range (statistics)

For n independent and identically distributed continuous random variables X₁, X₂, ..., X_n 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(X₁, X₂, ..., X_n)-

min(X₁, X₂, ..., X_n).

The range, T, has the cumulative distribution function{{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 }}{{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}}

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

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."{{R|gumbel|p=385 }}

If the distribution of each X_i 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.

The mean range is given by{{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 }}

::$n\; \backslash int\_0^1\; x(G)[G^\{n-1\}-(1-G)^\{n-1\}]\; \backslash ,\backslash text\{d\}G$

where x(G) is the inverse function. In the case where each of the X_i has a standard normal distribution, the mean range is given by{{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}}

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

For n nonidentically distributed independent continuous random variables X₁, X₂, ..., X_n with cumulative distribution functions G₁(x), G₂(x), ..., G_n(x) and probability density functions g₁(x), g₂(x), ..., g_n(x), the range has cumulative distribution function

::$F(t)\; =\; \backslash sum\_\{i=1\}^n\; \backslash int\_\{-\backslash infty\}^\backslash infty\; g\_i(x)\; \backslash prod\_\{j=1,\; j\; \backslash neq\; i\}^n\; [G\_j(x+t)-G\_j(x)]\; \backslash ,\; \backslash text\{d\}x.$

For n independent and identically distributed discrete random variables X₁, X₂, ..., X_n with cumulative distribution function G(x) and probability mass function g(x) the range of the X_i 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 X_i is {1,2,3,...,N} where N is a positive integer or infinity.{{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 }}{{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 }}

The range has probability mass function{{Cite journal | last1 = Abdel-Aty | first1 = S. H. | title = Ordered variables in discontinuous distributions | doi = 10.1111/j.1467-9574.1954.tb00442.x | journal = Statistica Neerlandica | volume = 8 | issue = 2 | pages = 61–82 | year = 1954 }}{{Cite journal | last1 = Siotani | first1 = M. | doi = 10.1007/BF02863574 | title = Order statistics for discrete case with a numerical application to the binomial distribution | journal = Annals of the Institute of Statistical Mathematics | volume = 8 | pages = 95–96 | year = 1956 | issue = 2 }}

::$f(t)=\backslash begin\{cases\}$

\sum_{x=1}^N[g(x)]^n & t=0 \\[6pt]

\sum_{x=1}^{N-t}\left(\begin{alignat}{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{alignat} \right)& t=1,2,3\ldots,N-1.

\end{cases}

If we suppose that g(x) = 1/N, the discrete uniform distribution for all x, then we find{{cite journal | author = Paul R. Rider | year = 1951 | title = The Distribution of the Range in Samples from a Discrete Rectangular Population | journal = Journal of the American Statistical Association | volume = 46 | issue = 255 | pages = 375–378 | jstor = 2280515 | doi=10.1080/01621459.1951.10500796}}

::$f(t)=\backslash begin\{cases\}$

\frac{1}{N^{n-1}} & t=0 \\[4pt]

\sum_{x=1}^{N-t}\left(\left[\frac{t+1}{N}\right]^n -2\left[\frac{t}{N}\right]^n +\left[\frac{t-1}{N}\right]^n

\right) & t=1,2,3\ldots ,N-1.

\end{cases}

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 $ng(x)$. The probability of another having a value t greater than x is:

:$(n-1)g(x+t).$

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

:$\backslash left(\backslash int\_x^\{x+t\}\; g(x)\backslash ,\backslash text\{d\}x\backslash right)^\{n-2\}\; =\; \backslash left(G(x+t)-G(x)\backslash right)^\{n-2\}.$

Combining the three together yields:

:$f(t)=\; n(n-1)\backslash int\_\{-\backslash infty\}^\backslash infty\; g(x)g(x+t)[G(x+t)-G(x)]^\{n-2\}\; \backslash ,\; \backslash text\{d\}x$

The range is a specific example of order statistics. In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation.

{{Portal|Mathematics}}

• Interdecile range
• Interquartile range
• Studentized range

{{Reflist}}

{{Statistics|descriptive}}

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Category:Statistical deviation and dispersion

Category:Scale statistics

Category:Summary statistics