Statistical dispersion

Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.

Measures of statistical dispersion

A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse.

Most measures of dispersion have the same units as the quantity being measured. In other words, if the measurements are in metres or seconds, so is the measure of dispersion. Examples of dispersion measures include:

• Standard deviation
• Interquartile range (IQR)
• Range
• Mean absolute difference (also known as Gini mean absolute difference)
• Median absolute deviation (MAD)
• Average absolute deviation (or simply called average deviation)
• Distance standard deviation

These are frequently used (together with scale factors) as estimators of scale parameters, in which capacity they are called estimates of scale. Robust measures of scale are those unaffected by a small number of outliers, and include the IQR and MAD.

All the above measures of statistical dispersion have the useful property that they are location-invariant and linear in scale. This means that if a random variable ${\displaystyle X}$ has a dispersion of ${\displaystyle S_X}$ then a linear transformation ${\displaystyle Y=aX+b}$ for real ${\displaystyle a}$ and ${\displaystyle b}$ should have dispersion ${\displaystyle S_Y=|a|S_X}$, where ${\displaystyle |a|}$ is the absolute value of ${\displaystyle a}$, that is, ignores a preceding negative sign ${\displaystyle -}$.

Other measures of dispersion are dimensionless. In other words, they have no units even if the variable itself has units. These include:

• Coefficient of variation
• Quartile coefficient of dispersion
• Relative mean difference, equal to twice the Gini coefficient
• Entropy: While the entropy of a discrete variable is location-invariant and scale-independent, and therefore not a measure of dispersion in the above sense, the entropy of a continuous variable is location invariant and additive in scale: If ${\displaystyle H(z)}$ is the entropy of a continuous variable ${\displaystyle z}$ and ${\displaystyle z=ax+b}$, then ${\displaystyle H(z)=H(x)+\log (a)}$.

There are other measures of dispersion:

• Variance (the square of the standard deviation) – location-invariant but not linear in scale.
• Variance-to-mean ratio – mostly used for count data when the term coefficient of dispersion is used and when this ratio is dimensionless, as count data are themselves dimensionless, not otherwise.

Some measures of dispersion have specialized purposes. The Allan variance can be used for applications where the noise disrupts convergence. The Hadamard variance can be used to counteract linear frequency drift sensitivity.

For categorical variables, it is less common to measure dispersion by a single number; see qualitative variation. One measure that does so is the discrete entropy.