A normal distribution is a distribution that is symmetric and is shaped like a bell:
Bell curves can represent lots of natural phenomena. For example, grades in class will often be bell-curved: most students will get C in class (the average), a smaller number of students will get a B or a D, and even fewer students will get an A or an F.
IQ scores naturally fall into a bell curve shape; the following graph shows that the majority of people have IQ scores between 85 and 115.
While the bell curve is useful, it would be tricky to try to figure out how many people have IQ scores between 110 and 130. Or perhaps you would want to know what percentage of students got grades slightly below passing (67-69). The math would get a little tricky, but not if you superimpose your information onto a normal distribution curve.
The area under the normal distribution curve represents 100% probability. In statistics and probability, 100% is written as a decimal: 1, so you will often see it mentioned that the total area under the curve is 1. The mean (the average) of a normal distribution is always zero. Why? Because it makes it easier to look up normal distribution probability (a z-score) using a z-table.
Normal distribution probability (and the associated z-scores) allow you to look up percentage probabilities for any set of data that is shaped like a bell-curve.
The above normal probability distribution shows what percentage of scores fall within a number of standard deviations from the mean. Standard deviations are the same as z-scores; a standard deviation of 1 corresponds to a z-score of 1 and a standard deviation of 2 corresponds to a z-score of 2. 68% of scores (IQ scores, grades, heights, weights and a host of other phenomena) will fall between standard deviations of -1 and +1 (z-scores of -1 and 1).
How to Calculate Normal Distribution Probability
Normal distribution probabilities can be calculated with a little basic arithmetic and use of a z-table. This article explains how to find normal distribution probabilities for a variety of different situations.