More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample's distribution.
In the calculation of the arithmetic mean, for example, the algorithm consists of summing all the data values and dividing this sum by the number of data items. Thus the arithmetic mean is a statistic.
Other examples of statistics include
- Sample mean and sample median
- Sample variance and sample standard deviation
- Sample quantiles besides the median, e.g., quartiles and percentiles
- t statistics, chi-square 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
On the other hand, the z-score is not a statistic, because it depends on the unknown parameters of the distribution, <math>\mu</math> and <math>\sigma.</math>
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. The average height of all (in the sense of genetically possible) 25-year-old North American men is a parameter and not a statistic.
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