What is the Central Limit Theorem?

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  • Written By: H.R. Childress
  • Edited By: Jenn Walker
  • Last Modified Date: 23 November 2019
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The central limit theorem in statistics states that the sum or mean of a large number random variables approximates the normal distribution. It can also be applied to binomial distributions. The larger the sample size, the closer the distribution will be to the normal distribution.

The normal distribution, which is approached by the central limit theorem, is shaped like a symmetrical bell curve. Normal distributions are described by the mean, which is represented by the Greek letter mu, and the standard deviation, represented by sigma. The mean is simply the average, and it is the point at which the bell curve peaks. Standard deviations indicate how spread out the variables in the distribution are — a lower standard deviation will result in a narrower curve.

How the random variables are distributed does not matter for the central limit theorem — the sum or mean of the variables will still approach a normal distribution if there is a large enough sample size. The sample size of the random variables is important because random samples are drawn from the population to get the sum or mean. Both the number of samples drawn and the size of those samples is important.


To calculate a sum from a sample drawn from random variables, first a sample size is chosen. The sample size can be as small as two, or it can be very large. It is drawn randomly and then the variables in the sample are added together. This procedure is repeated many times, and the results are graphed on a statistical distribution curve. If the number of samples and the sample size are large enough, the curve will be very close to the normal distribution.

Samples are drawn for means in the central limit theorem the same way as for sums, but instead of adding, the average of each sample is calculated. A larger sample size gives results closer to the normal distribution, and usually results in a smaller standard deviation as well. As for the sums, a larger number of samples gives a better approximation to the normal distribution.

The central limit theorem also applies to binomial distributions. Binomial distributions are used for events with only two possible outcomes, such as flipping a coin. These distributions are described by the number of trials performed, n, and the probability of success, p, for each trial. The mean and standard deviations for a binomial distribution are calculated using n and p. When n is very large, the mean and standard deviations will be the same for the binomial distribution as for the normal distribution.


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