In statistics, mean absolute deviation is a one of the ways to describe the variability of quantitative data. Mean absolute deviation is used to measure the average amount by which all values in a data set vary from their mean. The MAD can be used with any measure of central tendency, for example: we can calculate the mean absolute deviation of scores from the mean, from the mode, or even from the median. Mean absolute deviation is also called as average absolute deviation.

## Definition

The mean absolute deviation is the average of the absolute values of the deviations around the mean for a set of numbers. For a data set $y_1, y_2, y_3,......., y_n$, the mean absolute deviation is defined as the average of the absolute deviations from the mean of data. Mean absolute deviation can be abbreviated as MAD. In other words we can say that, the MAD is the mean of absolute values of the data set.

## Formula

The MAD is the arithmetic average of the absolute value of the difference between the data values and the arithmetic average of the data values.
Formula for the ungrouped data:

For a population:

$MAD$ = $\frac{\sum|y-\mu|}{N}$

For a sample:

$MAD$ = $\frac{\sum|y-\bar{y}|}{n}$
Where, $MAD$ = Mean absolute deviation
$\mu$ = $\bar{y}$ = Mean of the values
$n$ = Total number of values
and | | indicates the absolute value, i.e. the signs of deviations from the mean disregarded.
For a grouped frequency distribution

$MAD$ = $\frac{1}{\sum f}$ $\sum f|y - \bar{y}|$

Where $f$ is the frequency of the data values.

## Equation

As we know, the definition of mean is the point in a distribution of values, the sum of the deviations from which is equal to zero because of some negative and positive deviations. But while dealing with absolute deviation we disregard the sign of each deviation. If we calculate the mean of all the absolute deviations, we have an indicator of the dispersion of the distribution.
The mean absolute deviation equation for grouped data:

$MAD$ = $\frac{\sum f|x-\bar{x}|}{n}$

Where, $x$ = value of category
$\bar{x}$ = mean of values
$f$ = frequency in a category

## How to Find the Mean Absolute Deviation?

The mean absolute deviation (MAD) of a data set is the average distance between each value and the mean.
Follow the following steps to find the mean absolute deviation:
1) First find the mean for given data set.
2) Subtract mean from each data value.
3) Find the absolute deviation.
4) Divide the absolute deviation by total number of values.
While calculating the MAD, the median is also used to compute because the sum of the absolute values from the median is smaller than that from any other value.

## Mean Absolute Percentage Error

Mean absolute error is the average of the absolute values of errors of a set of estimates. In the context of population estimates, the arithmetic average of absolute percent differences between a set of estimate and corresponding census numbers. Mean absolute percent error is abbreviated as MAPE.
The mean absolute percent error is the average of the absolute percentage differences between the actual data and forecast data.Mean Absolute Percentage Error Formula:

$MAPE$ = $\frac{1}{n}$$\sum |\frac{A_v-F_v}{A_v}|$ $\times$ 100%

where $F_v$ is the forecast value and $A_v$ is the actual value.

## Mean Absolute Deviation vs Standard Deviation

Standard deviation is derived from the mean absolute deviation. The basic difference between both is that in the latter each deviation from the mean is squared. As squaring a negative number eliminates the negative sign, result of multiplying two negative numbers is again a positive number. In the case of standard deviation, the square root is taken of the result to compensate for the earlier squaring. This method gives the standard deviation a number of advantage over the MAD. Which really helpful to compare standard deviation across different types of distributions, whereas mean absolute deviation is not comparable.
We can calculate the values of mean absolute deviation and standard deviation by using below formulae (In the case of ungrouped data):

Mean deviation = $\frac{\sum (x-\bar{x})^2}{n}$

Standard deviation = $\sqrt{\frac{\sum (x-\bar{x})^2}{n}}$

## Example

Below is an example to illustrate the mean absolute deviation:

Example:
Find the mean absolute deviation for the data set: {10, 12, 13, 4, 15, 16, 5, 34, 21, 20}

Solution:

Given set of data is : {10, 12, 13, 4, 15, 16, 5, 34, 21, 20}

Total number of values (N) = 10

Step 1:
Mean of given data set

$\mu$ = $\frac{10+ 12+ 13+ 4+ 15+ 16+ 5+ 34+ 21+20}{10}$

= $\frac{150}{10}$ = 10

Step 2:

 y y - $\mu$ |y - $\mu$| 10 0 0 12 2 2 13 3 3 4 -6 6 15 5 5 16 6 6 5 -5 5 34 24 24 21 11 11 20 10 10 $\sum$ = 150 $\sum$ = 50 $\sum$ = 72

Formula for $MAD$ = $\frac{1}{N}$ $\sum|y - \mu|$

= $\frac{1}{10}$ $\times$ 72

= 7.2

=> $MAD$ = 7.2