# MathCS.org - Java

## The Central Limit Theorem

This applet illustrates the Central Limit Theorem (CLT). Students can explore and discover the theorem instead of being told what it says. You should also check out the closely related Hypothesis Testing applet.

• When the applet is loaded, check the "Slow Motion" checkbox
• Click on [Start] to select a random sample, compute its mean, and add it to a bar chart of sample means.
• Repeat that process until you understand how the blue bar chart is generated
• Now uncheck the "Slow Motion" checkbox to speed up the process

Next answer the following questions:

• Experiment with different distributions (click on [Pick] to choose another distribution). What shape does the distribution of the sample means (blue chart) have? Is that true regardless of the underlying population distribution (yellow chart)?
• What is the mean for the distribution of the sample means (blue chart) in relation to the mean of the distribution of the original distribution (yellow chart)?
• Is there a relation between the standard deviation of the sample means (blue chart) and that of the original population (yellow chart)? Experiment with sample sizes 16, 25, 36, 49, and 64 to find the relation

If you can answer these questions, you can make up a generalized statement along the following lines:

If you have a population with an arbitrary distribution with a given mean mu and standard deviation sigma, and you select random samples of size N from that population, then the distribution of those sample means has a ____________ distribution with mean ___________ and standard deviation ___________.

That theorem is called the Central Limit Theorem.