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.
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!