# Essay On Central Limit Theorem Demonstration

Everybody knows about the Central Limit Theorem, but have you ever seen a visual demonstration?

The *Central Limit Theorem* states that, given certain conditions, the mean of a large number of iterates of independent random variables will be approximately normally distributed, regardless of the underlying distribution.

Formally,

*Let {X_{1}, … , X_{n}} be a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ^{2}, then*

* has a distribution like that of a standard normal distribution N(0,1) for large values of n.*

Formulas are nice (if you can understand them!), but it’s always easier to learn things when given a visual demonstration. So let’s try!

As the theorem states, the underlying distribution is not a problem. Therefore, let’s choose an **exponential distribution** with labmda equal to two for our example.

We draw one thousand random sample of size two from this exponential distribution, take the mean of each pair of two, and plot the histogram of the results.

In this case, the **n** of theorem would be two, and as we can observe the distribution doesn’t look like a normal distribution:

If we take samples of size ten (**n** is now ten) and repeat the previous process, the distribution is a little bit more normal:

And as **n** gets larger, it’s easy to see how the distributions of the sample mean looks more like a normal distribution.

So that’s it! Here you have a nice, easy way to understand what the *Central Limit Theorem* says!

If you want a bit more fun understanding this theorem, go and visit this video from The New York Times.

**How the Central Limit Theorem tutorial fits into the typical statistics course:** WISE tutorials are modularized to allow instructors to pick or choose modules that best fit their course needs. Each module is a self-contained lesson that does not depend on any of the other modules, although some specific prerequisite information may be required.

The Central Limit Theorem (CLT) Module was designed with the assumption that students have some familiarity with basic elementary statistics, such as mean, standard deviation, variance, the normal curve, and sampling distributions. You may find it helpful for your students to complete the Sampling Distribution Module before the CLT Module. The CLT Module is intended to prepare students to learn about hypothesis testing and confidence intervals.

**When to use the CLTtutorial?** Instructors often introduce the Central Limit Theorem after they’ve discussed descriptive statistics and the *z*-probability distribution and before an introduction to formal hypothesis testing procedures. Some instructors may wish to use Activity 2 of this module for review later in the course. This relatively advanced component emphasizes conditions where it may not be appropriate to assume that sampling distributions are close to normal. This critical concept is relevant to students who have already learned the importance of the normality assumption for parametric hypothesis testing. You may consider having students return to this component later in the course, after *t*-tests and ANOVA have been introduced.

### Suggestions for Using the CLT Tutorial

**Class demonstration/Lecture aid****Lab assignment****Homework assignment****Review assignment**

There are many ways in which the CLT Module can be inserted into your lesson plan. Your choices may depend on students’ level of computer literacy, computer resources available at your school, and class time restrictions. Here are a few suggestions:

#### 1. Pre-lecture Assignment

Assign the module as homework to introduce the Central Limit Theorem to students. This will allow you to use more class time for in-depth discussions and activities instead of a full lecture.

#### 2. Live Demonstration

As part of either a lecture or guided lab assignment, the SDM applet itself may be used by the instructor to demonstrate visually different aspects of the sampling distribution and the Central Limit Theorem. Some instructors may choose to step through parts or all of the tutorial in a demonstration mode. This demonstration may serve as a stimulus for classroom discussion and/or introduction to an assignment for students. See our step-by-step guide for a live demonstration using the applet.

Access: SDM Applet | Demonstration Guide | Instructions for Using Applet

#### 3. Post-lecture Assignment

After your presentation of the Central Limit Theorem material, the module can be used to demonstrate lecture points and give students practice using the concepts. This applet allows students to gain a perspective on the concepts that complements a lecture or other presentations. The more perspectives students are exposed to in the course of instruction, the more likely they are to understand and retain the material.

For more information, see the Introduction to the tutorial.

#### 4. Evaluation

**Multiple-choice questions**– The main portion of the module is designed to give students feedback without evaluating their performance. The multiple-choice questions provide feedback on both correct and incorrect responses. However, no record is kept of student answers.**Essay questions**– There are follow-up questions after the main part of the module. These questions are multiple-choice and short-answer essays and are designed to examine conceptual understanding of the topic. You may want students to complete this portion of the module and hand in their responses for your evaluation. This will give you an opportunity to evaluate what your students have learned. We have not posted answers to these questions.

WISE modules are designed as self-contained lessons that students can use with little, if any, guidance. If you are concerned that students may not feel comfortable using web pages and applets, you may consider using the module as part of an in-class activity. Most students complete the module in 40 – 50 minutes.

### Your Feedback

We hope this tutorial is helpful for you and your students, and we welcome your feedback on this tutorial and other aspects of the WISE site. Please send your comments to wise@cgu.edu.

Questions, comments, difficulties? See our technical support page or contact us: wise@cgu.edu.

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