Population, sample, parameter, statistic

The six-part checklist for any described study — and which of the last two is fixed and which varies.

Six words the paper marks

Every inference question opens with a described study, and its first mark is for naming the pieces correctly. There are six.

Population — the complete set of units the question is about.
Unit — one member of the population (an account, an invoice, an employee).
Sample — the subset of units actually measured.
Variable — what is measured on each unit.
Parameter — a number that describes the population: $μ$ , $σ$ , $p$ . It is fixed and, in practice, unknown.
Statistic — the same kind of number computed from the sample: $\overset{x}{ˉ}$ , $s$ , $\overset{p}{^}$ . It is known once the sample is in hand, and it varies from sample to sample.

The parameter–statistic pair is the hinge of the whole course. A statistic estimates a parameter; the gap between them is what Modules 8–13 quantify.

Worked example — Grand River Credit Union

Grand River Credit Union holds 18,400 residential mortgage accounts. To estimate the proportion of accounts with at least one late payment in the past year, it draws a random sample of 250 accounts and finds 31 with a late payment.


Population	all 18,400 mortgage accounts at the credit union
Unit	one mortgage account
Sample	the 250 accounts drawn
Variable	whether the account had a late payment in the past year (yes / no)
Parameter	$p$ , the proportion of all 18,400 accounts with a late payment — fixed, unknown
Statistic	$\overset{p}{^} = 31/250 = 0.124$ — known, and it would differ in another sample of 250

Write $N = 18, 400$ for the population size and $n = 250$ for the sample size. Capital letters belong to the population; lower-case to the sample.

A second run — a payroll audit

An auditor at a Cambridge manufacturer with 1,250 hourly employees wants the mean overtime hours per employee last quarter, and pulls 40 timesheets at random: sample mean 6.8 hours. Population: the 1,250 employees' quarters. Unit: one employee. Sample: the 40. Variable: overtime hours (quantitative). Parameter: $μ$ , mean overtime across all 1,250. Statistic: $\overset{x}{ˉ} = 6.8$ .

Notice what changed: a yes/no variable produced a proportion; a measured variable produced a mean. The variable's type decides which row of the cheat sheet the question will use.

The trap: when the "sample" is the whole population

A Kitchener firm emails all 85 customers of its Kitchener branch and reports their mean satisfaction score, then says the result "applies to our Ontario customers". The 85 are not a sample of Ontario customers — they are a census of the Kitchener branch. Relative to the branch, the mean is a parameter (there is no sampling); relative to Ontario, the branch was never sampled at all, so no statistic estimates anything about Ontario. When a question hands you every unit of a group, ask which population the conclusion is claiming to reach.

Descriptive or inferential?

"31 of the 250 sampled accounts were late" is descriptive: it stops at the sample and is simply true. "About 12 % of the credit union's accounts had a late payment" is inferential: it reaches the population and must carry uncertainty — a margin of error, a confidence level. The paper marks the difference, and so does an auditor's report.

The same six rows answer both studies; the variable type (yes/no versus measured) decides whether the parameter is p or μ. The parameter is fixed; the statistic varies with the sample.