Means from non-normal populations
When the population is not a normal distribution, the sample mean does not have a normal distribution, though the earlier equations above still provide its mean and standard deviation:
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However an important theorem in statistics called the Central Limit Theorem states that...
For most non-normal population distributions, the distribution of the sample mean becomes close to normal when the sample size increases.
Simulation
The diagram below shows a population distribution with mean 4.0 and standard deviation 4.0 but is highly skew. A distribution of this form is sometimes appropriate for lifetimes of objects such as electric toasters or car windscreens. (It is less appropriate for lifetimes of biological organisms.)
A random sample of n = 2 values from the distribution is also shown.
Click Accumulate and take 50 or more samples. Observe that the sample means also have a skew distribution and that it is centred on the population mean, 4.0.
Use the pop-up menu to increase the sample size to 8 and take a further 50 samples. Observe that:
Theory
Advanced statistical theory can find the distribution of sample means for this type of distribution. The underlying theory is unimportant, but we use it in the diagram below to show the distributions of the sample means.
Use the slider to see how increasing sample size affects the distribution of the mean:
Even with a sample size of 20, the shape of the distribution is very close to normal.