# 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.

1) the sample minimum (smallest observation)

### Outlier

**outliersconservative estimateirregularities**

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.

1) the sample minimum

### Average absolute deviation

**mean absolute deviationmean deviationMAD**

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 statisticsorderedth-smallest of items**

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).

Sample maximum and minimum

### Prediction interval

**interval forecastsPI**

The sample maximum and minimum provide a non-parametric prediction interval:

In the same way, the probability that X n+1 will be the smallest is 1/(n + 1). 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 }. Thus, denoting the sample maximum and minimum by M and m, this yields an (n − 1)/(n + 1) prediction interval of [m, M].

### Mid-range

**midsummarymidrangehalf-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.

### Discrete uniform distribution

**uniform distributionuniformly distributeduniformly at random**

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.

where m is the sample maximum and k is the sample size, sampling without replacement.

### 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 testingstatistical test for the normality**

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.

### Maxima and minima

**maximumminimumlocal maximum**

* Maxima and minima

Sample maximum and minimum

### 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**

### Box plot

**boxplotbox and whisker plotadjusted boxplots**

### Robust statistics

**robustbreakdown pointrobustness**

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

### Extreme value theory

**Extreme Value Analysisextreme eventsextreme value distribution**

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

**quantilesquintiletertile**

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.