In arithmetic, the range of a set of data is the difference between the largest and smallest values.
Contents
- Independent identically distributed continuous random variables
- Distribution
- Moments
- Independent nonidentically distributed continuous random variables
- Independent identically distributed discrete random variables
- Example
- Related quantities
- References
However, in descriptive statistics, this concept of range has a more complex meaning. The range is the size of the smallest interval which contains all the data and provides an indication of statistical dispersion. It is measured in the same units as the data. Since it only depends on two of the observations, it is most useful in representing the dispersion of small data sets.
Independent identically distributed continuous random variables
For n independent and identically distributed continuous random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x).
Distribution
The range has cumulative distribution function
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 Xi 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
where x(G) is the inverse function. In the case where each of the Xi has a standard normal distribution, the mean range is given by
Independent nonidentically distributed continuous random variables
For n nonidentically distributed independent continuous random variables X1, X2, ..., Xn with cumulative distribution functions G1(x), G2(x), ..., Gn(x) and probability density functions g1(x), g2(x), ..., gn(x), the range has cumulative distribution function
Independent identically distributed discrete random variables
For n independent and identically distributed discrete random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability mass function g(x) the range of the Xi 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 Xi is {1,2,3,...,N} where N is a positive integer or infinity.
Distribution
The range has probability mass function
Example
If we suppose that g(x) = 1/N, the discrete uniform distribution for all x, then we find
Related quantities
The range is a simple function of the sample maximum and minimum and these are specific examples of order statistics. In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation.