What is the 'Central Limit Theorem - CLT'
The central limit theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Furthermore, all of the samples will follow an approximate normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample's size.
BREAKING DOWN 'Central Limit Theorem - CLT'According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data, and whether it is normal or non-normal. As a general rule, sample sizes equal to or greater than 30 are considered sufficient for the central limit theorem to hold, meaning the distribution of the sample means is fairly normally distributed.
The Central Limit Theorem in Finance
The central limit theorem is very useful when examining returns for a given stock or index because it simplifies many analysis procedures. An appropriate sample size depends on the data available, but generally speaking, having a sample size of at least 50 observations is sufficient. Due to the relative ease of generating financial data, it is often easy to produce much larger sample sizes. The central limit theorem is the basis for sampling in statistics, so it holds the foundation for sampling and statistical analysis in finance as well. Investors of all types rely on the central limit theorem to analyze stock returns, construct portfolios and manage risk.
Example of Central Limit Theorem
If an investor is looking to analyze the overall return for a stock index made up of 1,000 stocks, he can take random samples of stocks from the index to get an estimate for the return of the total index. The samples must be random, and at least 30 stocks must be evaluated in each sample for the central limit theorem to hold. Random samples ensure a broad range of stock across industries and sectors is represented in the sample. Stocks previously selected must also be replaced for selection in other samples to avoid bias. The average returns from these samples approximates the return for the whole index and are approximately normally distributed. The approximation holds even if the actual returns for the whole index are not normally distributed.
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.
- 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.
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 email@example.com.
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