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# Central Limit Theorem

Explanations > Social ResearchStatistical principles > Central Limit Theorem

## Description

If you measure a sample from a population, then you can find its middle point by calculating the average, or mean. If you measure multiple samples, then the mean of each sample will be different, as in the table below.

With larger samples, the distribution of the means of the samples quickly approximates a normal distribution. This is the central limit theorem, that the distribution of sample means will approximate a normal distribution.

Formally, this may be described as follows:

For any population with a mean m  and a standard deviation s, the distribution of sample means for a sample size n will have a mean m  and a standard deviation s/Ön and will approach a normal distribution as n approaches infinity.

## Example

 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 1 3 7 4 3 4 4 6 3 8 5 3 9 6 4 6 4 4 8 7 9 9 1 2 2 5 2 8 4 3 3 7 2 8 7 2 8 7 6 9 Means: 4.375 5 5.5 5.125 5.375 Mean of means: 5.075 Standard deviation of means: 0.438

## Discussion

The standard deviation of sample means is called the standard error, or standard error of the means. Note that this is calculated as s/Ön, which will always be smaller than s as n is always greater than 1. As n increases, so s gets smaller -- in other words the spread gets less. This is due to 'regression to the mean' where high (or low) scores that would cause greater spread are cancelled out by other scores within the sample which are low (or high).

Increasing the sample size thus quickly decreases the sample error.

Because a normal distribution curve is assured, this means you can use the standard deviation of the means to make accurate estimates of how likely any proportion of measurements are to fall into a given range of measurements.

For a single sample, the standard error of the mean can be calculated as the standard deviation of the sample, divided by the square root of the sample size.

Standard error