Parameter

By Jude Wallis · Published

A parameter is a fixed numerical value that describes a feature of an entire population, such as its true mean or proportion.

A parameter is a number attached to a population. It is fixed: it does not change when you draw a different sample, and it does not change because you failed to measure it. What is missing is your knowledge of the value, not the value. The symbols are mostly Greek: μ\mu (mu) for the population mean, σ\sigma (sigma) for the standard deviation, ρ\rho (rho) for the correlation. The proportion is the exception, since the population value is the Roman letter pp and the sample value is p^\hat{p} (p-hat).

A company has exactly five regional offices, staffed by 12, 15, 18, 20, and 25 people. If the population is those five offices, then μ=905=18\mu = \frac{90}{5} = 18 employees, and the population standard deviation divides the squared deviations by N=5N = 5, giving σ=985=4.43\sigma = \sqrt{\frac{98}{5}} = 4.43. Both are parameters and both are known, which is unusual and entirely legitimate.

Two sentences to unlearn: "a parameter is a number you cannot know" and "if I calculated it, it must be a statistic." The label comes from which group the number describes, not from whether the value is available and not from the group's size. Treat the same five offices as a random sample of 60 offices nationwide and the same data give xˉ=18\bar{x} = 18 and s=984=4.95s = \sqrt{\frac{98}{4}} = 4.95, dividing by n1=4n - 1 = 4. Same numbers in, different label out, and a different divisor for the spread.

Which one you hold depends on how the population was defined, and that comes from the question asked rather than from the spreadsheet in front of you. The mean of 500 rows is a parameter when those 500 are the whole group you want to describe and a statistic when they stand in for a larger one.

Inference is written about parameters: a confidence interval brackets one, and a hypothesis states a value for one, which is why H0:p=0.5H_0: p = 0.5 is well formed and H0:p^=0.5H_0: \hat{p} = 0.5 is not. The sample value is already in front of you, so there is nothing left to hypothesize.

Where this comes up

62 pages on the site use this term.

More collecting data and study design terms, or browse the full statistics glossary.