# Statistical dispersion

In mathematics, a (statistical) dispersion (also called statistical variability) of a set (list) of data is a measure of how observations in the data set are distributed across various categories. There are many different descriptive statistics that can be chosen as a measurement of the central tendency. In other words, dispersion is quantifiable variation of measurements of differing members of a population within the scale on which they are measured.

## Measures of statistical dispersion

A measure of statistical dispersion is a real number that is zero if all the data are identical, and increases as the data becomes more diverse. An important measure of dispersion is the standard deviation, which is the square root of the variance (which is itself a measure of dispersion).

Other such measures include the range, the interquartile range, the mean difference, the median absolute deviation 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. See also quartile coefficient of dispersion.

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 SX then a linear transformation Y = aX + b for real a and b should have dispersion SY = |a|SX. One of the forms in which statistical variability is realized in the empirical sciences is that of differences in repeated measurements of the same quantity.

## Sources of statistical dispersion

In the physical sciences, such variability may result only from random measurement errors: instrument measurements are often not perfectly precise - i.e., reproducible. One may assume that the quantity being measured is unchanging and stable, and that the variation between measurements is due to observational error.

In the biological sciences, this assumption is false: the variation observed might be intrinsic to the phenomenon: distinct members of a population differ greatly. This is also seen in the arena of manufactured products; even there, the meticulous scientist finds variation.

The simple model of a stable quantity is preferred when it is tenable. Each phenomenon must be examined to see if it warrants such a simplification. 