The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. Like all normal distributions, it follows a symmetric, bell-shaped curve centered at the mean, and it describes how data values are spread around that center.
The horizontal axis of a standard normal curve represents z-scores. Specific areas under the curve correspond to probabilities according to the Empirical Rule. For example, about 68% of the area lies within one standard deviation (between z = –1 and z = 1), about 95% within two, and about 99.7% within three.
The standard normal distribution is useful because it provides a common reference for comparing different normal distributions. By converting values from any normal distribution into z-scores, you can use a z-table for the standard normal distribution to calculate probabilities and percentiles.
One of the most powerful uses of the standard normal distribution is that it allows analysts to standardize values from different datasets or distributions using z-scores. A z-score tells you how many standard deviations a particular value is from the mean of its distribution. By converting raw scores into z-scores, you can place data from different sources onto a common scale, regardless of the original units or spread.
Standardizing values in this manner allows you to compare values across different variables or populations. For example, comparing test scores from exams with different averages and standard deviations. Once standardized, these values can be interpreted using the standard normal distribution.
For example, if a student scores a 75 on a test where the mean is 70 and the standard deviation is 5, their z-score is (75 – 70) / 5 = +1. This means the student scored one standard deviation above average. Using the a z-table for the standard normal distribution, you can estimate what percentage of students scored lower.
« Back to Glossary Index