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Part III: Distributions and the Central Limit Theorem
Noting the mean and standard deviation is important so you can properly inter-

pret numbers located on a particular normal distribution. For example, you can
compare where the number 120 falls on each of the normal distributions in
Figure 9-1. In Figure 9-1a, the number 120 is one standard deviation above the
mean (because the standard deviation is 30, you get 90 + 1 ∗ 30 = 120). So on this
first distribution, the number 120 is the upper value for the range where about
68% of the data are located, according to the Empirical Rule (see Chapter 5).
In Figure 9-1b, the number 120 lies directly on the mean, where the values
are most concentrated. In Figure 9-1c, the number 120 is way out on the
rightmost fringe, 3 standard deviations above the mean (because the stan-
dard deviation this time is 10, you get 90 + 3[10]=120). In Figure 9-1c, values
beyond 120 are very unlikely to occur because they are beyond the range
where about 99.7% of the values should be, according to the Empirical Rule.
Meeting the Standard Normal
(Z-) Distribution
One very special member of the normal distribution family is called the
standard normal distribution, or Z-distribution. The Z-distribution is used to
help find probabilities and percentiles for regular normal distributions (X). It
serves as the standard by which all other normal distributions are measured.
Checking out Z
The Z-distribution is a normal distribution with mean zero and standard
deviation 1; its graph is shown in Figure 9-2. Almost all (about 99.7%) of its
values lie between –3 and +3 according to the Empirical Rule. Values on the
Z-distribution are called z-values, z-scores, or standard scores. A z-value
represents the number of standard deviations that a particular value lies
above or below the mean. For example, z = 1 on the Z-distribution represents
a value that is 1 standard deviation above the mean. Similarly, z = –1 repre-
sents a value that is one standard deviation below the mean (indicated by the
minus sign on the z-value). And a z-value of 0 is — you guessed it — right on
the mean. All z-values are universally understood.
If you refer back to Figure 9-1 and the discussion regarding where the number
120 lies on each normal distribution in “Exploring the Basics of the Normal
Distribution,” you can now calculate z-values to get a much clearer picture.
In Figure 9-1a, the number 120 is located one standard deviation above the
mean, so its z-value is 1. In Figure 9-1b, 120 is equal to the mean, so its z-value
is 0. Figure 9-1c shows that 120 is 3 standard deviations above the mean, so
its z-value is 3.

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