Central limit theorem | Inferential statistics | Probability and Statistics | Khan Academy

Summary

The video introduces the central limit theorem (CLT), which explains why normal distributions appear so often in statistics—even when the sampled data do not follow a normal distribution.

Demonstration with an unusual distribution

The instructor imagines a discrete random variable with possible values from 1 to 6. Some outcomes are impossible, while others have different probabilities, so the original distribution is deliberately not bell-shaped.

For a sample of size 4, four observations are drawn and averaged. For example:

• A sample of 1, 1, 3, and 6 has a mean of 2.75.
• A sample of 3, 4, 3, and 1 also has a mean of 2.75.
• A sample of 1, 1, 6, and 6 has a mean of 3.5.

Each sample mean is recorded. Repeating this process many times produces a sampling distribution of the sample mean: a distribution showing how frequently different sample means occur.

Main ideas of the central limit theorem

• If the original distribution has a well-defined mean and variance, the distribution of sample means tends to become approximately normal as the sample size increases.
• The original data do not need to be normally distributed. Even the unusual distribution in the example produces an increasingly bell-shaped distribution of sample means.
• The sampling distribution’s mean remains the same as the original distribution’s mean.
• As the sample size increases, sample means cluster more closely around that mean, so their standard deviation becomes smaller.
• The theorem applies to sums as well as means.
• This helps explain why normal distributions are useful for describing sums or averages of many underlying processes, even when the individual processes are complicated or not normally distributed.

Process illustrated

1. Choose a sample size (n).
2. Draw (n) observations from the original distribution.
3. Calculate the sample mean or, alternatively, the sum.
4. Record the result.
5. Repeat the process many times.
6. Plot the recorded means or sums as a frequency distribution.
7. Observe that the distribution becomes more nearly normal as (n) grows.

The instructor notes that sample sizes such as 10 or 20 can often provide a useful approximation, though the accuracy depends on the situation. The video closes by previewing a follow-up demonstration of the effect.

Speakers and sources

• One unnamed instructor/narrator presents the explanation and examples. The video is from Khan Academy.