Standard Normal Distribution
Because we are dealing with a theoretical distribution, we will use μ and σ , rather than and s , when
referring to the normal curve. If X is a variable that has a normal distribution with mean μ and standard
deviation s (we say “X has N (μ ,s )”), there is a related distribution we obtain by standardizing the data
in the distribution to produce the standard normal distribution . To do this, we convert the data to a set
of z -scores, using the formula
The algebraic effect of this, as we saw earlier, is to produce a distribution of z -scores with mean 0
and standard deviation 1. Computing z -scores is just a linear transformation of the original data, which
means that the transformed data will have the same shape as the original distribution. In this case then, the
distribution of z -scores is normal. We say z has N (0,1). This simplifies the defining density function to
For the standardized normal curve, the 68-95-99.7 rule says that approximately 68% of the terms lie
between z = 1 and z = –1, 95% between z = –2 and z = 2, and 99.7% between z = –3 and z = 3. (Trivia
for calculus students: one standard deviation from the mean is a point of inflection .)
Because many naturally occurring distributions are approximately normal (heights, SAT scores, for
example), we are often interested in knowing what proportion of terms lie in a given interval under the
normal curve. Problems of this sort can be solved either by use of a calculator or a table of Standard
Normal Probabilities (Table A in the appendix to this book). In a typical table, the marginal entries are z -
scores, and the table entries are the areas under the curve to the left of a given z -score. All statistics texts
have such tables.