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
with elements. Let it be the median of the sample. Then is called
the mean absolute deviation from the median. The notation can also be found as .
example
The sample is given
- ,
so it is . As a sorted sample you get
- ,
thus is the median
- .
It follows
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
- ,
see this example .
properties
Looking at the mean absolute deviation from any value , that is
- ,
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.
Individual evidence
- ↑ 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 .
- ^ 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 .
- ↑ 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 .