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170 Part III: Distributions and the Central Limit Theorem

Case 1: The distribution of X is normal

If X has a normal distribution, then does too, no matter what the sample
size n is. In the example regarding the amount of time (X) for a clerical
worker to complete a task (refer to the section “Sample size and standard
error”), you knew X had a normal distribution (refer to the lowest curve in
Figure 11-2). If you refer to the other curves in Figure 11-2, you see the aver-
age times for samples of n = 10 and n = 50 clerical workers, respectively, also
have normal distributions.
When X has a normal distribution, the sample means also always have a
normal distribution, no matter what size samples you take, even if you take
samples of only 2 clerical workers at a time.

The difference between the curves in Figure 11-2 is not their means or their
shapes, but rather their amount of variability (how close the values in the
distribution are to the mean). Results based on large samples vary less and
will be more concentrated around the mean than results from small samples
or results from the individuals in the population.

Case 2: The distribution of X is not normal —
enter the Central Limit Theorem

If X has any distribution that is not normal, or if its distribution is unknown,
you can’t automatically say the sample mean ( ) has a normal distribution.
But incredibly, you can use a normal distribution to approximate the distribu-
tion of — if the sample size is large enough. This momentous result is due
to what statisticians know and love as the Central Limit Theorem.
The Central Limit Theorem (abbreviated CLT) says that if X does not have a
normal distribution (or its distribution is unknown and hence can’t be deemed
to be normal), the shape of the sampling distribution of is approximately
normal, as long as the sample size, n, is large enough. That is, you get an
approximate normal distribution for the means of large samples, even if the
distribution of the original values (X) is not normal.
Most statisticians agree that if n is at least 30, this approximation will be rea-
sonably close in most cases, although different distribution shapes for X have
different values of n that are needed. The larger the sample size (n), the closer
the distribution of the sample means will be to a normal distribution.

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