 computes binomial probabilities and quantiles for the distribution of girl births and compares them to probabilities computed approximately using a normal approximation.

Exercise #2