Range (statistics)

The range provides an indication of statistical dispersion. Robust measures of range include the interdecile range and the interquartile range.

Range of continuous IID random variables

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).

Distribution

The range, T, has the cumulative distribution function

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

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."

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.

Moments

The mean range is given by

${\displaystyle n{\int }_0^{1}x(G)[G^{n-1}-(1-G{)}^{n-1}]\text{d}G}$

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

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

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 ${\displaystyle m,M}$ denote respectively the min and max of the random variables ${\displaystyle X_1\dots X_n}$.

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

• the index of the minimum value
• and the value ${\displaystyle x}$ of the minimum.

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

1. ${\displaystyle X_i}$ is the minimum,
2. and ${\displaystyle X_i=x}$,
3. and the range is smaller than ${\displaystyle T}$,

is:$${\displaystyle g(x){\left[G(x+T)-G(x)\right]}^{n-1}}$$Summing over the indices and integrating over ${\displaystyle x}$ yields the total probability of the event: "the range is smaller than ${\displaystyle T}$" which is exactly the cumulative density function of the range:$${\displaystyle F(t)=n{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(x){\left[G(t+x)-G(x)\right]}^{n-1}\text{d}x}$$which concludes the proof.