Population Parameter

Statistics separates the group of interest, the population, from the subset actually observed, the sample. A parameter is any numerical feature of the population: its mean, its variance, the share of its members with some characteristic, the correlation between two of its variables. Because it describes the entire population, a parameter is a single fixed number, even if no one knows its value.

In investment work the population is often abstract. The mean return of a stock is not the average of the returns that happened to occur, but the center of the distribution those returns were drawn from. That distribution can never be observed in full, so its parameters are known only through estimates.

Estimation is the bridge. A sample statistic, such as the sample mean X̄ or the sample standard deviation s, is computed from the data and used as an estimate of the corresponding parameter, μ or σ. Unlike the parameter, the statistic varies from one sample to another, which makes it a random variable with a distribution of its own, the sampling distribution.

Good estimators share three properties: unbiasedness, so that on average they hit the parameter; efficiency, so that their sampling variance is small; and consistency, so that they converge on the parameter as the sample grows. The sample variance divides by n − 1 rather than n precisely so that it estimates the population variance without bias.

Because the parameter is unknown, statements about it are probabilistic. A confidence interval gives a range built so that, across repeated samples, a stated share of such intervals would contain the parameter. A hypothesis test asks whether the sample evidence is strong enough to reject a claim about the parameter, such as a mean return of zero.

Both tools rely on the standard error, the standard deviation of the statistic's sampling distribution. When the population standard deviation σ is known, the standard error of the mean is σ / √n; when it is not, as is usual, it is estimated as s / √n, and the t-distribution takes the place of the standard normal.

Exam questions often hinge on notation and wording. Greek letters signal parameters and Latin letters signal statistics, with exceptions such as N for population size. A question that gives σ describes a known population standard deviation; one that gives s describes a sample estimate, and the formulas for tests and intervals change accordingly.

Candidates should also recognize that a parameter has no standard error, since it does not vary, while every statistic has one. A statement such as the population mean fell this year is a sign that a sample statistic has been confused with the parameter it estimates.