How to Find Mean Absolute Deviation?

To understand the mean of absolute deviation, let us split both words and try to understand their meaning. “Mean” refers to the average of observations, and deviation means departure or variation from a preset standard. Now, we can define mean deviation as the mean distance of each observation from the mean of the data.

A step-by-step guide to finding mean absolute deviation

The mean absolute deviation is the average deviation of data points from a central point. The center point can be the mean, median, mode, or any random point. The average is often considered the center point.

The formula of mean absolute deviation

There are two formulas for finding the mean absolute deviation. One is for ungrouped data, and the other is for grouped data.

Let \(x_1\), \(x_2\), …. \(x_n\) be the data set and let \(μ\) be its average of the ungrouped data. And, \(f\) is the frequency of the data point \(x_i\)_, for the grouped data.  The mean absolute deviation formulas for the two types of data are as follows:

Mean absolute deviation for grouped data \(\color{blue}{=\frac{1}{n}\sum _{i=1}^n\:\left|x_i-μ\right|}\)

Mean absolute deviation for ungrouped data \(\color{blue}{=\frac{\sum f\left|x-x_i\right|\:}{\sum f\:}}\)

Finding Mean Absolute Deviation – Example 1:

Find mean absolute deviation for the following data set: \(300, 142, 356, 560, 459, 217, 220\)

Solution:

First, find mean of the data \((μ)=\frac{sum \ of \ the \ data}{total \ number \ of \ data \ entires}\)

\(=\frac{300+142+ 356+ 560+ 459+ 217+ 220}{7}\)

\(=\frac{2254}{7}=322\)

Now, using mean deviation formula: \(\color{blue}{=\frac{1}{n}\sum _{i=1}^n\:\left|x_i-μ\right|}\)

\(=\frac{|300-322|+|142-322|+|356-322|+|560-322|+|459-322|+|217-322|+|220-322|}{7}\)

\(=\frac{|-22|+|-180|+|34|+|238|+|137|+|-105|+|-102|}{7}\)

\(=\frac{22+180+34+238+137+105+102}{7}\)

\(=\frac{818}{7}\)

\(=116.86\)

Exercises for Finding Mean Absolute Deviation

Find the mean absolute deviation of the data.

1. \(\color{blue}{86, 93, 88, 85, 89, 95, 85, 83}\)
2. \(\color{blue}{29, 24, 15, 29, 41, 35, 65, 49, 46}\)

1. \(\color{blue}{3.25}\)
2. \(\color{blue}{11.78}\)