Editing Range (statistics)

{{Short description|Concept in statistics}}
{{Distinguish|Mid-range}}
{{other uses|Range (disambiguation)#Mathematics}}

In [[descriptive statistics]], the '''range''' of a set of data is size of the narrowest [[interval (statistics)|interval]] which contains all the data.
It is calculated as the difference between the largest and smallest values (also known as  the [[sample maximum and minimum]]).<ref>{{cite book|title=An Introduction to Statistics|author=George Woodbury|page=74|isbn=0534377556|publisher=Cengage Learning|year=2001}}</ref> 
It is expressed in the same [[units (physics)|units]] as the data.

The range provides an indication of [[statistical dispersion]]. Closely related alternative measures are the [[Interdecile range]] and the [[Interquartile range]].

==Range of continuous IID random variables==

For ''n'' [[independent and identically distributed random variables|independent and identically distributed continuous random variables]] ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> with the [[cumulative distribution function]]  G(''x'') and a [[probability density function]] g(''x''), let T denote the range of them, that is, T= max(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>)-
min(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>).

===Distribution===

The range, T,  has the cumulative distribution function<ref name="gumbel">{{cite journal | author = E. J. Gumbel | author-link = E. J. Gumbel | year = 1947 | title = The Distribution of the Range | journal = The Annals of Mathematical Statistics | volume = 18 | issue = 3 | pages = 384–412 | jstor = 2235736 | doi=10.1214/aoms/1177730387| doi-access = free }}</ref><ref name="tsimashenka">{{Cite book | last1 = Tsimashenka | first1 = I. | last2 = Knottenbelt | first2 = W. | last3 = Harrison | first3 = P. | author-link3 = Peter G. Harrison| doi = 10.1007/978-3-642-30782-9_12 | chapter = Controlling Variability in Split-Merge Systems | title = Analytical and Stochastic Modeling Techniques and Applications | series = Lecture Notes in Computer Science | volume = 7314 | pages = 165 | year = 2012 | isbn = 978-3-642-30781-2 | url = http://www.doc.ic.ac.uk/~wjk/publications/tsimashenka-knottenbelt-harrison-asmta-2012.pdf}}</ref>
::<math>F(t)= n \int_{-\infty}^\infty g(x)[G(x+t)-G(x)]^{n-1} \, \text{d}x.</math>
[[Emil Julius Gumbel|Gumbel]] notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express ''G''(''x''&nbsp;+&nbsp;''t'') by ''G''(''x''), and that the numerical integration is lengthy and tiresome."{{R|gumbel|p=385 <!-- (PDF p.&nbsp;2) -->}}

If the distribution of each ''X''<sub>''i''</sub> 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]].<ref name="gumbel" />

===Moments===

The mean range is given by<ref>{{cite journal | author1 = H. O. Hartley | author-link1 = H. O. Hartley | author2 = H. A. David | year = 1954 | title = Universal Bounds for Mean Range and Extreme Observation | journal = The Annals of Mathematical Statistics | volume = 25 | issue = 1 | pages = 85–99 | jstor = 2236514 | doi=10.1214/aoms/1177728848| doi-access = free }}</ref>
::<math>n \int_0^1 x(G)[G^{n-1}-(1-G)^{n-1}] \,\text{d}G</math>
where ''x''(''G'') is the inverse function. In the case where each of the ''X''<sub>''i''</sub> has a [[standard normal distribution]], the mean range is given by<ref>{{cite journal | author = L. H. C. Tippett | author-link = L. H. C. Tippett | year = 1925 | title = On the Extreme Individuals and the Range of Samples Taken from a Normal Population | journal = Biometrika | volume = 17 | issue = 3/4 | pages = 364–387 | jstor = 2332087 | doi=10.1093/biomet/17.3-4.364}}</ref>
::<math display="block">\int_{-\infty}^\infty (1-(1-\Phi(x))^n-\Phi(x)^n ) \,\text{d}x.</math>

===Derivation of the distribution===

Please note that the following is an informal derivation of the result. It is a bit loose with the calculation of the probabilities.

Let <math>m, M</math> denote respectively the min and max of the random variables <math>X_1 \dots X_n</math>.

The event that the range is smaller than <math>T</math> can be decomposed into smaller events according to:

* the index of the minimum value
* and the value <math>x</math> of the minimum.

For a given index <math>i</math> and minimum value <math>x</math>, the probability of the joint event:

# <math>X_i</math> is the minimum,
# and <math>X_i=x</math>,
# and the range is smaller than <math>T</math>,

is:<math display="block">
g(x) \left[ G(x+T) - G(x) \right]^{n-1}
</math>Summing over the indices and integrating over <math>x</math> yields the total probability of the event: "the range is smaller than <math>T</math>" which is exactly the cumulative density function of the range:<math display="block">
F(t) = n \int_{-\infty}^{\infty} g(x) \left[G(t+x)-G(x) \right]^{n-1} \, \text{d}x
</math>which concludes the proof.

==The range in other models==

Outside of the IID case with continuous random variables, other cases have explicit formulas. These cases are of marginal interest.

* non-IID continuous random variables.<ref name="tsimashenka" />
* Discrete variables supported on <math>\mathbb N</math>.<ref name="evans">{{Cite journal | last1 = Evans | first1 = D. L. | last2 = Leemis | first2 = L. M. | last3 = Drew | first3 = J. H. | title = The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping | doi = 10.1287/ijoc.1040.0105 | journal = INFORMS Journal on Computing | volume = 18 | pages = 19–30 | year = 2006 }}</ref><ref>{{cite journal | author = Irving W. Burr | year = 1955 | title = Calculation of Exact Sampling Distribution of Ranges from a Discrete Population | journal = The Annals of Mathematical Statistics  | volume = 26 | issue = 3 | pages = 530–532 | jstor = 2236482 | doi=10.1214/aoms/1177728500| doi-access = free }}</ref> A key difficulty for discrete variables is that the range is discrete. This makes the derivation of the formula require [[combinatorics]].



==Related quantities==
The range is a specific example of [[order statistic]]s. In particular, the range is a linear function of order statistics, which brings it into the scope of [[L-estimator|L-estimation]].

==See also==
{{Portal|Mathematics}}
*[[Interdecile range]]
*[[Interquartile range]]
*[[Studentized range]]

==References==
{{Reflist}}

{{Statistics|descriptive}}

{{DEFAULTSORT:Range (Statistics)}}
[[Category:Statistical deviation and dispersion]]
[[Category:Scale statistics]]
[[Category:Summary statistics]]