Sampling Distributions and CLT Cheat Sheet

Key Facts

• A sampling distribution is the distribution of a statistic, such as $\bar{x}$ or $\hat{p}$, from all possible samples of the same size $n$.
• The sampling distribution of the sample mean has mean $\mu_{\bar{x}} = \mu$ when samples are random and independent.
• The standard error of the sample mean is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ when the population standard deviation $\sigma$ is known.
• The Central Limit Theorem states that if $n$ is large, then $\bar{x}$ is approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$.
• For a sample proportion, the sampling distribution has mean $\mu_{\hat{p}} = p$ and standard error $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.
• A sample proportion is approximately normal when $np \ge 10$ and $n(1-p) \ge 10$.
• A sample mean is approximately normal if the population is normal or if the sample size is large, often $n \ge 30$.
• Increasing sample size reduces standard error because both $\frac{\sigma}{\sqrt{n}}$ and $\sqrt{\frac{p(1-p)}{n}}$ get smaller as $n$ increases.

Vocabulary

Sampling distribution

A sampling distribution is the distribution of a statistic calculated from many random samples of the same size.

Statistic

A statistic is a number calculated from a sample, such as $\bar{x}$ or $\hat{p}$.

Parameter

A parameter is a number that describes an entire population, such as $\mu$, $\sigma$, or $p$.

Standard error

Standard error is the standard deviation of a sampling distribution and measures how much a statistic varies from sample to sample.

Central Limit Theorem

The Central Limit Theorem states that sampling distributions of means become approximately normal as sample size increases.

Unbiased estimator

An unbiased estimator is a statistic whose sampling distribution is centered at the true population parameter.