A statistic (singular) is a single measure of some attribute of a sample (e.g. its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items comprising the sample which are known together as a set of data.

More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample's distribution; that is, the function can be stated before realization of the data. The term statistic is used both for the function and for the value of the function on a given sample.

A statistic is distinct from a statistical parameter, which is not computable because often the population is much too large to examine and measure all its items. However a statistic, when used to estimate a population parameter, is called an estimator. For instance, the sample mean is a statistic which estimates the population mean, which is a parameter.

  Examples

In calculating the arithmetic mean of a sample, for example, the algorithm works by summing all the data values observed in the sample then divides this sum by the number of data items. This single measure, the mean of the sample, is called a statistic and its value is frequently used as an estimate of the mean value of all items comprising the population from which the sample is drawn. The population mean is also a single measure however it is not called a statistic; instead it is called a population parameter.

Other examples of statistics include

• Sample mean discussed in the example above and sample median
• Sample variance and sample standard deviation
• Sample quantiles besides the median, e.g., quartiles and percentiles
• Test statistics, such as t statistics, chi-squared statistics, f statistics
• Order statistics, including sample maximum and minimum
• Sample moments and functions thereof, including kurtosis and skewness
• Various functionals of the empirical distribution function

  Properties

  Observability

A statistic is an observable random variable, which differentiates it from a parameter that is a generally unobservable quantity describing a property of a statistical population. A parameter can only be computed exactly if the entire population can be observed without error; for instance, in a perfect census or for a population of standardized test takers.

Statisticians often contemplate a parameterized family of probability distributions, any member of which could be the distribution of some measurable aspect of each member of a population, from which a sample is drawn randomly. For example, the parameter may be the average height of 25-year-old men in North America. The height of the members of a sample of 100 such men are measured; the average of those 100 numbers is a statistic. The average of the heights of all members of the population is not a statistic unless that has somehow also been ascertained (such as by measuring every member of the population). The average height of all (in the sense of genetically possible) 25-year-old North American men is a parameter and not a statistic.

  Statistical properties

Important potential properties of statistics include completeness, consistency, sufficiency, unbiasedness, minimum mean square error, low variance, robustness, and computational convenience.

  Information of a statistic

Information of a statistic on model parameters can be defined in several ways. The most common one is the Fisher information which is defined on the statistic model induced by the statistic. Kullback information measure can also be used.

  See also

Look up statistic in Wiktionary, the free dictionary.

• Statistics
• Statistical theory
• Descriptive statistics
• Statistical hypothesis testing
• Well-behaved statistic

  References

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