Mean Absolute Deviation Formula

Absolute deviation is defined as the mean average deviation of data points from a central point. The center point can be a random location, such as the mean, median, or mode. The mean is frequently used as the center point. The mean absolute deviation formula is used to calculate the mean absolute deviation (MAD), which is the average of the data points' absolute deviation (distance) from the data set's mean. A data set's average absolute deviation (AAD) is the sum of absolute departures from a central point.

It's a statistical dispersion or variability summary statistic. The central point can be a mean, mode, median or the result of any other measure of central tendency or any reference value related to the given data set in its most general form. The AAD includes the mean absolute deviation and the median absolute deviation (both abbreviated as MAD).

Mean Absolute Deviation Formula

The mean absolute deviation can be calculated using one of two formulae. One has ungrouped data, while the other contains grouped data. Let the data set $x_1, x_2,.... x_n $ be the data set, and the mean of the ungrouped data be. For the grouped data, f is the frequency of the data point xi. The following are the mean absolute deviation formula for the two types of data. 

The average absolute deviation formula for grouped data is as follows,

Mean Absolute Deviation $=\dfrac{1}{n} \sum_{i=1}^{n}\left|x_{i}-\mu\right|$

The average absolute deviation formula for ungrouped data is,

Mean Absolute Deviation $=\dfrac{\sum f\left|x-x_{i}\right|}{\sum f}$

Where

$x_i$ = Input data values

$\mu$ = Mean value for a given set of data,

$n$ = Number of data values

$f$ = Frequency of given observations

Steps to Find Mean Absolute Deviation

To finding the mean absolute deviation (MAD), follow the steps below:

• Calculate the average of the provided data set.

• Calculate the absolute value by finding the difference between each value in the data set and the mean.

• Calculate the mean absolute deviation by taking the average of all the absolute values of the difference between the data set and the mean (MAD).

Solved Examples

Ex.1. Find mean absolute deviation for the following data set: 202, 170, 252, 463, 465, 315, 223.

Solution:

Given: Data set $= {202, 170, 252, 463, 465, 315, 223}$

Mean of the data $(\mu) = \dfrac{(202 + 170 + 252 + 463 + 465 + 315 + 223 + 205)}{8} = 286.87$

Using the mean absolute deviation equation,

Mean Absolute Deviation $=\dfrac{1}{n} \sum_{i=1}^{n}\left|x_{i}-\mu\right|$

Mean Absolute Deviation

$=\dfrac{|202-286.87|+|170-286.87|+|252-286.87|+|463-286.87|+|465-286.87|+|315-286.87|+|223-286.87|+|205-286.87|}{8}$

$=\dfrac{84.87+116.87+34.87+176.13+178.13+28.13+63.87+81.87}{8}$

$=95.5925$

Ex.2.  Find the mean absolute deviation for the following data set by using the mean average deviation formula.

Solution: By using the mean average deviation formula we get,

Mean Absolute Deviation $= \dfrac{\sum f\left|x-x_{i}\right|}{\sum f}$

The average absolute deviation formula for ungrouped data is given as,

Mean absolute Deviation $= \dfrac{\sum f|x-x_i|}{\sum f}$

Now, put the calculated value in the above formula,

Mean Absolute Deviation $= \dfrac{70.4286}{28}$

Mean Absolute Deviation = 2.5153

Hence the mean absolute deviation value is 2.5153.

Conclusion

The mean absolute deviation (MAD) is a measure of variability that shows how far apart observations are from their mean. MAD interprets data in its original units, which makes it easier to understand. Larger values indicate that the data points are more scattered than the average.