Range (statistics)

In descriptive statistics, the range of a set of data is size of the narrowest 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).^[1] It is expressed in the same units as the data.

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

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^[2][3]

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

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."^[2]: 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.^[2]

The mean range is given by^[4]

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

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^[5]

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

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

Let ${\displaystyle m,M}$ 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:$${\displaystyle F(t)=n\int _{-\infty }^{\infty }g(x)\left[G(t+x)-G(x)\right]^{n-1}\,{\text{d}}x}$$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 ^[3].
• Discrete variables supported on ${\displaystyle \mathbb {N} }$ ^[6][7]. 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.

• Interdecile range
• Interquartile range
• Studentized range