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# Absolute deviations

### Average absolute deviation

The average absolute deviation is a measure of statistical dispersion. It is the average of the absolute deviations from a specified measure of central tendency (usually the mean or the median). When computed as the average of the absolute deviations from the mean, it is commonly termed the mean absolute deviation. It is less than or equal to the standard deviation.

#### Algebraically speaking -

 Mean absolute deviation = Σ | -Y| n

Where :

• is the sample mean,
• -Y are the deviations from that mean (both -ve and +ve),
• | -Y| are the 'absolute' deviations from the mean, that is with all the negative deviations made positive, and
• n is the number of observations the mean was calculated from,
so Σ| -Y| is the total of the absolute deviations from the mean.

The mean absolute deviation was used as a measure of dispersion in the past, but then fell into disuse. It has the disadvantage that, unlike the standard deviation (σ), it cannot be readily 'plugged' into the normal distribution formulae. Moreover, Fisher (1920) noted that the standard deviation was a more efficient measure of dispersion (that is, it had the smaller probable error as an estimate of the population parameter), at least when there is no measurement error and when distributions are normal. ### Median absolute deviation

The median absolute deviation (denoted by MAD) is the median of the absolute deviations from the median. It is a robust measure of statistical dispersion, which is related to the standard deviation by a scale factor, the value of which depends on the distribution. For a normal distribution, the relationship is as follows:

#### Algebraically speaking - 