Unit 5 (AP Statistics): Sampling Variability and What It Means for Inference

Sampling distribution

The distribution of a statistic (e.g., $x̄$ or $\hat{p}$) across many repeated samples of the same size from the same population using the same sampling method; it describes how the statistic varies from sample to sample.

Statistic

A numerical value computed from a sample (random because the sample is random), such as x̄, p̂, or s.

Population distribution

The distribution of values for all individuals in the population (fixed, but usually unknown).

Sample data distribution

The distribution of values within one particular sample; it changes from sample to sample.

Sampling distribution of a statistic

The distribution of the statistic’s values over many possible samples (not the distribution of the raw data and not the distribution of “possible samples”).

Parameter

A numerical value describing a population (fixed, usually unknown), such as μ, p, or σ.

Point estimate

A statistic used to estimate a population parameter (e.g., x̄ estimates μ; p̂ estimates p).

Sampling variability

The natural sample-to-sample variation in a statistic (like x̄ or p̂) when taking different random samples from the same population with the same method and n.

Simple Random Sample (SRS)

A sampling method where every possible sample of size n has an equal chance of being selected; commonly assumed in sampling distribution formulas.

Center (of a sampling distribution)

The mean of the sampling distribution; it tells what value the statistic tends to hit on average across repeated samples.

Spread (of a sampling distribution)

The standard deviation of the sampling distribution; it tells how much the statistic typically varies from sample to sample.

Mean of p̂ (μ_p̂)

The center of the sampling distribution of the sample proportion: μ_p̂ = p (so p̂ is centered at the true proportion).

Standard deviation of $\hat{p}$ ($\sigma_{\hat{p}}$)

The spread (standard error) of the sampling distribution of the sample proportion: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, assuming (approximate) independence.

10% condition

When sampling without replacement, independence is approximately reasonable if the sample size n is no more than 10% of the population size (n ≤ 0.10N).

Standard error

The standard deviation of a sampling distribution (sampling-to-sampling variability), such as $\sigma_{x̄}$ or $\sigma_{\hat{p}}$; it typically depends on $n$ and often unknown parameters.

Mean of $x̄$ ($\mu_{x̄}$)

The center of the sampling distribution of the sample mean: $\mu_{x̄} = \mu$ (so $x̄$ is centered at the true mean).

Standard deviation of $x̄$ ($\sigma_{x̄}$)

The spread (standard error) of the sampling distribution of the sample mean: $\sigma_{x̄} = \frac{\sigma}{\sqrt{n}}$, assuming (approximate) independence.

Independence (in sampling distribution formulas)

A condition (exact or approximate) that allows standard sampling distribution formulas for $\sigma_{\hat{p}}$ and $\sigma_{x̄}$; often justified by random sampling and the 10% condition when sampling without replacement.

Square-root rule

Sampling variability shrinks like 1/√n; multiplying n by 4 cuts the standard error in half (it does not shrink like 1/n).

Population standard deviation ($\sigma$)

A parameter describing variability of individual values in the population (fixed but usually unknown); it helps determine $\sigma_{x̄}$ via $\sigma_{x̄} = \frac{\sigma}{\sqrt{n}}$.

Sample standard deviation (s)

A statistic describing variability within one sample; it can differ from sample to sample and is not the same thing as σ_x̄.

Unbiased estimator

An estimator whose sampling distribution is centered at the true parameter (mean of the sampling distribution equals the parameter), e.g., $\mu_{x̄} = \mu$ and $\mu_{\hat{p}} = p$.

Biased estimator

An estimator whose sampling distribution is not centered at the true parameter; it tends to overshoot or undershoot in the long run, and increasing n does not automatically remove this bias.

Bias–variability tradeoff

The idea that a good estimator/procedure depends on both center (bias) and spread (variability): an unbiased estimator can still be poor if it has high variability, while a slightly biased one might be preferable if it greatly reduces variability.

Biased sampling method (study design bias)

A data-collection problem where the sampling process systematically favors certain outcomes (e.g., undercoverage, voluntary response, nonresponse), potentially producing statistics that are consistently off-target even if the statistic is theoretically unbiased under SRS.