# Range (statistics)

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In statistics, the range of a set of data is the difference between the largest and smallest values. [1]

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

## Contents

However, in descriptive statistics, this concept of range has a more complex meaning. The range is the size of the smallest interval (statistics) which contains all the data and provides an indication of statistical dispersion. It is measured in the same units as the data. Since it only depends on two of the observations, it is most useful in representing the dispersion of small data sets. [2]

A descriptive statistic is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential statistics, in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently nonparametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups, and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, the proportion of subjects with related comorbidities, etc.

In statistics, dispersion is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

## For continuous IID random variables

For n independent and identically distributed continuous random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x).

In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d. or iid or IID. Herein, i.i.d. is used, because it is the most prevalent.

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

### Distribution

The range has cumulative distribution function [3] [4]

${\displaystyle F(t)=n\int _{-\infty }^{\infty }g(x)[G(x+t)-G(x)]^{n-1}{\text{d}}x.}$

Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome." [3]

Emil Julius Gumbel was a German mathematician and political writer.

If the distribution of each Xi is limited to the right (or left) then the asymptotic distribution of the range is equal to the asymptotic distribution of the largest (smallest) value. For more general distributions the asymptotic distribution can be expressed as a Bessel function. [3]

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation

### Moments

The mean range is given by [5]

${\displaystyle n\int _{0}^{1}x(G)[G^{n-1}-(1-G)^{n-1}]\,{\text{d}}G}$

where x(G) is the inverse function. In the case where each of the Xi has a standard normal distribution, the mean range is given by [6]

${\displaystyle \int _{-\infty }^{\infty }(1-(1-\Phi (x))^{n}-\Phi (x)^{n})\,{\text{d}}x.}$

## For continuous non-IID random variables

For n nonidentically distributed independent continuous random variables X1, X2, ..., Xn with cumulative distribution functions G1(x), G2(x), ..., Gn(x) and probability density functions g1(x), g2(x), ..., gn(x), the range has cumulative distribution function [4]

${\displaystyle F(t)=\sum _{i=1}^{n}\int _{-\infty }^{\infty }g_{i}(x)\prod _{j=1,j\neq i}^{n}[G_{j}(x+t)-G_{j}(x)]\,{\text{d}}x.}$

## For discrete IID random variables

For n independent and identically distributed discrete random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability mass function g(x) the range of the Xi is the range of a sample of size n from a population with distribution function G(x). We can assume without loss of generality that the support of each Xi is {1,2,3,...,N} where N is a positive integer or infinity. [7] [8]

### Distribution

The range has probability mass function [7] [9] [10]

{\displaystyle f(t)={\begin{cases}\sum _{x=1}^{N}[g(x)]^{n}&t=0\\[6pt]\sum _{x=1}^{N-t}\left({\begin{alignedat}{2}&[G(x+t)-G(x-1)]^{n}\\{}-{}&[G(x+t)-G(x)]^{n}\\{}-{}&[G(x+t-1)-G(x-1)]^{n}\\{}+{}&[G(x+t-1)-G(x)]^{n}\\\end{alignedat}}\right)&t=1,2,3\ldots ,N-1.\end{cases}}}

#### Example

If we suppose that g(x) = 1/N, the discrete uniform distribution for all x, then we find [9] [11]

${\displaystyle f(t)={\begin{cases}{\frac {1}{N^{n-1}}}&t=0\\[4pt]\sum _{x=1}^{N-t}\left([{\frac {t+1}{N}}]^{n}-2[{\frac {t}{N}}]^{n}+[{\frac {t-1}{N}}]^{n}\right)&t=1,2,3\ldots ,N-1.\end{cases}}}$

## Derivation

The probability of having a specific range value, t, can be determined by adding the probabilities of having two samples differing by t, and every other sample having a value between the two extremes. The probability of one sample having a value of x is ${\displaystyle n*g\left(x\right)}$. The probability of another having a value t greater than x is:

${\displaystyle \left(n-1\right)g\left(x+t\right)}$.

The probability of all other values lying between these two extremes is:

${\displaystyle \left(\int _{x}^{x+t}g\left(x\right){\text{d}}x\right)^{n-2}=\left(G\left(x+t\right)-G\left(x\right)\right)^{n-2}}$.

Combining the three together yields:

${\displaystyle f(t)=n\left(n-1\right)\int _{-\infty }^{\infty }g\left(x\right)g\left(x+t\right)\left[G\left(x+t\right)-G\left(x\right)\right]^{n-2}{\text{d}}x}$

The range is a simple function of the sample maximum and minimum and these are specific examples of order statistics. In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation.

## Related Research Articles

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