Mean absolute deviation from the arithmetic mean

The mean absolute deviation from the arithmetic mean , usually short mean absolute deviation called (English mean deviation or mean absolute deviation ) is a measure of dispersion in the descriptive statistics and are similar to the empirical variance in how much the sample to the arithmetic mean scatters. In contrast to the empirical variance, however, in the case of the mean absolute deviation, the distance to the arithmetic mean is not weighted squarely, but only according to the amount . Large deviations from the arithmetic mean are therefore not so significant.

definition

Given a sample with elements and be ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$ ${\displaystyle n}$

${\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$

the arithmetic mean , hereinafter referred to as mean. Then the mean absolute deviation is defined as

${\displaystyle d_{\overline {x}}(x)={\frac {1}{n}}\sum _{i=1}^{n}|x_{i}-{\overline {x}}|}$.

In addition to the notation with, there is also an abbreviation for the English term M ean absolute D eviation. ${\displaystyle d_{\overline {x}}}$ ${\displaystyle \operatorname {MD} }$

example

The sample is given

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

so it is . For the mean results ${\displaystyle n=5}$

${\displaystyle {\overline {x}}={\frac {1}{5}}(10+9+13+15+16)={\frac {63}{5}}=12{,}6}$.

So is

${\displaystyle {\begin{aligned}d_{\overline {x}}&={\frac {1}{5}}\left(|10-12{,}6|+|9-12{,}6|+|13-12{,}6|+|15-12{,}6|+|16-12{,}6|\right)\\&={\frac {1}{5}}\left(|-2{,}6|+|-3{,}6|+|0{,}4|+|2{,}4|+|3{,}4|\right)\\&={\frac {1}{5}}\left(2{,}6+3{,}6+0{,}4+2{,}4+3{,}4\right)\\&={\frac {1}{5}}\cdot {\frac {62}{5}}={\frac {62}{25}}\\&=2{,}48\end{aligned}}}$

In particular, the mean absolute deviation from the arithmetic mean generally does not agree with the mean absolute deviation from the median . If the sample is identical, this provides the value

${\displaystyle {\tilde {d}}_{0{,}5}=2{,}4}$,

see this example .

See also

• Mean square deviation

Individual evidence

1. ^ Reinhold Kosfeld, Hans Friedrich Eckey, Matthias Türck: Descriptive statistics . Basics - methods - examples - tasks. 6th edition. Springer Gabler, Wiesbaden 2016, ISBN 978-3-658-13639-0 , p. 118 , doi : 10.1007 / 978-3-658-13640-6 .
2. ↑ ^{a b} Eric W. Weisstein : Mean Deviation . In: MathWorld (English).
3. ↑ Norbert Henze: Stochastics for beginners . An introduction to the fascinating world of chance. 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 32 , doi : 10.1007 / 978-3-658-03077-3 .