Median absolute deviation

In

For a univariate data set X₁, X₂, ..., X_n, the MAD is defined as the

absolute deviations

from the data's median ${\displaystyle {\tilde {X}}=\operatorname {median} (X)}$:

${\displaystyle \operatorname {MAD} =\operatorname {median} (|X_{i}-{\tilde {X}}|)}$

that is, starting with the

Because the MAD is a more robust estimator of scale than the sample variance or standard deviation, it works better with distributions without a mean or variance, such as the Cauchy distribution.

Relation to standard deviation

The MAD may be used similarly to how one would use the deviation for the average. In order to use the MAD as a consistent estimator for the estimation of the standard deviation ${\displaystyle \sigma }$, one takes

${\displaystyle {\hat {\sigma }}=k\cdot \operatorname {MAD} ,}$

where ${\displaystyle k}$ is a constant

For normally distributed data ${\displaystyle k}$ is taken to be

${\displaystyle k=1/\left(\Phi ^{-1}(3/4)\right)\approx 1/0.67449\approx 1.4826,}$

i.e., the reciprocal of the quantile function ${\displaystyle \Phi ^{-1}}$ (also known as the inverse of the

${\displaystyle Z=(X-\mu )/\sigma }$

Derivation

The argument 3/4 is such that ${\displaystyle \pm \operatorname {MAD} }$ covers 50% (between 1/4 and 3/4) of the standard normal cumulative distribution function, i.e.

${\displaystyle {\frac {1}{2}}=P(|X-\mu |\leq \operatorname {MAD} )=P\left(\left|{\frac {X-\mu }{\sigma }}\right|\leq {\frac {\operatorname {MAD} }{\sigma }}\right)=P\left(|Z|\leq {\frac {\operatorname {MAD} }{\sigma }}\right).}$

Therefore, we must have that

${\displaystyle \Phi \left(\operatorname {MAD} /\sigma \right)-\Phi \left(-\operatorname {MAD} /\sigma \right)=1/2.}$

Noticing that

${\displaystyle \Phi \left(-\operatorname {MAD} /\sigma \right)=1-\Phi \left(\operatorname {MAD} /\sigma \right),}$

we have that ${\displaystyle \operatorname {MAD} /\sigma =\Phi ^{-1}(3/4)=0.67449}$, from which we obtain the scale factor ${\displaystyle k=1/\Phi ^{-1}(3/4)=1.4826}$.

Another way of establishing the relationship is noting that MAD equals the half-normal distribution median:

${\displaystyle \operatorname {MAD} =\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(1/2)\approx 0.67449\sigma .}$

This form is used in, e.g., the probable error.

In the case of complex values (X+iY), the relation of MAD to the standard deviation is unchanged for normally distributed data.

MAD using geometric median

Analogously to how the median generalizes to the geometric median (gm) in multivariate data, MAD can be generalized to MADGM (median of distances to gm) in n dimensions. This is done by replacing the absolute differences in one dimension by euclidian distances of the data points to the geometric median in n dimensions.^[5] This gives the identical result as the univariate MAD in 1 dimension and generalizes to any number of dimensions. MADGM needs the geometric median to be found, which is done by an iterative process.

The population MAD

The population MAD is defined analogously to the sample MAD, but is based on the complete distribution rather than on a sample. For a symmetric distribution with zero mean, the population MAD is the 75th percentile of the distribution.

Unlike the

The earliest known mention of the concept of the MAD occurred in 1816, in a paper by Carl Friedrich Gauss on the determination of the accuracy of numerical observations.^[6][7]

See also

• Deviation (statistics)
• Interquartile range
• Probable error
• Robust measures of scale
• Relative mean absolute difference
• Average absolute deviation
• Least absolute deviations

Notes