﻿

# Median absolute deviation

In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.

For a univariate data set X1X2, ..., Xn, the MAD is defined as the median of the absolute deviations from the data's median: $\operatorname{MAD} = \operatorname{median}_{i}\left(\ \left| X_{i} - \operatorname{median}_{j} (X_{j}) \right|\ \right), \,$

that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values.

## Example

Consider the data (1, 1, 2, 2, 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1, 1, 2, 4, 7)). So the median absolute deviation for this data is 1.

## Uses

The median absolute deviation is a measure of statistical dispersion. It is a more robust estimator of scale than the sample variance or standard deviation. It thus behaves better with distributions without a mean or variance, such as the Cauchy distribution.

For instance, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so on average, large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the magnitude of the distances of a small number of outliers is irrelevant.

## Relation to standard deviation

In order to use the MAD as a consistent estimator for the estimation of the standard deviation σ, one takes $\hat{\sigma}=K\cdot \operatorname{MAD}, \,$

where K is a constant scale factor, which depends on the distribution.

For normally distributed data K is taken to be 1/Φ−1(3/4) $\approx$ 1.4826, where Φ−1 is the inverse of the cumulative distribution function for the standard normal distribution, i.e., the quantile function. This is because the MAD is given by: $\frac 12 =P(|X-\mu|\le \operatorname{MAD})=P\left(\left|\frac{X-\mu}{\sigma}\right|\le \frac {\operatorname{MAD}}\sigma\right)=P\left(|Z|\le \frac {\operatorname{MAD}}\sigma\right).$

Therefore we must have that Φ(MAD/σ) − Φ(−MAD/σ) = 1/2. Since Φ(−MAD/σ) = 1 − Φ(MAD/σ) we have that MAD/σ = Φ−1(3/4) from which we obtain the scale factor K = 1/Φ−1(3/4).

Hence $\sigma \approx 1.4826\ \operatorname{MAD}. \,$

In other words, the expectation of 1.4826 times the MAD for large samples of normally distributed Xi is approximately equal to the population standard deviation. Other distributions behave differently: for example for large samples from a uniform continuous distribution, this factor is about 1.1547 (the square root of 4/3).

The population MAD is defined analogously to the sample MAD, but is based on the complete distribution rather than on a sample. For a symmetric distribution with zero mean, the population MAD is the 75th percentile of the distribution.

Unlike the variance, which may be infinite or undefined, the population MAD is always a finite number. For example, the standard Cauchy distribution has undefined variance, but its MAD is 1.

The earliest known mention of the concept of the MAD occurred in 1816, in a paper by Carl Friedrich Gauss on the determination of the accuracy of numerical observations.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Absolute deviation — 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 point from which the deviation is measured is a measure of central tendency, most often the median… …   Wikipedia

• Median — This article is about the statistical concept. For other uses, see Median (disambiguation). In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability… …   Wikipedia

• Deviation (statistics) — In mathematics and statistics, deviation is a measure of difference for interval and ratio variables between the observed value and the mean. The sign of deviation (positive or negative), reports the direction of that difference (it is larger… …   Wikipedia

• Mittlere absolute Abweichung — Unter Streuung fasst man in der deskriptiven Statistik verschiedene Maßzahlen zusammen zur Beschreibung der Streubreite von Werten einer Häufigkeitsverteilung um ihren Mittelwert. Die verschiedenen Berechnungsmethoden unterscheiden sich… …   Deutsch Wikipedia

• Standard deviation — In probability and statistics, the standard deviation is a measure of the dispersion of a collection of values. It can apply to a probability distribution, a random variable, a population or a data set. The standard deviation is usually denoted… …   Wikipedia

• mean deviation — noun the arithmetic mean of the absolute values of deviations from the mean of a distribution • Syn: ↑mean deviation from the mean • Topics: ↑statistics • Hypernyms: ↑deviation * * * noun …   Useful english dictionary

• mean deviation — Statistics. a measure of dispersion, computed by taking the arithmetic mean of the absolute values of the deviations of the functional values from some central value, usually the mean or median. Also called average deviation. [1890 95] * * * …   Universalium

• mean deviation — mean′ devia′tion n. sta a measure of dispersion, computed by taking the arithmetic mean of the absolute values of the deviations of the functional values from some central value, usu. the mean or median Also called average deviation Etymology:… …   From formal English to slang

• mean deviation — noun Date: 1858 the mean of the absolute values of the numerical differences between the numbers of a set (as statistical data) and their mean or median …   New Collegiate Dictionary

• Robust statistics — provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions. Contents 1 Introduction 2 Examples of robust and non robust… …   Wikipedia