Seeing and Describing the Predictable Pattern: The Central Limit Theorem

This page is authored by Shirley J. Alt, based on an original activity by Joan Garfield and Bob delMas, both at the University of Minnesota.

Author Profile

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: Apr 5, 2007

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


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 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).

Goals for the Lesson:

Materials Needed:

The Process and Task of Having Students Make and Test Conjectures

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:

Helpful Hints:

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.


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