5.4 Approximating the Binomial by the Normal Distribution


10605 twnormalize ()
illustrates the approximation of the binomial by the normal distribution

This quantlet illustrates that the normal distribution provides a good approximation to the binomial distribution for large $ n$, as well as why it is important and natural to subtract the mean and divide by the standard deviation when doing such an approximation.

The mathematical formula used for the graphs is the probability function of the binomial distribution:

$\displaystyle P(X=x)= f(x) = {n\choose x}\, p^x \,(1-p)^{n-x}\,.$

To activate this quantlet, the user should type in the following:

  twnormalize()
After this, the following windows should be displayed:

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The Display window displays graphs of the distribution functions of three binomial distributions, all with $ p = 0.7$, with $ n = 10$, 20, and 40. The graphs on the left are the original graphs, those on the right (identical at first to the ones on the left) will become the transformed graphs.

In the Choose transformation window, the user is asked to select which transformation (subtract the mean or divide by the standard deviation) he/she wishes to apply to the data. Additionally, the Normal distribution choice will superimpose a graph of the standard normal distribution over the transformed data in red. Clicking on the OK button applies these transformations to the data, resulting in the transformed data on the right side of the Display window. For example, choosing the Subtract mean option will result in the right side of the Display window showing the data subtracted from the mean. By subtracting the mean from the data, dividing it by the standard deviation, and then superimposing the normal distribution over it, the user can see that the transformed data look very similar to the standard normal distribution. Furthermore, the user can see that as the sample size increases, the transformed data (i.e. subtracted from the mean and divided by the standard deviation) look more and more similar to the standard normal distribution. Thus, the standard normal distribution can be used to approximate the binomial distribution for large $ n$.