In statistics, the range of a set of data is the difference between the largest and smallest values.
Difference here is specific, the range of a set of data is the result of subtracting the sample maximum and minimum.
However, in descriptive statistics, this concept of range has a more complex meaning. The range is the size of the smallest interval (statistics) 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.
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.": 385
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.
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 
The probability of having a specific range value, t, can be determined by adding the probabilities of having two samples differing by t, and every other sample having a value between the two extremes.
The probability of one sample having a value of x is . The probability of another having a value t greater than x is:
The probability of all other values lying between these two extremes is:
Combining the three together yields:
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.
^George Woodbury (2001). An Introduction to Statistics. Cengage Learning. p. 74. ISBN 0534377556.
^Carin Viljoen (2000). Elementary Statistics: Vol 2. Pearson South Africa. pp. 7–27. ISBN 186891075X.
^ abcE. J. Gumbel (1947). "The Distribution of the Range". The Annals of Mathematical Statistics. 18 (3): 384–412. doi:10.1214/aoms/1177730387. JSTOR 2235736.
^ abTsimashenka, I.; Knottenbelt, W.; Harrison, P. (2012). "Controlling Variability in Split-Merge Systems". Analytical and Stochastic Modeling Techniques and Applications(PDF). Lecture Notes in Computer Science. 7314. p. 165. doi:10.1007/978-3-642-30782-9_12. ISBN 978-3-642-30781-2.
^H. O. Hartley; H. A. David (1954). "Universal Bounds for Mean Range and Extreme Observation". The Annals of Mathematical Statistics. 25 (1): 85–99. doi:10.1214/aoms/1177728848. JSTOR 2236514.
^L. H. C. Tippett (1925). "On the Extreme Individuals and the Range of Samples Taken from a Normal Population". Biometrika. 17 (3/4): 364–387. doi:10.1093/biomet/17.3-4.364. JSTOR 2332087.
^ abEvans, D. L.; Leemis, L. M.; Drew, J. H. (2006). "The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping". INFORMS Journal on Computing. 18: 19. doi:10.1287/ijoc.1040.0105.
^Irving W. Burr (1955). "Calculation of Exact Sampling Distribution of Ranges from a Discrete Population". The Annals of Mathematical Statistics. 26 (3): 530–532. doi:10.1214/aoms/1177728500. JSTOR 2236482.
^ abAbdel-Aty, S. H. (1954). "Ordered variables in discontinuous distributions". Statistica Neerlandica. 8 (2): 61–82. doi:10.1111/j.1467-9574.1954.tb00442.x.
^Siotani, M. (1956). "Order statistics for discrete case with a numerical application to the binomial distribution". Annals of the Institute of Statistical Mathematics. 8: 95–96. doi:10.1007/BF02863574.