The purpose of the sample may be to estimate properties of a larger population, such as a lot, pile or shipment, the percentage of some constituent, the fraction of the items that fail to meet (or meet) a specified requirement, the average characteristic or quality of an item, the total weight of the shipment, or the probable maximum or minimum content of, say, some chemical.

The purpose may be the rational disposition of a lot or shipment without the intermediate step of the formation of an estimate.

The purpose may to provide aid toward rational action concerning the production process that generated the lot, pile or shipment.

Whatever the purpose of the sample, adhering to the principles of probability sampling will allow the uncertainties, such as bias and variance of estimates or the risks of the rational disposition or action, to be calculated objectively and validly from the theory of combinatorial probabilities. This assumes, of course, that the sampling operations themselves ere carried out properly, as well. For example, that any random numbers required were generated properly, the units to be sampled from were correctly identified, located, and drawn, and the measurements were made with measurement error at a level not exceeding the required purposes.

Determination of bias and variance and of risks can be calculated when the selection was only partially determined by random numbers and a frame, but they then require suppositions and assumptions which may be more or less mistaken or require additional data which may introduce experimental error.