# Using an Applet to Demonstrate a Sampling Distribution

This material is replicated on a number of sites
as part of the
SERC Pedagogic Service Project

#### Summary

This in-class demonstration combines real world data collection with the use of the applet to enhance the understanding of sampling distribution. Students will work in groups to determine the average date of their 30 coins. In turn, they will report their mean to the instructor, who will record these. The instructor can then create a histogram based on their sample means and explain that they have created a sampling distribution. Afterwards, the applet can be used to demonstrate properties of the sampling distribution. The idea here is that students will remember what they physically did to create the histogram and, therefore, have a better understanding of sampling distributions.

## Learning Goals

- Define a "Sampling Distribution."
- Explain that the distribution of the sample average is less varied than the distribution of the individuals from which they are calculated.
- Explain that even if the population is very skewed, the average from a large random sample will be normally distributed.

## Context for Use

This demonstration can be used as a first introduction to the concept of sampling distributions. It has been applied at the undergraduate level, but could also be used at the advanced placement level as well. Including the time spent for viewing the applet, this activity can be completed in a 50 minute class period.

Prior to this activity students should be able to calculate the mean and should have seen distributions of many different shapes.

## Description and Teaching Materials

In this activity students are exposed to the concept of sampling distributions by creating a sampling distribution as a class. By having the students assemble a sampling distribution, they can more readily understand that a sampling distribution is made up of a collection of sample statistics from different samples.

- Students are asked to think about the distribution of the dates on U.S. coins and describe what they believe it will be. This distribution should be skewed to the left. Based on the description in the handout, most students will get this idea. If not, an instructor may need to guide them to this point. This idea will form the basis for illustrating the Central Limit Theorem in later steps.
- The students then calculate the average of 30 coins to create their own sample statistic.
- The instructor "assembles" a dot plot or histogram of the sample means created by each student or group. This is done by calling on each group (or individual student) to give their sample mean. The instructor should then mark these on a dotplot or histogram on the board or an overhead.
- When this histogram is complete, the instructor should point out that
- the distribution is made up of the means of different samples from the population of interest.
- the distribution is more compact than the original distribution. For example, some students may have coins that date back to the 1940s, but the averages will typically be between 1989 and 1995.
- the distribution of means is more symmetric than the distribution of the individual coin dates. The instructor should also point out that there are too few means to clearly see the distribution. This serves as a lead into using the applet. The applet can serve as a method to more quickly take samples from the population.

- The instructor should then open the applet and construct a left skewed distribution resembling the distribution of the individual coin dates.
- The instructor should take samples of n=25 from this distribution one at a time. Point out that these would be equivalent to each sample of coins the class put together. The instructor should point out both the values of the individuals in the second graph and the mean of the sample that appears in blue on the third graph.
- The instructor will continue to take samples one at a time until the same number have been taken as were in the histogram assembled by hand. The instructor can point out that this histogram is crude, due to the small number of samples from which it is constructed.
- The instructor can use the "1000" button to take a large number of samples and allow the students to watch the sampling distribution build into an approximately normal distribution. The instructor should point out that even though the parent population is skewed to the left, the distribution of the mean coin dates is approximately normal.
- The instructor can repeat the sampling process with different shapes of parent populations. A drop-down menu allows the choice of skewed right, uniform and normal distributions. For each situation, the instructor should point out that regardless of the shape of the parent population, the sampling distribution should be approximately normal.
- Finally, the instructor can illustrate how different sample sizes influence the normality of the distribution by repeating the exercise with samples of 5 and 10. Again, the instructor should point out the shape of the distribution.
- The activity should conclude with a summary of the demonstration that can be written in their notes. It is a good idea is to have the students summarize what they have done by filling in the blanks of these sentences:

- Even if a parent population's distribution is "____________," the distribution of the sample mean is approximately "______________," and becomes even more so as the sample size "_____________."

- Even if a parent population's distribution is "____________," the distribution of the sample mean is approximately "______________," and becomes even more so as the sample size "_____________."

**Materials Needed**- Students should bring 10 coins to class if working in groups of 3, or 30 coins if working individually.
- The instructor should have a computer with projector.
- Sampling Distribution Applet
- The activity document for students, Activity Worksheet (Microsoft Word 28kB Jul6 06).

## Teaching Notes and Tips

This demonstration is appropriate for classes as small as 20 students to those large as 200 students. In either, case you want to have at least 20 samples of 30 coins. For small class sizes students should work on individually. For a large class size, break students into groups of up to 3.

Careful planning will make this demonstration go smoothly. Some specific tips include:

- The activity sheet should be updated to reflect the last full year. If it is currently 2010 the sheet should be updated to say 2009.
- The instructor should bring some extra coins. Even if the students are asked to bring coins, some students may not bring them.
- Worksheets should be pre-numbered to facilitate calling on the groups.
- Students should bring a calculator to class to use in calculating averages.
- As the students are calculating their averages, the instructor should prepare a histogram on the board or a transparency.
- The instructor should stress the connection between the histogram created with the average coin dates and the online simulation of the sampling distribution.

## Assessment

This concept is very deep and can be assessed on several levels. Students can be given a scenario and asked to describe the sampling distribution. Students should also be prompted to explain what makes up the sampling distribution. At the most basic level students should be able to choose a histogram that reflects the sampling distribution of a sample mean. An example of such a question can be found in the file: Sampling distribution questions. (Microsoft Word 201kB May2 07)