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}}

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

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

Let $m,\; M$ denote respectively the min and max of the random variables $X\_1\; \backslash dots\; X\_n$.

The event that the range is smaller than $T$ can be decomposed into smaller events according to:

• the index of the minimum value
• and the value $x$ of the minimum.

For a given index $i$ and minimum value $x$, the probability of the joint event:

1. $X\_i$ is the minimum,
2. and $X\_i=x$,
3. and the range is smaller than $T$,

is:$$$$

g(x) \left[ G(x+T) - G(x) \right]^{n-1}

Summing over the indices and integrating over $x$ yields the total probability of the event: "the range is smaller than $T$" which is exactly the cumulative density function of the range:$$$$

F(t) = n \int_{-\infty}^{\infty} g(x) \left[G(t+x)-G(x) \right]^{n-1} \, \text{d}x

which concludes the proof.

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.
• Discrete variables supported on $\backslash mathbb\; N$.{{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 }} A key difficulty for discrete variables is that the range is discrete. This makes the derivation of the formula require combinatorics.

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}}

{{DEFAULTSORT:Range (Statistics)}}

Category:Statistical deviation and dispersion

Category:Scale statistics

Category:Summary statistics