Wiki Page, Math 13 - Perez Sandbox : Module 22 - Distribution of Sample Means (3b of 4)

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2026-01-17T21:15:36-08:00

Is the Size of the Population Important?

Notice that the size of the population is not mentioned in our discussion of the center, spread and shape of the sampling distribution for sample means.

From our previous discussion, we know the following:

The mean of the distribution of sample means is the population mean, .

The standard deviation of the distribution of sample means is where is the population's standard deviation of individual measurements.
The larger the sample size n, the smaller the standard deviation. In other words, the means from larger samples have less variability, so larger samples give more accurate estimates of the population mean.

The shape of the distribution of sample means is approximately normal when n 30. This is true even when the variable has a skewed distribution for individuals in the population. If the variable is normally distributed in the population, then the distribution of samples means will be normal regardless of sample size.

These statements are true regardless of the size of the population as long as the population is large. (In this course, we work only with large populations.)

To illustrate this point, we compare a distribution of sample means from two populations of different sizes:

Population A has 10,000 newborns.

Population B has 20,000 newborns.

For each population, the mean and standard deviation of individual birth weights is the same: and .

To create the sampling distribution, we selected 525 random samples of 100 babies from each population and made a histogram of the sample means. We did this twice for population A, so two of the histograms represent 525 samples from the same population. As expected, there are some differences in the samples collected due to random chance. Comparing these two histograms gives us a sense of how much variation we can expect from the process of selecting random samples.

Notice that the histogram of sample means from the larger population B has a similar shape, center, and spread to the histograms from population A. The size of population B did not affect the sampling distribution.

What’s the Main Point?

The size of the population does not affect the variability of the sample means. Size matters if we are talking about sample size for random samples, but size does not matter if we are talking about population size as long as the population is large.

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