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Sampling bias is a systematic distortion arising when the sampling mechanism makes the observed sample unrepresentative of the target statistical population for the estimand of interest. It is distinct from random sampling variability and from other causes of the bias of an estimator.

Suppose a population has equal-sized groups with values zero and ten, so its mean is five. If independent sampling with replacement chooses the first group with probability $0.9$, the sample mean has expectation one and converges to one as the sample grows. More data reduce its random variation without removing the selection bias. Known selection probabilities can support a design-based correction, but a group never sampled cannot be recovered merely by assigning larger weights to observed units.

A sampling frame is the operational list or other device identifying the units from which a statistical sample can be selected. The intended statistical population, the sampling frame and the responding units can differ. A register of residential addresses, for example, can serve as a sampling frame for a household survey, but may miss newly built homes and include demolished buildings.

If 100 of 1000 target households are absent from the sampling frame, randomly selecting 90 of the 900 listed households gives each listed household inclusion probability $0.1$ and each omitted household inclusion probability zero. Increasing the sample among listed households cannot by itself correct that lack of coverage. Duplicate entries create a different problem: units with more entries can be more likely to be selected. A sound design therefore documents how frame units map to population units and distinguishes coverage errors from sampling variability.

Sampling variability is the variation of a statistic over random samples generated by a specified sampling design or model. It concerns the statistic before the sample is realized, even when the observed dataset is later treated as fixed.

For independent identically distributed observations with variance ${\sigma }^2$, the sample mean has variance ${\sigma }^2/n$ and standard error $\sigma /\sqrt{n}$. Thus increasing $n$ from 100 to 400 halves that standard error under the same model. A sampling distribution describes this variation; the bias of an estimator describes a displacement of the estimator's expectation from its target. Correlated observations or an informative selection mechanism can invalidate the simple independent-sample formula.

Statistical sampling selects observations from a target statistical population through an explicit selection procedure or statistical model. It links observed values to the larger collection about which conclusions are sought. The accessible sampling frame and the statistical population need not coincide, which can cause sampling bias even in a large statistical sample.

For a finite statistical population with values $y_1,\dots ,y_N$, suppose every unit has known positive inclusion probability ${\pi }_i$. If $s$ is the selected statistical sample, thenis an unbiased estimator of the total $Y={\sum }_i{y}_i$. Indeed, writing the sum with inclusion indicator functions gives $\mathbb{E}\hat{Y}={\sum }_i{y}_i{\pi }_i/{\pi }_i=Y$. Under simple random sampling without replacement of size $n$, ${\pi }_i=n/N$, so $\hat{Y}/N$ is the sample mean. The selection procedure also determines dependence between selected units and the resulting sampling variability; unbiasedness alone does not specify its size.