WSU Lexicon: Quantitative Methods

Sample Statistic

A sample statistic is a number calculated from a sample, such as the sample mean X̄ or the sample standard deviation s, and used to estimate the corresponding population parameter. Because each sample differs, the statistic varies from sample to sample and has a probability distribution of its own.

It estimates a population parameter; its own variability is measured by the standard error.

The entry

Numbers Computed From the Data

Any quantity calculated from observed data is a statistic: the sample mean, the median, the standard deviation, a correlation coefficient, a regression slope. When it is used to learn about the population the data come from, the formula is called an estimator, and the value it takes in a particular sample is an estimate. The sample mean X̄, for example, estimates the population mean μ, and the sample standard deviation s estimates σ.

The defining feature of a statistic is that it depends on which observations happened to be drawn. A different set of twelve monthly returns from the same fund would produce a different sample mean, even though the fund's true expected return has not changed.

The entry

Sampling Distributions and the Central Limit Theorem

Since a statistic varies across samples, it has a probability distribution. For the sample mean, that distribution is centered on the population mean, and its standard deviation, the standard error, equals σ / √n. The central limit theorem adds a powerful result: for a reasonably large sample from almost any population with finite variance, the distribution of the sample mean is approximately normal.

That is what makes inference possible. Knowing the shape and spread of the sampling distribution, an analyst can say how far a sample mean is likely to fall from the true mean, build confidence intervals and test hypotheses about the parameter, even when the population itself is far from normal.

The entry

Bias in How Samples Are Drawn

A statistic is only as good as its sample. Survivorship bias arises when a database includes only funds or companies still in existence, inflating average returns because the failures have dropped out. Look-ahead bias uses information that was not available at the time, and time-period bias draws conclusions from a stretch of history that may not be representative. Data snooping, testing many ideas on the same data until one looks significant, produces statistics that do not hold up out of sample.

None of these problems is cured by a larger sample. They shift the estimate away from the parameter rather than adding noise around it, so the standard error understates how wrong the statistic may be.

The entry

Exam Focus

Candidates should compute common sample statistics, apply the n − 1 divisor for the sample variance and standard deviation, and keep notation straight: Latin letters for statistics, Greek letters for parameters, with recognized exceptions. They should also be able to explain why a statistic has a standard error and a parameter does not.

Questions frequently describe a data set and ask which statistic or test is appropriate. The first step is always to establish whether a given number is a sample result or a population value, because that choice decides whether to use σ or s, the normal or the t-distribution, and which formula for the standard error applies.

Common trap

“Sample variance is the sum of squared deviations divided by the sample size n.”

Dividing by n understates the population variance, because deviations are measured from the sample mean, which sits closer to the data than the true mean does. Dividing by n − 1 corrects that bias and gives an unbiased estimator.

In practice

How the US unemployment rate is measured

The monthly US unemployment rate comes from the Current Population Survey, a sample of about 60,000 households. The published rate is a sample statistic estimating the true rate for the whole labor force, and it is reported with a margin of sampling error.

Candidates ask

Questions on this term

What is the difference between a statistic and a parameter?A parameter describes the whole population and is fixed but usually unknown. A statistic is computed from a sample, is known once the data are in, and changes when a different sample is drawn.
Why does a sample statistic have a distribution?Because the sample is one of many that could have been drawn. Each possible sample gives a slightly different mean or standard deviation, and the spread of those values across samples is the sampling distribution of the statistic. Its standard deviation, the standard error, shows how precisely the statistic pins down the parameter.
Does a bigger sample always give a better statistic?A larger random sample reduces the standard error, so the estimate becomes more precise, roughly in proportion to one over the square root of the sample size. But size does not fix bias. A large sample drawn only from surviving funds, or from an unusual period, will estimate the wrong population precisely. Representative sampling matters as much as sample size.

Related entries

Central Limit TheoremThe central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size grows, whatever the shape of the underlying population, provided observations are independent with finite variance.
Population ParameterA population parameter is a fixed number that describes a whole population, such as its true mean μ or true standard deviation σ.
Sampling ErrorSampling error is the difference between a sample statistic and the corresponding population parameter it estimates: for example, the sample mean minus the true population mean.
Standard ErrorThe standard error is the standard deviation of a sample statistic across repeated samples.

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Sample Statistic

It estimates a population parameter; its own variability is measured by the standard error.