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This quantlet illustrates that the normal distribution
provides a good approximation to the binomial distribution
for large , 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:
To activate this quantlet, the user should type in the following:
twnormalize()After this, the following windows should be displayed:
The Display window displays graphs of the distribution
functions of three binomial distributions, all with ,
with
, 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 .