Central limit theorem

Definition

The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution. This holds true provided the samples are independent and identically distributed (i.i.d.).

5 Must Know Facts For Your Next Test

1. The central limit theorem applies to sample means, not individual data points.
2. As sample size increases, the sampling distribution of the mean becomes approximately normal even if the population distribution is not normal.
3. The mean of the sampling distribution equals the population mean.
4. 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: $\sigma_{x̄} = \frac{\sigma}{\sqrt{n}}$.
5. The central limit theorem allows for inference about population parameters using sample statistics.

Review Questions

• What does the central limit theorem imply about large sample sizes?
• How does sample size affect the shape of the sampling distribution according to the central limit theorem?
• What is the relationship between population standard deviation and standard error in context with this theorem?

Related terms

Sampling Distribution:

A probability distribution of a statistic obtained from a larger number of samples drawn from a specific population.

Standard Error:

The standard deviation of a sampling distribution, calculated as $\frac{\sigma}{\sqrt{n}}$ where $\sigma$ is population standard deviation and $n$ is sample size.

Law of Large Numbers:

A principle stating that as a sample size grows, its mean gets closer to the average of the whole population.