Introducing Statistics: What Can We Learn from Data?

A statistical study collects data from a sample to answer an investigative question about a larger population - larger being the point, since the population is measured by sampling exactly when it cannot be measured whole. The population is every unit the question is about, size $N$; the sample is the subset actually measured, size $n$. A numerical summary of the population is a parameter, normally unknown; the same summary computed from the sample is a statistic, and it is the basis for inference about the parameter rather than a substitute for it. The question itself is fixed before the data arrive, posed about a defined population with a variable someone could record.

The trap in this topic has one shape with many faces: treating sample and population as interchangeable. It appears as a class survey announced as what the school thinks, as calling a measured sample proportion a parameter because it was directly computed, as writing $p$ for the sample's number, and as expecting the statistic to equal the parameter and calling the gap a mistake. It also hides in the belief that a random or very large sample somehow becomes the population; size and randomness improve an estimate, but the number it produces remains a statistic.