Sample maximum and minimum

sample maximumsample minimumMaximum
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.wikipedia
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Five-number summary

five-number summaries
They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot.

Outlier

If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low.
Outliers, being the most extreme observations, may include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low.

Seven-number summary

Bowley's seven-figure summaryseven-figure summary
They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot.

Average absolute deviation

mean absolute deviationMaximum absolute deviationmean deviation
They also realize the maximum absolute deviation: one of them is the furthest point from any given point, particularly a measure of center such as the median or mean.
While not strictly a measure of central tendency, the maximum absolute deviation can be found using the formula for the average absolute deviation as above with, where \max(X) is the sample maximum.

Order statistic

order statisticsk'th-smallest of n itemsordered
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).

Prediction interval

interval forecastsPIpredictive performance
The sample maximum and minimum provide a non-parametric prediction interval:
The other (n − 1)/(n + 1) of the time, X n+1 falls between the sample maximum and sample minimum of the sample {X 1, ..., X n }.

Mid-range

Midrangemidsummaryhalf-range
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.
The sample maximum and sample minimum, together with sample size, are a sufficient statistic for the population maximum and minimum – the distribution of other samples, conditional on a given maximum and minimum, is just the uniform distribution between the maximum and minimum and thus add no information.

Range (statistics)

rangerangingsample range
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.
The range is a simple function of the sample maximum and minimum and these are specific examples of order statistics.

Normality test

Non-normality of errorsnormality testingNormality tests
The sample extrema can be used for a simple normality test, specifically of kurtosis: one computes the t-statistic of the sample maximum and minimum (subtracts sample mean and divides by the sample standard deviation), and if they are unusually large for the sample size (as per the three sigma rule and table therein, or more precisely a Student's t-distribution), then the kurtosis of the sample distribution deviates significantly from that of the normal distribution.
Simple back-of-the-envelope test takes the sample maximum and minimum and computes their z-score, or more properly t-statistic

German tank problem

a statistical analysis of the serial numbers on the road wheelsstatistical analysis
If only the top endpoint is unknown, the sample maximum is a biased estimator for the population maximum, but the unbiased estimator (where m is the sample maximum and k is the sample size) is the UMVU estimator; see German tank problem for details.
where m is the largest serial number observed (sample maximum) and k is the number of tanks observed (sample size).

Sufficient statistic

sufficient statisticssufficientsufficiency
For sampling without replacement from a uniform distribution with one or two unknown endpoints (so 1,2,\dots,N with N unknown, or with both M and N unknown), the sample maximum, or respectively the sample maximum and sample minimum, are sufficient and complete statistics for the unknown endpoints; thus an unbiased estimator derived from these will be UMVU estimator.
If X 1, ...., X n are independent and uniformly distributed on the interval [0,θ], then T(X) = max(X 1, ..., X n ) is sufficient for θ — the sample maximum is a sufficient statistic for the population maximum.

Minimum-variance unbiased estimator

minimum variance unbiased estimatorbest unbiased estimatorUMVU
For sampling without replacement from a uniform distribution with one or two unknown endpoints (so 1,2,\dots,N with N unknown, or with both M and N unknown), the sample maximum, or respectively the sample maximum and sample minimum, are sufficient and complete statistics for the unknown endpoints; thus an unbiased estimator derived from these will be UMVU estimator.

Maxima and minima

maximumminimumlocal maximum
* Maxima and minima

Statistics

statisticalstatistical analysisstatistician
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.

Sample (statistics)

samplesamplesstatistical sample
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.

Summary statistics

summary statisticSummarizationdata summarization
They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot.

Descriptive statistics

descriptivedescriptive statisticstatistics
They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot.

Box plot

They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot.

Robust statistics

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

Extreme value theory

rare eventextreme eventsextreme value analysis
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.

Quantile

quantilestertilequintile
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.

Percentile

percentiles50th percentile85th percentile speed
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.

Smooth maximum

A smooth maximum, for example,

Exchangeable random variables

exchangeabilityexchangeableexchangeable sequence
in a sample from a population, or more generally an exchangeable sequence of random variables, each observation is equally likely to be the maximum or minimum.

Estimator

estimatorsestimateestimates
Due to their sensitivity to outliers, the sample extrema cannot reliably be used as estimators unless data is clean – robust alternatives include the first and last deciles.