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Part III: Distributions and the Central Limit Theorem
You take your x-value, subtract the mean of X, and divide by the standard
deviation of X. This gives you the corresponding standard score (z-value or
z-score).
Standardizing is just like changing units (for example, from Fahrenheit to
Celsius). It doesn’t affect probabilities for X; that’s why you can use the
Z-table to find them!
You can standardize an x-value from any distribution (not just the normal)

using the z-formula. Similarly, not all standard scores come from a normal
distribution.

Because you subtract the mean from your x-values and divide everything by
the standard deviation when you standardize, you are literally taking the mean
and standard deviation of X out of the equation. This is what allows you to
compare everything on the scale from –3 to +3 (the Z-distribution) where nega-
tive values indicate being below the mean, positive values indicate being
above the mean, and a value of 0 indicates you’re right on the mean.
Standardizing also allows you to compare numbers from different distribu-
tions. For example, suppose Bob scores 80 on both his math exam (which has
a mean of 70 and standard deviation of 10) and his English exam (which has a
mean of 85 and standard deviation of 5). On which exam did Bob do better, in
terms of his relative standing in the class?
Bob’s math exam score of 80 standardizes to a z-value of . That
tells us his math score is one standard deviation above the class average. His
English exam score of 80 standardizes to a z-value of , putting
him one standard deviation below the class average. Even though Bob scored
80 on both exams, he actually did better on the math exam than the English
exam, relatively speaking.
To interpret a standard score, you don’t need to know the original score, the
mean, or the standard deviation. The standard score gives you the relative
standing of a value, which in most cases is what matters most. In fact, on most
national achievement tests, they won’t even tell you what the mean and stan-
dard deviation were when they report your results; they just tell you where
you stand on the distribution by giving you your z-score.
Finding probabilities for Z with the Z-table
A full set of less-than probabilities for a wide range of z-values is in the Z-table
(Table A-1 in the appendix). To use the Z-table to find probabilities for the
standard normal (Z-) distribution, do the following:

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