The normal distribution is central to the theory of inferential statistics. It also provides a model for many empirical distributions that approximate normality.
In all normal or nearly normal curves, we find a constant proportion of the area under the curve lying between the mean and any given distance from the mean when measured in standard deviation units.
The standard normal distribution is a normal distribution represented in standard scores, or Z scores, with mean = 0 and standard deviation = 1.Z scores express the number of standard deviations that a given score is above or below the mean. The proportions corresponding to any Z score or its fraction are organized into a special table called the standard normal table.