AP Statistics Unit 4: Sampling distribution of a sample mean

The sampling distribution of a sample mean ( $\bar{x}$ ) bridges probability and statistical inference for quantitative data, explaining how sample averages behave across repeated random samples. On the AP Statistics exam, this topic is tested in multiple-choice questions comparing individual observations to sample means, and it can appear in free-response questions (for example Question 2 or 4) when you describe sampling distributions or calculate probabilities for $\bar{x}$ .

What the exam asks

Calculate and interpret the mean $\mu_{\bar{x}}$ and standard deviation $\sigma_{\bar{x}}$ of the sampling distribution of $\bar{x}$ in context.
Justify why the randomization and $10\%$ conditions are met before using the standard deviation formula.
Describe the shape, center, and variability of the sampling distribution of $\bar{x}$ for a given sample size $n$ .
Calculate and interpret probabilities involving $\bar{x}$ using a normal distribution when population parameters $\mu$ and $\sigma$ are known.

Core ideas

Center and variability of $\bar{x}$

Suppose we take all possible random samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$ . The sampling distribution of the sample mean $\bar{x}$ is the distribution of the values of $\bar{x}$ from all such samples.

On the AP Statistics formula sheet, the parameters of the sampling distribution of $\bar{x}$ are:

\mu_{\bar{x}} = \mu

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

Unbiased estimator: Because $\mu_{\bar{x}} = \mu$ , the sample mean $\bar{x}$ is an unbiased estimator of the population mean .

Conditions for the sampling distribution

Each parameter and shape property depends on a specific condition:

Feature	Condition	What to check in the problem
Unbiased center ( $\mu_{\bar{x}} = \mu$ )	Randomization	Data come from a random sample from the population (or random assignment in an experiment).
Standard deviation ( $σ_{\overset{}{x}}$ )

Sampling distribution of a sample mean

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What the exam asks

Core ideas

Center and variability of $\bar{x}$

Conditions for the sampling distribution

By the end you’ll be able to

Upgrade to Pro to unlock the rest

Still to come

What the exam asks

Core ideas

Center and variability of xˉ\bar{x}xˉ

Conditions for the sampling distribution

Center and variability of $\bar{x}$