Wiki Page, Math 13 - Perez Sandbox : Module 22 - Wrap Up "Distribution of Sample Means"

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2026-01-17T21:15:38-08:00

Let's Summarize

When data is quantitative, it is summarized with a mean and standard deviation.

When selecting random samples of size n from a population with mean µ and standard deviation , the resulting distribution of sample means is our probability model. It has the following properties:

The mean of the sampling distribution of means is .

The standard deviation of the sampling distribution is called the standard error. It is equal to .

Notice that as n grows, the standard error of the sampling distribution of means shrinks. It means that larger samples give more accurate estimates of population means.

The central limit theorem states that for large enough sample sizes, the sampling distribution of means is approximately normal, even if the population is not normal.

If a variable has a skewed distribution for individuals in the population, a larger sample size is needed to ensure that the sampling distribution has a normal shape.

The general rule is that if n is more than 30, the sampling distribution of means will be approximately normal. However, if the population is already normal, then any sample size will produce a normal sampling distribution.

When the sampling distribution is approximately normal, we can calculate a Z-score and use the Standard Normal Curve to calculate probabilities.

Convert a sample mean into a z-score:

Use technology to find a probability associated with a given range of z-scores.

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