# Summary

Probability distributions show the pattern of how data are distributed around a central value and help visualize the expected frequency of different values. For example, if researchers survey people’s heights in the general population, a probability distribution allows them to see the relative occurrence of a random person’s height within that population.

Different types of probability distributions are used to model different types of data. When a variable is *continuous* (e.g., variables like height, weight, etc.), data derived from that variable in the natural world often fit a *normal distribution*.

# What is a normal distribution?

A normal distribution — also called a Gaussian distribution — is a type of probability distribution that is often referred to as a “bell curve” because it looks like a bell. The data points are symmetrically distributed around the central value, with the probability of finding extreme values decreasing sharply with distance from the central value. The central value represents the average value, which is called the *mean* in a normal distribution. In a normal distribution, the mean is also equal to the *median*, which is the middle value, or 50th percentile. Figure 1 shows an example of normally distributed data.

Normally distributed data show that most data points cluster around the mean. The further from the mean, the fewer data points. In fact, in a normal distribution, approximately 68% of the data points fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations.

To conceptualize this pattern, imagine taking a random sample of the general population and plotting people’s heights against the number of times those heights are observed. The plot will show a distribution of heights around the mean (the average) and fewer heights in the extremes. The majority (99.7%) of heights will fall within 3 standard deviations of the mean (average) height. The National Health and Nutrition Examination Survey (NHANES) shows that the average (mean) height of male adults in the U.S. is 5’9” (175.3 cm) and the standard deviation is 5.4 in (13.6 cm).^{[1]} According to a normal distribution, nearly all (99.7%) male adult heights will fall within 3 standard deviations (16 in or 40.8 cm) of 5’9”, or within the range of 5’–6’5” (154.9–195.7 cm).

The normal distribution arises when the variable being measured results from the sum of several independent random processes. This phenomenon is frequently observed in nature. For example, height arises from a combination of several genetic and environmental factors,^{[2]} which explains why height from random samples of a population is usually normally distributed.

## References

- ^Fryar CD, Carroll MD, Gu Q, Afful J, Ogden CLAnthropometric Reference Data for Children and Adults: United States, 2015-2018.Vital Health Stat 3.(2021 Jan)
- ^Lui JC, Palmer AC, Christian PNutrition, Other Environmental Influences, and Genetics in the Determination of Human Stature.Annu Rev Nutr.(2024 May 17)