Constants of the population, estimates from the sample
A parameter is a number describing a population: the true mean height of all adults in a city, the exact share of all voters who approve. It is fixed, usually unknown, and the thing studies try to learn. A statistic is a number computed from a sample: the mean height of the 200 people you measured. Statistics estimate parameters, and the gap between them is sampling error, which good design shrinks but never removes.
| Quantity | Parameter (population) | Statistic (sample) |
|---|---|---|
| Mean | μ | x̄ |
| Standard deviation | σ | s |
| Variance | σ² | s² |
| Proportion | p | p̂ |
| Correlation | ρ | r |
| Size | N | n |
A college has 1,200 enrolled students. The registrar's full records give a true mean GPA: that number, whatever it is, is a parameter (μ). You survey 60 students and compute a mean GPA of 3.1: a statistic (x̄ = 3.1), your estimate of μ. A second survey of a different 60 would give a slightly different statistic, while μ sits unchanged. Inference is the discipline of saying how far x̄ plausibly sits from μ.
Why the distinction runs the whole subject
Hypothesis tests state claims about parameters (H0: μ = 40) and judge them with statistics (x̄ = 44.5, so t = 2.68). Confidence intervals use a statistic to bracket a parameter. Even the sample variance's n − 1 divisor exists because s² must estimate σ² honestly. When a report blurs the two ("the average is 3.1", said of a whole college from 60 surveys), it hides the uncertainty that separates a finding from a guess.
Treating a big sample's statistic as the parameter. Sampling error shrinks with n but reaches zero only at a census. A statistic from 10,000 people is a sharp estimate; it is still an estimate, and its standard error says how sharp.
Frequently asked questions
Is the sample mean a parameter or a statistic?
A statistic: it is computed from a sample. The population mean it estimates is the parameter. The Greek/Latin symbol split (mu vs x-bar) keeps the roles visible.
Can a parameter ever be known exactly?
Yes, when the population is fully measured: a census, complete company records, every item off a production line that gets inspected. Most research settings cannot afford that, which is why estimation exists.
What is a point estimate?
A single statistic offered as the best guess of a parameter: x-bar for mu, p-hat for p. Intervals add the honesty: a range plus the confidence level, instead of one naked number.