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Probability Distributions 53FIGURE 2.5 Galton’s boardaverages of samples of a given size, taken such that each sampled item is
selected independently of the others, will yield a distribution that is close
to Normal. This is an extremely powerful fact, for it means that you can
generalize about an actual random process from averages computed using
sample data.
Thus, we can state thatif N random samples are drawn from a popu-
lation, then the sums (or averages) of the samples will be approximately
Normally distributed, regardless of the distribution of the population
from which the samples are drawn. The closeness to the Normal Dis-
tribution improves as N (the number of samples) increases.
As an example, consider the distribution of numbers from 1 to 100.
This is what is known as auniform distribution: All elements (numbers
in this case) occur only once. The number 82 occurs once and only once,
as does 19, and so on. Suppose now that we take a sample of five elements
and we take the average of these five sampled elements (we can just as