# Seeing and Describing the Predictable Pattern: The Central Limit Theorem

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

#### Summary

This activity is designed to develop student understanding of how sampling distributions behave by having them make and test conjectures about distributions of means from different random samples; from three different theoretical populations (normal, skewed, and multimodal).

Students will investigate the impact of sample size and population shape on the shape of the sampling distribution, and learn to distinguish between sample size and number of samples. Students then apply the Empirical Rule (when appropriate) to estimate the probability of sample means occurring in a specific interval.

## Learning Goals

The goal is to enable students to discover the Central Limit Theorem and come to understand that it describes the predictable pattern they have seen when generating empirical distributions of sample means. Students will also learn to describe this pattern in terms of its shape, center and spread and how it allows us to estimate percentages or probabilities for a particular sample statistic. They will also come to understand how we determine if a result is surprising.

## Context for Use

This activity usually follows an activity where students physically take samples (e.g., cups of 25 Reeses Pieces candies) and study the variability between samples. This activity can also follow simulations of sampling such as those at Rossmanchance.com. The lesson takes approximately 75 minutes and is conducted in a computer lab where each student or pair of students has access to a computer. The activity may easily be adapted for junior high, high school, and college-level instruction.

## Description and Teaching Materials

This lesson plan uses a student handout, a sheet of stickers showing normal, skewed, and multimodal populations, and a free downloadable computer software program called Sampling SIM (see resources below).

Instructor Lesson Plan

This lesson moves students from noticing a predictable pattern when they generate distributions of sample statistics to describing that pattern using mathematical theory (i.e., the Central Limit Theorem).

• A copy of the lesson plan as a word document can be accessed by clicking the link.

Lesson Plan (Microsoft Word 34kB Mar1 07)

Goals for the Lesson:

• To enable students to discover the Central Limit Theorem by examining the characteristics of sampling distributions.
• See that the Central Limit Theorem describes the predictable pattern that students have seen when generating empirical distributions of sample means.
• Describe this pattern in terms of shape, center, and spread; contrasting these characteristics of the population to the distribution of sample means.
• See how this pattern allows us to estimate percentages or probabilities for a particular sample statistic, using the Normal Distribution as a model.
• Understand how we determine if a result is surprising.

Materials Needed:

• A copy of the student handout can be accessed by clicking the link below.

Student Handout (Microsoft Word 174kB Oct20 06)
• A copy of the scrapbook page to be used with the student handout can be accessed by clicking the link below.

Scrapbook Page (Microsoft Word 62kB Oct20 06)
• A sheet of stickers and instructions for making them can be accessed by clicking the link below.

Stickers (Microsoft Word 73kB Oct20 06)
• A sample page of what the scrapbook sheet should look like can be accessed by clicking the link below.

Sample Scrapbook Page (Microsoft Word 94kB Oct20 06)

The Process and Task of Having Students Make and Test Conjectures

• Open with a review of variability, standard deviations around the mean, and the normal distribution.
• For each distribution have students make a prediction about which picture (from a page of stickers) is most likely to represent 500 sample means for a particular sample size.
• Have the students see how accurate they are with their predictions by simulating the data using the Sampling SIM program. Each time they simulated the data and see whether their predictions were correct or not, they affix the correct sticker from the sticker sheet to their scrapbook page at the end of the student handout.
• After having students compare their three predictions for each distribution with the simulated data, one at a time, have students add up their score to see how many of their predictions were correct.
• Ask the students what three things they saw happening as they repeatedly sampled from the same population, increasing their sample size each time (center, shape, spread). They can do this by looking at the stickers on their completed scrapbook.
• As a wrap up:
Ask the students how we distinguish between populations, samples and sampling distributions; what is similar, what is different, and why.

Ask the students how we can distinguish between the Law of Large Numbers and the Central Limit Theorem; how are they related, how are they different.

Ask the students how we can use the Central Limit Theorem and the Empirical Rule to assess the rareness of a particular sample statistic in the distribution of sample statistic.

As how we can describe the sampling distribution of sample means without running simulations.

Ask what the shape, center and spread of the distribution of sample means would be for random samples size 100 from a population of human body temperatures.

## Teaching Notes and Tips

For each population, students are asked to predict which sticker (showing a histogram of 500 sample means) is the one for a particular sample size. They then run a simulation to generate a distribution of 500 sample means for the specified population, compare it to their prediction, and then select the sticker that best matches their simulated data to enter in a scrapbook. They repeat this for larger and larger samples sizes, ending up with a progression of three stickers showing distributions of sample means for small, medium and large sample sizes. They repeat this process for two additional populations. The resulting scrapbook allows them to discover what happens when you increase the sample size as you plot distributions of sample means from differently shaped populations.

Time Involved:

• 5 minutes to introduce the activity
• 15 minutes for students to work individually
• 15 minutes for students to work in groups
• 10 minutes for instructor led discussions
• 5 minutes for follow-up questions

For the first few rounds of making and testing predictions, students need help matching their simulated data distributions with a sticker on their sticker sheet. It helps to draw their attention to the highest and lowest values on the graphs and the height of the bars, to make this match. After going through the first few simulations as a class, students can proceed on their own.

## Assessment

Here are two items that can be used to assess student understanding of sampling distributions after using this activity.

To access the Assessment Activity, click link text (Microsoft Word 61kB Oct20 06)

## References and Resources

delMas, R. C., Garfield, J. B., & Chance, B. L. (1999). A Model of Classroom Research in Action: Developing Simulation Activities to Improve Student Statistical Reasoning.

Our findings demonstrate that while software can provide the means for a rich classroom experience, computer simulations alone do not guarantee conceptual change.

To read more: Developing Simulation Activites

Sampling SIM Software This website has the software along with activities and assessment items.

Chance, B., delMas, R., & Garfield, J. (2004). Reasoning About Sampling Distributions. In D. Ben-Zavi & J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking. Kluwer Academic Publishers; Dordrecht, The Netherlands