Normal distribution | Definition, Examples, Graph, & Facts (2025)

statistics

verifiedCite

While every effort has been made to follow citation style rules, there may be some discrepancies.Please refer to the appropriate style manual or other sources if you have any questions.

Select Citation Style

Feedback

Thank you for your feedback

Our editors will review what you’ve submitted and determine whether to revise the article.

External Websites

Also known as: Gaussian distribution

Written and fact-checked by

The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.

The Editors of Encyclopaedia Britannica

Last Updated: Jun 17, 2024•Article History

normal distribution

See all media

Also called:: Gaussian distribution

Related Topics:: distribution function; standard normal distribution

See all related content →

normal distribution, the most common distribution function for independent, randomly generated variables. Its familiar bell-shaped curve is ubiquitous in statistical reports, from survey analysis and quality control to resource allocation.

The graph of the normal distribution is characterized by two parameters: the mean, or average, which is the maximum of the graph and about which the graph is always symmetric; and the standard deviation, which determines the amount of dispersion away from the mean. A small standard deviation (compared with the mean) produces a steep graph, whereas a large standard deviation (again compared with the mean) produces a flat graph. See the figure.

Britannica QuizDefine It: Math Terms

The normal distribution is produced by the normal density function, p(x)=e^{−(x−μ)²/2σ²}/σSquare root of√2π. In this exponential function e is the constant 2.71828…, is the mean, and σ is the standard deviation. The probability of a random variable falling within any given range of values is equal to the proportion of the area enclosed under the function’s graph between the given values and above the x-axis. Because the denominator (σSquare root of√2π), known as the normalizing coefficient, causes the total area enclosed by the graph to be exactly equal to unity, probabilities can be obtained directly from the corresponding area—i.e., an area of 0.5 corresponds to a probability of 0.5. Although these areas can be determined with calculus, tables were generated in the 19th century for the special case of =0 and σ=1, known as the standard normal distribution, and these tables can be used for any normal distribution after the variables are suitably rescaled by subtracting their mean and dividing by their standard deviation, (x−μ)/σ. Calculators have now all but eliminated the use of such tables. For further details see probability theory.

The term “Gaussian distribution” refers to the German mathematician Carl Friedrich Gauss, who first developed a two-parameter exponential function in 1809 in connection with studies of astronomical observation errors. This study led Gauss to formulate his law of observational error and to advance the theory of the method of least squares approximation. Another famous early application of the normal distribution was by the British physicist James Clerk Maxwell, who in 1859 formulated his law of distribution of molecular velocities—later generalized as the Maxwell-Boltzmann distribution law.

The French mathematician Abraham de Moivre, in his Doctrine of Chances (1718), first noted that probabilities associated with discretely generated random variables (such as are obtained by flipping a coin or rolling a die) can be approximated by the area under the graph of an exponential function. This result was extended and generalized by the French scientist Pierre-Simon Laplace, in his Théorie analytique des probabilités (1812; “Analytic Theory of Probability”), into the first central limit theorem, which proved that probabilities for almost all independent and identically distributed random variables converge rapidly (with sample size) to the area under an exponential function—that is, to a normal distribution. The central limit theorem permitted hitherto intractable problems, particularly those involving discrete variables, to be handled with calculus.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Adam Augustyn.

FAQs

Normal distribution | Definition, Examples, Graph, & Facts? ›

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The normal distribution appears as a "bell curve" when graphed.

Why is the normal distribution important in real life? ›

for practical purpose normal distribution is good enough to represent the distribution of continuous variable like-height,weight,blood pressure etc.. often used to aproximate other distribution. normal distribution has significant use in statistical quality control.

Get More Info ›

Why is normal distribution so common? ›

The Normal Distribution (or a Gaussian) shows up widely in statistics as a result of the Central Limit Theorem. Specifically, the Central Limit Theorem says that (in most common scenarios besides the stock market) anytime “a bunch of things are added up,” a normal distribution is going to result.

Get More Info Here ›

What is the best graph for normal distribution? ›

A bell curve is a graph depicting the normal distribution, which has a shape reminiscent of a bell. The top of the curve shows the mean, mode, and median of the data collected.

Learn More ›

What are the 3 rules for normal distribution? ›

Properties of Normal Distribution

In a normal distribution, mean (average), median (midpoint), and mode (most frequent observation) are equal. These values represent the peak or highest point.

View Details ›

What are the main properties of the normal distribution? ›

Some important properties of normal distribution are,

For normal distribution of data, mean, median, and mode are equal, (i.e., Mean = Median = Mode). Total area under the normal distribution curve is equal to 1. Normally distributed curve is symmetric at the center along the mean.

Keep Reading ›

What are the advantages of the normal distribution? ›

Answer. The first advantage of the normal distribution is that it is symmetric and bell-shaped. This shape is useful because it can be used to describe many populations, from classroom grades to heights and weights.

Explore More ›

How do you know if data is normally distributed? ›