Distribution of a sample mean
In an earlier
section, we explained that the mean of a random sample, ,
has a distribution whose mean and standard deviation depend on the population
mean, µ,
and standard deviation, σ,
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= μ |
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= | ![]() |
The standard deviation of the sample mean decreases as n increases.
Also, irrespective of the population distribution, the shape of the distribution approaches a normal distribution as the sample size, n, increases (Central Limit Theorem).
Sum of values in a random sample
A sample mean is often the most descriptive summary statistic for a random sample, but occasionally the sum of the sample values is more useful. For example, if the individual values in a data set are the amounts paid by customers in a supermarket during one day,
The sum of sample values is n times their mean, so its distribution is a scaled version of the distribution of the mean — the same shape but different mean and standard deviation.
Its distribution also approaches a normal distribution as n increases. It is important to note that, in contrast with the sample mean,
The standard deviation of the sample sum increases as n increases.
Simulation to illustrate distributions of sample mean and sum
The diagram below allows samples of different sizes to be selected from a standard normal distribution (with mean 0 and standard deviation 1).
The theoretical normal distribution of the sample mean is shown in blue, and that of the sum is shown in green. Observe that the sample mean has lower spread than that of the sample sum.
Click the checkbox Accumulate then click Take sample a few times to select different samples of size 4. Observe that the sampling distributions match these theoretical distributions reasonably well.
Repeat with different sample sizes.