The Central Limit Theorem

Definition

The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as the sample size grows, regardless of the original population's distribution. This theorem is fundamental in inferential statistics because it allows for making predictions about population parameters.

5 Must Know Facts For Your Next Test

1. The CLT applies to sample means from any population with a finite variance.
2. As the sample size increases, the sampling distribution of the mean becomes approximately normal.
3. The standard deviation of the sampling distribution (standard error) is equal to the population standard deviation divided by the square root of the sample size: $\sigma/\sqrt{n}$.
4. For practical purposes, a sample size of 30 or more is generally considered sufficient for the CLT to hold.
5. The CLT justifies using normal probability models in hypothesis testing and confidence interval estimation.

Review Questions

• What does the Central Limit Theorem state about the distribution of sample means?
• How does increasing sample size affect the shape of the sampling distribution according to CLT?
• What is the formula for standard error and how does it relate to sample size?

Related terms

Sample Mean: The average value calculated from a set of data points drawn from a larger population.

Standard Error: $\sigma/\sqrt{n}$, representing how much variability can be expected in the sample mean.

Normal Distribution: $N(\mu, \sigma^2)$, a continuous probability distribution characterized by its bell-shaped curve.