Stimulus–response model

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. In arithmetic, the range of a set of data is the difference between the largest and smallest values.

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, ..., X_n with cumulative distribution function G(x) and probability density 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).

Distribution

The range has cumulative distribution function

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

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

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

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

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

Independent nonidentically distributed continuous random variables

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

$F(t)=\sum _{i=1}^n{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}_i(x)\prod _{j=1,j\ne i}^n[G_j(x+t)-G_j(x)]\text{d}x.$

Independent identically distributed discrete random variables

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.

Distribution

The range has probability mass function

$f(t)=\{\begin{array}{cc}\sum _{x=1}^N[g(x){]}^n& t=0\\ \sum _{x=1}^{N-t}\left(\begin{array}{cc}& [G(x+t)-G(x-1){]}^n\\ & -[G(x+t)-G(x){]}^n\\ & -[G(x+t-1)-G(x-1){]}^n\\ & +[G(x+t-1)-G(x){]}^n\end{array}\right)& t=1,2,3\dots ,N-1.\end{array}$

Example

If we suppose that g(x)=1/N, the discrete uniform distribution for all x, then we find

$f(t)=\{\begin{array}{cc}\frac{1}{N^{n-1}}& t=0\\ \sum _{x=1}^{N-t}([\frac{t+1}{N}{]}^n-2[\frac{t}{N}{]}^n+[\frac{t-1}{N}{]}^n)& t=1,2,3\dots ,N-1.\end{array}$

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.

See also

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• Mid-range
• Interquartile range
• Studentized range

References

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External links

• APPL, a Maple script for computing the range of independent identically discrete random variables