# Simulating the Effect of Sample Size on the Sampling Distribution of the Mean

### This activity has been undergone anonymous peer review.

This activity was anonymously reviewed by educators with appropriate statistics background according to the CAUSE review criteria for its pedagogic collection.

This page first made public: May 17, 2007

This material was originally developed through CAUSE

as part of its collaboration with the SERC Pedagogic Service.

#### Summary

This activity allows students to explore the relationship between sample size and the variability of the sampling distribution of the mean. Students use a Java applet to specify the shape of the "parent" distribution and two sample sizes. The simulation then samples from the parent distribution to approximate the sampling distributions for the two sample sizes. Students can see both sampling distributions at the same time making them easy to compare. The activity also allows students to determine the probability of extreme sample means for the different sample sizes so that they can discover that small sample sizes are much more likely than large samples to produce extreme values.

Keywords: sampling distribution, sample size, simulation## Learning Goals

## Context for Use

This activity works well when each student has access to a computer, but can also be used when students share a computer and/or there is one computer with a screen projected for the whole class to see. This activity can be used with high school and college students. If done carefully, it can be adapted for middle school children.

Time involved:

5 minutes to introduce the activity

15 minutes to work, either individually or in groups

15 minutes for discussion and application of the concepts.

## Description and Teaching Materials

A set of questions is available for students to attempt to answer before they interact with the simulation. No feedback is given the first time students answer the questions other than an overall score. They are then asked to use the simulation to help them discover the answers to the questions. The second time through the questions feedback and explanations are given.

The simulation is available at http://onlinestatbook.com/chapter7/SampDist_v2.html. It can be downloaded for local from http://onlinestatbook.com.

## Teaching Notes and Tips

The simulation provides general instructions and step-by-step instructions. It is preferable for the students to use only the general instructions. However, if they get stuck they should refer to the step-by-step instructions.

More advanced students may wish to contrast the effect of sample size on the sampling distribution of the range with the sampling distribution of the mean.

Following the work on the simulation it can be valuable to discuss the general principle that extreme outcomes are more likely with small sample sizes than with large sample sizes. For example, students can discuss whether it is more likely that a sample of two people or a sample of five people would have a mean height over six feet.

## Assessment

For a period of one year, two hospitals, the larger one having about 45 births per day and the smaller about 15 births per day, recorded the days on which more than 60% of the babies born were boys (given a gender ratio of 50:50). Participants were asked which hospital recorded more such days.

Since transfer of learning can be difficult, don't expect particularly good performance on this question. You may wish to introduce this problem (without discussion) before introducing the activity.

How is the variability of the sampling distribution of the mean affected by sample size?

How does sample size affect the probability that a sample mean will be extremely far from the population mean?

## References and Resources

It can be downloaded for local use from http://onlinestatbook.com.