A measure of

**statistical dispersion**must yield a number that is zero if all the data are identical, and must increase as the data are more diverse. A very important measure of dispersion is the standard deviation, the square root of the variance.

Other such measures include the statistical range, the interquartile range, and the average absolute deviation, and, in the case of categorical random variables, the discrete entropy. None of these can be negative; their least possible value is zero.

A measure of statistical dispersion is particularly useful if it is location invariant, and linear in scale. So if a random variable *X* has a dispersion of *S _{X}* then a linear transformation

*Y*=

*aX*+

*b*for real

*a*and

*b*should have dispersion

*S*=

_{Y}*aS*.

_{X}See also summary statistics.