Average absolute deviation

mean absolute deviationmean deviationMADmaximum absolute deviation
The average absolute deviation (or mean absolute deviation) of a data set is the average of the absolute deviations from a central point.wikipedia
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Median

averagesample medianmedian-unbiased estimator
In this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency.
When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation.

Standard deviation

standard deviationssample standard deviationsigma
The mean absolute deviation from the mean is less than or equal to the standard deviation; one way of proving this relies on Jensen's inequality.
It is algebraically simpler, though in practice less robust, than the average absolute deviation.

Deviation (statistics)

deviationabsolute deviationdeviations
The average absolute deviation (or mean absolute deviation) of a data set is the average of the absolute deviations from a central point.
Average absolute deviation, is the sum of absolute values of the deviations divided by the number of observations.

Statistical dispersion

dispersionvariabilityspread
It is a summary statistic of statistical dispersion or variability.
Average absolute deviation (or simply called average deviation)

Central tendency

Localitycentral locationcentral point
In this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency.

Sample maximum and minimum

sample maximumsample minimumMaximum
While not strictly a measure of central tendency, the maximum absolute deviation can be found using the formula for the average absolute deviation as above with, where \max(X) is the sample maximum.
They also realize the maximum absolute deviation: one of them is the furthest point from any given point, particularly a measure of center such as the median or mean.

Mean absolute error

Mean absolute error
The mean absolute error is a common measure of forecast error in time series analysis, sometimes used in confusion with the more standard definition of mean absolute deviation.

Bias of an estimator

unbiasedunbiased estimatorbias
The mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population.
A minimum-average absolute deviation median-unbiased estimator minimizes the risk with respect to the absolute loss function (among median-unbiased estimators), as observed by Laplace.

Mid-range

midsummarymidrangehalf-range
L ∞ norm statistics: the mid-range minimizes the maximum absolute deviation
While the mean of a set of values minimizes the sum of squares of deviations and the median minimizes the average absolute deviation, the midrange minimizes the maximum deviation (defined as ): it is a solution to a variational problem.

Least absolute deviations

LADleast absolute deviationleast absolute deviation (LAD) regression
Least absolute deviations
Average absolute deviation

Average

Rushing averageReceiving averagemean
The average absolute deviation (or mean absolute deviation) of a data set is the average of the absolute deviations from a central point.
Average absolute deviation

Absolute value

modulusabsolutemagnitude
The average absolute deviation (or mean absolute deviation) of a data set is the average of the absolute deviations from a central point.

Summary statistics

summary statisticSummarizationdata summarization
It is a summary statistic of statistical dispersion or variability.

Arithmetic mean

meanaveragearithmetic
In this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency.

Mode (statistics)

modemodalmodes
In this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency.

Jensen's inequality

See here
The mean absolute deviation from the mean is less than or equal to the standard deviation; one way of proving this relies on Jensen's inequality.

Square root

square rootssquareradical
Since both sides are positive, and the square root is a monotonically increasing function in the positive domain:

Inequality (mathematics)

inequalityinequalitiesless than
Since both sides are positive, and the square root is a monotonically increasing function in the positive domain:

Normal distribution

normally distributednormalGaussian
For the normal distribution, the ratio of mean absolute deviation to standard deviation is.

Mean squared error

mean square errormean-squared errorsquared error loss
This method's forecast accuracy is very closely related to the mean squared error (MSE) method which is just the average squared error of the forecasts.

Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
This is the maximum likelihood estimator of the scale parameter b of the Laplace distribution.

Laplace distribution

Laplacedouble exponentialLaplace distributed
This is the maximum likelihood estimator of the scale parameter b of the Laplace distribution.

Robust measures of scale

Qn estimatorrobust estimator of dispersionrobust measure of scale
It is a robust estimator of dispersion.