Normal Approximation to the Binomial

For certain values of n and p, the binomial distribution can be approximated by the normal distribution with good results. 

Suppose we are conducting a binomial experiment with 50 trials and the probability of a "success" is 0.4 (n = 50 and p = 0.4).  In one experiment we would expect to get np = (50)(0.4) = 20 successes, but since each experiment is random, some experiments will produce less than 20 successes and some will produce more.

An Excel spreadsheet was used to replicate this binomial experiment 100 times.  The results of are shown in the graph below:

 

Note that the distribution of successes appears to be normal.  In fact, it has been shown that if we have npq > 5 the normal distribution can be used to approximate the binomial distribution.  Note that, with this criteria, if we have p = 0.5, then the sample size only has to be 20 or more for the normal approximation to be good.  If we have p = 0.01, then the sample size has to be 505 or greater.

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