**Title:** It depends on what “mean” means: a lesson study on sampling distributions

**Discipline(s) or Field(s):** Statistics

**Authors: **Abdulaziz Elfessi, Heather Hulett, Dan Nordman, University of Wisconsin – La Crosse

**Submission Date:** August 29, 2006

**Executive Summary:** In this activity, we hope to help students differentiate and explain three statistical terms at the heart of statistical inference: the mean of a population, the mean of a sample of observations, and the mean of the sample means.

Past experience indicates that term “mean” can be very confusing for students in an Elementary Statistics class, especially when the same word choice may be applied in all three situations above, with different meanings in each case. Understanding the differences, as well as the connection, between the three types of means above is important for the most basic hypothesis tests in statistics: testing if the population mean equals a certain value by looking at just one random sample. The idea that data from a small sample can be used to estimate the mean of an entire population, which cannot be obtained directly, is critical to statistical applications in many, many fields.

The specific learning goals of the lesson are as follows:

- Students will practice applying statistical techniques to data collected from samples.
- Students will see and explain sample variability and how sample size decreases the variability of sample means.
- Students will see graphically that the “typical/central/mean/expected” value of a sample mean is the same as the population mean.

In this lesson students will take random samples of different sizes and calculate their averages. They will then put their averages on Post-It notes and place them in the correct spot on the chalkboard to make histograms that will represent the sampling distributions. By comparing their sample means, the mean of the histogram (the mean of all the samples), and the population mean (which will be revealed at the right moment), they will hopefully get a fuller appreciation of the three different uses of the word “mean”.

The activity was successful in several ways. Students enjoyed the short exercise in drawing random samples and were surprised by some of the sample means obtained. As the histograms were formed, they saw clearly how the variability decreased as sample size increased. Finally, they got to see how most sample means gave close approximations to the true population mean.

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