Central limit theorem for means

Definition

The Central Limit Theorem for Sample Means states that the distribution of sample means will approximate a normal distribution, regardless of the population's distribution, provided the sample size is sufficiently large. This approximation improves as the sample size increases.

5 Must Know Facts For Your Next Test

1. The theorem applies to samples with a sufficiently large size, typically $n \geq 30$.
2. The mean of the sampling distribution will be equal to the population mean ($\mu$).
3. The standard deviation of the sampling distribution (standard error) is given by $\sigma / \sqrt{n}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
4. The shape of the sampling distribution becomes increasingly normal as sample size increases, even if the original population distribution is not normal.
5. It allows statisticians to make inferences about population parameters using sample statistics.

Review Questions

• What conditions must be met for the Central Limit Theorem for Sample Means to hold true?
• How does increasing the sample size affect the standard error of the sampling distribution?
• If a population has a mean $\mu$ and standard deviation $\sigma$, what are these values for its sampling distribution?

Related terms

Sampling Distribution: A probability distribution of a statistic obtained through a large number of samples drawn from a specific population.

Standard Error: The standard deviation of a sampling distribution, often represented as $\sigma / \sqrt{n}$.

Law of Large Numbers: A principle that states that as a sample size grows, its mean will get closer to the average of the whole population.