Normal Distribution

TL;DR

A common continuous probability distribution that is symmetric around its mean (the “bell curve”).
Most observations lie within a few standard deviations of the mean.
Frequently used as a model for real-world measurements and for statistical procedures like hypothesis testing and confidence intervals.

Definition

Normal distribution, also known as the bell curve, is a type of statistical distribution that is symmetrical around the mean, with the majority of the data falling within a few standard deviations of the mean. It is a continuous distribution, which means that it is possible for any value within the range of the distribution to occur.

Explanation

The normal distribution models data that cluster symmetrically around an average value, with probabilities tapering off as values move further from the mean. Because many real-world measurements tend to concentrate near an average and spread out in a predictable way, the normal distribution is often used as a model for such data. When sample data follow a normal distribution, statisticians can use standard statistical tests to assess hypotheses and compute confidence intervals, enabling inferences about the broader population from which the sample was drawn.

Examples

Heights of adult men

According to the Centers for Disease Control and Prevention, the average height of adult men in the United States is around 5 feet 9 inches, with a standard deviation of around 3 inches. This means that the majority of men will fall within a few inches of the mean height, with a smaller percentage of men being either shorter or taller. The distribution of men’s heights will resemble a bell curve, with the majority of the data falling within a few standard deviations of the mean.

Distribution of test grades

If a test is designed to be of average difficulty, and the majority of students are of average ability, then it is likely that the grades on the test will follow a normal distribution. The mean grade will be around the average, with most of the grades falling within a few standard deviations of the mean. This means that there will be a small percentage of students who score very high or very low, but the majority of the grades will be clustered around the average.

Use cases

Modeling real-world measurements that tend to cluster around an average.
Statistical testing: using sample data that follow a normal distribution to assess the likelihood of hypotheses.
Calculation of confidence intervals: deriving ranges within which population parameters are likely to fall when sample data are approximately normal.

Bell curve
Mean
Standard deviation
Continuous distribution
Statistical testing
Confidence intervals
Hypothesis
Sample
Population