The normal distribution can be used to approximate the binomial distribution. This demonstration allows you to explore the accuracy of the approximation under a variety of conditions.

In this demonstration, you specify the number of trials (N) and the probability of a success on any given trial (p). Note that in the text the Greek letter π is used for the probability of success.

The default values for the demonstration are N = 10 and p = 0.5. The demonstration shows that the mean and standard deviation of the binomial distribution with these parameters are 5 and 1.58 respectively.

The demonstration lets you specify the range of values for which you wish to compute the probability of occurring. The default is to compute the probability of fewer than 3 successes. The calculation based on the normal approximation to the binomial is shown in green at the bottom of the screen and is equal to 0.1714. The actual binomial probability of 0.1719 is shown in red. The approximation is quite good for these parameters. The graph displays the binomial probabilities as vertical lines. The lines included in the probability calculation are shown in red, the others are shown in blue. The normal approximation is shown as the area shaded in green. Note that the area below 3.5 is shaded to calculate the probability of 3 or fewer. This is done to correct for the fact that the normal distribution is a continuous distribution and the binomial distribution is a discrete distribution.

By clicking the appropriate radio button, you can choose to calculate the probability above a specified value or between two values.

Try out various values of N and p to calculate various approximations to the binomial to explore the accuracy of the approximation.