# 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.