# Order statistic

**order statisticsorderedth-smallest of items**

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.wikipedia

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### Uniform distribution (continuous)

**uniform distributionuniformuniformly distributed**

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. In this section we show that the order statistics of the uniform distribution on the unit interval have marginal distributions belonging to the Beta distribution family.

Let X (k) be the kth order statistic from this sample.

### Bapat–Beg theorem

Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat–Beg theorem.

In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables.

### Exploratory data analysis

**explorative data analysisexploratorydata analysis**

A similar important statistic in exploratory data analysis that is simply related to the order statistics is the sample interquartile range.

Francis Galton emphasized order statistics and quantiles.

### Exponential distribution

**exponentialexponentially distributedexponentially**

For random samples from an exponential distribution with parameter λ the order statistics X (i) for i = 1,2,3, ..., n each have distribution

Let denote the corresponding order statistics.

### Range (statistics)

**rangerangingsample range**

The sample range is the difference between the maximum and minimum.

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

### Rankit

**rankits**

Rankit

In statistics, rankits of a set of data are the expected values of the order statistics of a sample from the standard normal distribution the same size as the data.

### Beta distribution

**betabeta of the first kindBeta PDF**

In this section we show that the order statistics of the uniform distribution on the unit interval have marginal distributions belonging to the Beta distribution family.

Also, the kth order statistic of n uniformly distributed variates is, so an alternative if α and β are small integers is to generate α + β − 1 uniform variates and choose the α-th smallest.

### L-estimator

**L-estimation**

L-estimator – linear combinations of order statistics

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements.

### Selection algorithm

**selection problemmedian-findingSelection**

Selection algorithm

In computer science, a selection algorithm is an algorithm for finding the kth smallest number in a list or array; such a number is called the kth order statistic.

### Sample maximum and minimum

**sample maximumsample minimumMaximum**

Sample maximum and minimum

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).

### Concomitant (statistics)

**concomitant**

Concomitant (statistics)

If the sample is ordered by the X i, then the Y-variate associated with X r:n will be denoted by Y [r:n] and termed the concomitant of the r th order statistic.

### Median

**averagesample medianmedian-unbiased estimator**

Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

operations, selection algorithms can compute the 'th-smallest of items with only

### Percentile

**percentiles50th percentile85th percentile speed**

Percentile

Given the order statistics

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.

### Nonparametric statistics

**non-parametricnonparametricnon-parametric statistics**

Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.

### Statistical inference

**inferenceinferential statisticsinferences**

Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.

### Maxima and minima

**maximumminimumlocal maximum**

Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

### Quantile

**quantilesquintiletertile**

Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

### Probability theory

**theory of probabilityprobabilityprobability theorist**

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

### Sampling (statistics)

**samplingrandom samplesample**

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

### Cumulative distribution function

**distribution functionCDFcumulative probability distribution function**

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

### Time series

**time series analysistime-seriestime-series analysis**

A case when the order is significant is when the observations are part of a time series.

### Interquartile range

**inter-quartile rangebelowinterquartile**

A similar important statistic in exploratory data analysis that is simply related to the order statistics is the sample interquartile range.

### Parity (mathematics)

**even numberodd numberodd**

of observations is odd.

### Realization (probability)

**realizationrealizationsactual observed results**

Given any random variables X 1, X 2 ..., X n, the order statistics X (1), X (2), ..., X (n) are also random variables, defined by sorting the values (realizations) of X 1, ..., X n in increasing order.