# Life Is a Bell Curve

## Sampling Distribution and the Central Limit Theorem

William Thompson, Diana Gedye, Will Thompson, David Thomas | Published: June 18th, 2024 by K20 Center

• Subject Mathematics
• Course AP Statistics
• Time Frame 3-4 class period(s)
• Duration 180 minutes

### Summary

In this lesson, students will explore the concepts of sampling distributions and the central limit theorem through hands-on activities and practical application in the Deadly Distribution digital game-based learning (DGBL) module.

### Essential Question(s)

What is central limit theorem? Why is an understanding of the central limit theorem essential to statistics?

### Snapshot

Engage

Students use the Anchor Chart strategy to demonstrate what they already know about the concepts of sampling distribution and the central limit theorem.

Explore

Students play the first two missions of the Deadly Distributions DGBL module to get introduced to its mechanics and start exploring the core statistical concepts.

Explain

Students use the Inside Out strategy to produce deeper thinking about the concepts before explaining the concepts in greater depth.

Extend

Students extend their understanding of the concepts by playing the third mission of the Deadly Distribution DGBL module.

Evaluate

Students use the Always, Sometimes, Never True strategy to demonstrate knowledge of the statistical concepts.

### Materials

• Deadly Distribution Instructor’s Guide (attached)

• K20 Game Portal accounts or iPad apps of Deadly Distribution for each student

• Game Portal Guide v1.2 (attached)

• Always, Sometimes, Never True Worksheet - Teacher (attached)

• Always, Sometimes, Never True Worksheet - Student (attached)

• Inside Out handout (attached)

• Student devices with internet access

• Paper

• Whiteboard or large poster paper

• Writing utensils

### Engage

10 Minute(s)

To start the lesson, ask your students to write down anything they know about the central limit theorem, specifically focusing on terms and concepts and how they relate to each other. You can have them do this individually or in small groups.

Give them around 5 minutes to do this, and then discuss the terms and concepts they've come up with as a class. Using the Anchor Chart strategy, draw a chart of the concepts your students have come up with on the board or on a large piece of poster paper. Keep this chart available and visible for your students throughout the rest of the lesson.

Discuss concepts such as sample, mean and median, standard deviation, margin of error, confidence interval, and of course, the actual central limit theorem.

### Explore

45 Minute(s)

After this initial introduction and exploration of your students' knowledge of the topics, introduce them to the Deadly Distributions DGBL module. Click here to learn more about the game. It is recommended that you play through the game at least once before teaching with it so you have a general understanding of the story and how the game's mechanics function.

Have your students play through the first two missions of the game. This should take them 30-45 minutes. The first mission is a short tutorial that introduces most of the game’s mechanics, and the second mission begins to introduce the actual learning content.

### Explain

45 Minute(s)

Using a modified version of the Inside Out strategy, gauge your students' understanding and set them up for some deeper explanations. Give each student a copy of the Inside Out handout available in the Attachments section.

Prompt them with a question about what the central limit theorem means. Have them write their thoughts in the innermost circle of the worksheet. Once all of the students have done this, have them find a partner and share this information, copying their partner's responses in the second circle.

Once all of your students have finished the worksheet, discuss some of their thoughts focusing on their understanding of the central limit theorem, making sure to watch for and correct any misconceptions. Some definitions that may be useful are listed below.

• Sample Size: the number of observations or recordings that will be taken when collecting a statistical sample

• Random Sampling: the random selection of a subset of individuals within a statistical population, thus allowing for a smaller subset of samples to represent a larger total population for statistical inference

• Sampling Distribution: the set of all mean values of possible samples

• Central Limit Theorem: regardless of the underlying distribution of the population, with a large enough sample size, the means, or proportions, of all samples from the same population will be equal to the mean, or proportion, of the population and that the distribution of the sample means, or proportions, will be a normal distribution

### Extend

45 Minute(s)

Now, have your students to go back and play mission three of Deadly Distribution so they can continue to apply what they have learned so far. This should take around another 30-45 minutes to complete. As mentioned previously, it is not required that players complete mission four, but if you have students who complete mission three very quickly, you can have them continue on to the final mission. It is much more challenging, however, and takes roughly another 30-45 minutes to complete.

### Evaluate

Give each student an Always, Sometimes, Never True Worksheet - Student and have them determine if each statement on the worksheet is always, sometimes, or never true. Make sure that they write a justification for their choice that is based on information presented over the course of the lesson. Once everyone has completed the worksheet, discuss each statement as a class. Read a statement and then have students raise their hands to show whether they decided if the statement is always, sometimes, or never true. Discuss the right answer for each statement, and if any students have a wrong answer, have them explain their thinking.