Shape of the mean's distribution
Whatever the population distribution, the sample mean has a distribution whose mean and standard deviation are closely related to those of the population.
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= μ | ![]() |
= | ![]() |
Although we can easily find the centre and spread of the sample mean's distribution using these formulae, the exact shape of its distribution depends on the shape of the population distribution. For example, skewness in the population distribution leads to some skewness in the distribution of the mean.
Samples from normal populations
This simplifies greatly if the samples come from a normal population.
When the population distribution is normal, the sample mean also has a normal distribution.
This can be expressed as:
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~ normal (μ , | ![]() |
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Distribution of sample mean from normal population
The diagram below illustrates the theory. The top half of the diagram shows a normal population with mean 12 and standard deviation 2, and the bottom half shows the distribution of a sample mean.
Use the slider to display the distribution of the sample mean for different sample sizes. Observe that: