Mean absolute deviation from median

The mean absolute deviation from the median is a measure of dispersion in descriptive statistics and indicates how far a sample deviates from the median on average .

definition

A sample is given

${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$

with elements. Let it be the median of the sample. Then is called ${\displaystyle n}$ ${\displaystyle {\tilde {x}}}$

${\displaystyle {\tilde {d}}_{0{,}5}={\frac {1}{n}}\sum _{i=1}^{n}|x_{i}-{\tilde {x}}|}$

the mean absolute deviation from the median. The notation can also be found as . ${\displaystyle \operatorname {MAD} }$

example

The sample is given

${\displaystyle x=(10;9;13;15;16)}$,

so it is . As a sorted sample you get ${\displaystyle n=5}$

${\displaystyle x_{\text{sort}}=(9,10,13,15,16)}$,

thus is the median

${\displaystyle {\tilde {x}}=13}$.

It follows

${\displaystyle {\begin{aligned}{\tilde {d}}_{0{,}5}&={\frac {1}{5}}\left(|10-13|+|9-13|+|13-13|+|15-13|+|16-13|\right)\\&={\frac {1}{5}}\left(|-3|+|-4|+0+|2|+|3|\right)\\&={\frac {1}{5}}(3+4+2+3)\\&={\frac {12}{5}}=2{,}4\end{aligned}}}$

In particular, the mean absolute deviation from the median almost always differs from the mean absolute deviation from the arithmetic mean . This delivers the value for the same sample

${\displaystyle d_{\overline {x}}=2{,}48}$,

see this example .

properties

Looking at the mean absolute deviation from any value , that is ${\displaystyle m}$

${\displaystyle A(m)={\frac {1}{n}}\sum _{i=1}^{n}|x_{i}-m|}$,

so is minimal if and only if the median is. An analogous result also applies to the mean square deviation from a value : it becomes minimal exactly when the arithmetic mean is. In this sense, the mean absolute deviation is a natural spread around the median, just as the mean square deviation is a natural spread around the arithmetic mean. ${\displaystyle A(m)}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle m}$

Individual evidence

1. ↑ Helge Toutenburg, Christian Heumann: Descriptive statistics . 6th edition. Springer-Verlag, Berlin / Heidelberg 2008, ISBN 978-3-540-77787-8 , pp. 74 , doi : 10.1007 / 978-3-540-77788-5 .
2. ^ Thomas Cleff: Descriptive Statistics and Exploratory Data Analysis . A computer-aided introduction with Excel, SPSS and STATA. 3rd, revised and expanded edition. Springer Gabler, Wiesbaden 2015, ISBN 978-3-8349-4747-5 , doi : 10.1007 / 978-3-8349-4748-2 .
3. ↑ Ehrhard Behrends: Elementary Stochastics . A learning book - co-developed by students. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-8348-1939-0 , pp. 275 , doi : 10.1007 / 978-3-8348-2331-1 .