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.

### Standard Normal Distributions and *Z* Scores

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!

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