# Central limit theorem

You can use the central limit theorem when sampling from a population that is not normally distributed.

Most of the time the population from which the samples are selected is not going to be normally distributed.

However, as the sample size increases, the sampling distribution of  x̄ will approach a normal distribution.

Central limit theorem

For a large sample, usually when the sample is bigger or equal to 30, the sample distribution is approximately normal. This is true regardless of the
shape of the population distribution.

The mean and standard deviation of the sampling distribution of x̄ are

$$\mu_{\overline{x} = \mu }$$ $$\sigma_{\overline{x}} = \frac{\sigma }{\sqrt{n} }$$
• Keep in mind that the shape of the sampling distribution is not exactly normal, but approximately normal for a large sample size (n ≥ 30). As the sample size increases the approximation becomes more accurate.
• You can only use the central limit theorem when n ≥ 30 since the theorem applies to large samples only.
• Finally, in the formula for the standard deviation, n/N must be less than or equal to 0.05.

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