Absolute Deviation

An Absolute Deviation is an absolute difference between any given element in a dataset and its mean .

• Context:
  • It can be defined as [math]\displaystyle{ \Delta\;x_i = |x_i-\bar{x}| }[/math] where [math]\displaystyle{ x_i }[/math] is i-th elements of dataset, [math]\displaystyle{ \bar{x} }[/math] is the mean value of the dataset and |A| denotes the absolute value of A.
  • It can range from being a deviation to being average absolute deviation.
• Counter-Example(s):
  • Average Absolute Deviation.
  • Median Absolute Deviation.
  • Deviation.
• See: Deviation, Average Absolute Deviation, Mean Absolute Error, Median Absolute Deviation.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Absolute_deviation Retrieved 2016-07-10
  • In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the deviation is reckoned from the central value, being construed as some type of average, most often the median or sometimes the mean of the data set.

[math]\displaystyle{ D_i = |x_i-m(X)| }[/math]

where

D_i is the absolute deviation,

x_i is the data element

and m(X) is the chosen measure of central tendency of the data set — sometimes the mean ([math]\displaystyle{ \overline{x} }[/math]), but most often the median.

2016

• (Eric W. Weisstein, 2016) ⇒ Weisstein, Eric W. (1999 - 2016) "Absolute Deviation.” From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AbsoluteDeviation.html Retrievd 2016-07-10
  • Let [math]\displaystyle{ \bar{u} }[/math] denote the mean of a set of quantities [math]\displaystyle{ u_i }[/math], then the absolute deviation is defined by

[math]\displaystyle{ \Delta u_i\equiv|u_i-\bar{u}| }[/math]

2016

• (GNU Scientic Library, 2016) ⇒ http://www.gnu.org/software/gsl/manual/html_node/Absolute-deviation.html
  • 21.2 Absolute deviation
    • Function: double gsl_stats_absdev (const double data[], size_t stride, size_t n)

This function computes the absolute deviation from the mean of data, a dataset of length n with stride stride. The absolute deviation from the mean is defined as,

absdev = (1/N) \sum |x_i - \Hat\mu|

where x_i are the elements of the dataset data. The absolute deviation from the mean provides a more robust measure of the width of a distribution than the variance. This function computes the mean of data via a call to gsl_stats_mean.

Function: double gsl_stats_absdev_m (const double data[], size_t stride, size_t n, double mean)

This function computes the absolute deviation of the dataset data relative to the given value of mean,

absdev = (1/N) \sum |x_i - mean|

This function is useful if you have already computed the mean of data (and want to avoid recomputing it), or wish to calculate the absolute deviation relative to another value (such as zero, or the median).

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