The normal distribution is defined by a mathematical formula. Although normal
distributions may have different means and standard deviations, all normal distributions
are "bell-curve" shaped, symmetrical with the greatest height at the mean. Tails of a normal distribution are asymptotic, indefinitely
decreasing but never touching the *x*-axis. The total area under the curve sums to
100%.

The normal distribution may characterize either distributions of individual data points in a population of scores or the theoretical distribution of sample statistics such as the mean.

*Important note*: Before we use the normal distribution to
compute probabilities, we must verify that the distribution of interest
is very close to normal. Although a distribution of scores in a sample of
*N* cases may be quite far from normal, the distribution of means
for all possible samples of *N* cases may be quite close to normal. This fact, as described in the
Central Limit Theorem, is critical for many applications of statistical
inference.

A normal distribution that is standardized (so that it has a mean of 0 and a *S.D.* of 1)
is called the standard normal distribution, which represents a distribution of *z*-scores. The formula to convert a sample mean, *X*,
to a *z*-score, is:

where *m* is the population mean,
*s* is the population standard deviation, and
*N* is the sample size.

- Note that converting values, such as sample means, to z scores does NOT change the shape of the distribution. The distribution of z scores is normal if and only if the distribution of the values is normal.
- Depending upon the sample size and the
shape of the population distribution, the
sampling distribution of means may be very close to a
normal distribution even when the population
distribution

By converting normally distributed values into *z*-scores, we can ascertain the
probabilities of obtaining specific ranges of scores using
either a table for the standard normal distribution (i.e., a z-table) or a calculator like
the WISE
*p*-*z* converter. *Caution*: It is not appropriate to use the *z*-table to
find probabilities unless you are confident that the shape of your distribution of interest is very
close to the normal distribution!