Sample maximum

In statistics, the sample maximum and sample minimum, also called the largest observation, and smallest observation, are the values of the greatest and least elements of a sample. They are basic summary statistics, used in descriptive statistics such as the five-number summary and seven-number summary and the associated box plot.

The minimum and the maximum value are the first and last order statistics (often denoted X₍₁₎ and X_(n) respectively, for a sample size of n).

If there are outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum need not be outliers, if they are not unusually far from other observations.

The sample maximum and minimum are the least robust statistics: they are maximally sensitive to outliers.

This can either be an advantage or a drawback: if extreme values are real (not measurement errors), and of real consequence, as in applications of extreme value theory such as building dikes or financial loss, then outliers (as reflected in sample extrema) are important. On the other hand, if outliers have little or no impact on actual outcomes, then using non-robust statistics such as the sample extrema simply cloud the statistics, and robust alternatives should be used, such as other quantiles: the 10th and 90th percentiles (first and last decile) are more robust alternatives.

In addition to being a component of every statistic that uses all elements of the sample, the sample extrema are important parts of the range, a measure of dispersion, and mid-range, a measure of location. They also realize the maximum absolute deviation: they are the furthest points from any given point, particularly a measure of center such as the median or mean.

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