Lesson 7.1 Description of the Central Limit Theorem

The Central Limit Theorem

The Central Limit Theorem is one of the most powerful theorems in all of statistics. There are two alternatives of this theorem: means (or averages) and sums. Both alternatives are concerned with drawing finite samples of size n from a population with a known mean μ and a known standard deviation σ.

For the first alternative, we collect samples of size n and calculate the mean or average of each sample. Then, we construct a histogram of the sample means. If n is "large enough," then the histogram will tend to have an approximate bell shape.

For the second alternative, we collect samples of size n and calculate the sum of each sample. Then, we construct a histogram of the sample sums. If n is "large enough," then the histogram will tend to have an approximate bell shape.

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Facts About the Central Limit Theorem

Sampling is done with replacement.

We may or may not know the probability distribution of the original population. We do not need to know it.

If the original population is far from normal, then we must take more observations for each sample and n will be large.

Watch what happens to the graph as n is increased. Close the window when you are done viewing the example. You will return here.

Abbreviation for the Central Limit Theorem

The abbreviation for the Central Limit Theorem is CLT.

Please continue to the next section of this lesson.

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Up » 7.1 Central Limit Theorem » 7.2 Central Limit Theorem for Averages » 7.3 Central Limit Theorem for Sums