The central limit theorem states that, given a distribution with a mean μ and variance σ², the sampling distribution of the mean x-bar approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases.

The amazing and counter-intuitive thing about the central limit theorem is that no matter the shape of the original distribution, x-bar approaches a normal distribution.

• If the original variable X has a normal distribution, then x-bar will be normal regardless of the sample size.
• If the original variable X does not have a normal distribution, then x-bar will be normal only if N ≥ 30. This is called a distribution-free result. This means that no matter what distribution X has, it will still be normal for sufficiently large n.

Keep in mind that N is the sample size for each mean and not the number of samples. Remember that in a sampling distribution the number of samples is assumed to be infinite. The sample size is the number of scores in each sample; it is the number of scores that goes into the computation of each mean.

Two things should be noted about the effect of increasing N:

• The distributions become more and more normal.
• The spread of the distributions decreases.

Based on the central limit theorem, when the sample size is large, you can:

• use the sample mean to infer the population mean.
• construct confidence intervals for the population mean based on the normal distribution.

Note that the central limit theorem does not prescribe that the underlying population must be normally distributed. Therefore, the central limit theorem can be applied to a population with any probability distribution.