# Mean absolute deviation calculator with examples

## Mean absolute deviation Calculator

Use this calculator to find the Mean Absolute Deviation using frequency distribution (uniform or discrete),frequencies and grouped data

Mean absolute deviation is another measure of dispersion. MAD is an absolute measure of dispersion.

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The mean of $X$ is denoted by $\overline{x}$ and is given by

 $$\begin{eqnarray*} \overline{x}& =\frac{1}{N}\sum_{i=1}^{n}f_ix_i \end{eqnarray*}$$

The mean absolute deviation about mean is given by

$$MAD =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|$$

where,

• $N$ total number of observations
• $\overline{x}$ sample mean

Mean absolute deviation is minimum when taken about median.

## Coefficient of Mean absolute deviation

The coefficient of mean absolute deviation is a relative measure of dispersion. The coefficient of quartile deviation is used to study and compare the degree of variation for two or more data set having different units of measurements.

The coefficient of mean absolute deviation about mean is given by

 \begin{aligned} \text{Coeff. of MAD about mean}&= \frac{MAD \text{ about mean}}{mean}\\ &=\frac{MAD \text{ about mean}}{\overline{X}} \end{aligned}

where,

• MAD about mean mean absolute deviation about mean
• $\overline{x}$ sample mean

## Mean Absolute Deviation Calculator for grouped data

Mean Absolute Deviation Calculator
Type of Frequency Distribution DiscreteContinuous
Enter the Classes for X (Separated by comma,)
Enter the frequencies (f) (Separated by comma,)
Results
Number of Observation (n):
Sample Mean : ($\overline{x}$)

## How to use mean absolute deviation calculator for grouped data?

Step 1 - Select type of frequency distribution (Discrete or continuous)

Step 2 - Enter the Range or classes (X) seperated by comma (,)

Step 3 - Enter the Frequencies (f) seperated by comma

Step 4 - Click on "Calculate" for mean absolute deviation

Step 5 - Gives output as number of observation (N)

Step 6 - Calculate the sample mean

Step 7 - Calculate mean absolute deviation

## Example #1 - Calculate Mean absolute deviation

A librarian keeps the records about the amount of time spent (in minutes) in a library by college students. Data is as follows:

Time spent 30 32 35 38 40
No. of students 8 12 20 10 5

Calculate mean absolute deviation about mean.

#### Solution

$x_i$ $f_i$ $f_i*x_i$ $(x_i-xb)$ $|x_i-xb|$ $f_i|x_i-xb|$
30 8 240 -4.62 4.62 36.95
32 12 384 -2.62 2.62 31.42
35 20 700 0.38 0.38 7.64
38 10 380 3.38 3.38 33.82
40 5 200 5.38 5.38 26.91
Total 55 1904 136.74

The mean absolute deviation about mean is given by

 \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}| \end{aligned}

where,

• $N$ total number of observations
• $\overline{x}$ sample mean

The mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{1904}{55}\\ &=34.62 \text{ minutes} \end{aligned}

The mean absolute deviation about mean is

 \begin{aligned} MAD & =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\\ &= \frac{136.74}{55}\\ &= 2.49 \text{ minutes} \end{aligned}

## Example #2 - Find Mean absolute deviation

The number of defective items in successive groups of fifteen items were counted as they came off a production line. The results ccan be summarized as follows:

No. of defective 0 1 2 3 4 $>$ 4
Frequency 57 57 18 5 3 0

Calculate mean absolute deviation about mean.

#### Solution

$x_i$ $f_i$ $f_i*x_i$ $(x_i-xb)$ $|x_i-xb|$ $f_i|x_i-xb|$
0 57 0 -0.86 0.86 48.86
1 57 57 0.14 0.14 8.14
2 18 36 1.14 1.14 20.57
3 5 15 2.14 2.14 10.71
4 3 12 3.14 3.14 9.43
Total 140 120 97.71

The mean absolute deviation about mean is given by

 \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}| \end{aligned}

where,

• $N$ total number of observations
• $\overline{x}$ sample mean

The mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{120}{140}\\ &=0.86 \text{ } \end{aligned}

The mean absolute deviation about mean is


\begin{aligned} MAD & =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\ &= \frac{97.71}{140}\ &= 0.7 \text{ } \end{aligned}

## Example #3 - Compute Mean absolute deviation

The following table gives the distribution of weight (in pounds) of 100 newborn babies at certain hospital in 2012.

Weight (in pounds) 3-5 5-7 7-9 9-11 11-13
No.of babies 10 30 28 18 14

Calculate mean absolute deviation about mean.

#### Solution

Class Interval $x_i$ $f_i$ $f_i*x_i$ $x_i-xb$ $|x_i-xb|$ $f_i|x_i-xb|$
3-5 4 10 40 -3.92 3.92 39.2
5-7 6 30 180 -1.92 1.92 57.6
7-9 8 28 224 0.08 0.08 2.24
9-11 10 18 180 2.08 2.08 37.44
11-13 12 14 168 4.08 4.08 57.12
Total 100 792 193.6

The mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{792}{100}\\ &=7.92 \text{ pounds} \end{aligned} 

The mean absolute deviation about mean is

 \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\\ &= \frac{193.6}{100}\\ &= 1.94 \text{ pounds} \end{aligned} 

## Example #4 - Get Mean absolute deviation for frequency distribution

The following data shows the distribution of maximum loads in short tons supported by certain cables produced by a company:

9.25-9.75 2
9.75-10.25 5
10.25-10.75 12
10.75-11.25 17
11.25-11.75 14
11.75-12.25 6
12.25-12.75 3
12.75-13.25 1

Compute mean absolute deviation about mean for the above frequency distribution.

#### Solution

Class Interval $x_i$ $f_i$ $f_i*x_i$ $x_i-xb$ $|x_i-xb|$ $f_i|x_i-xb|$
9.25-9.75 9.5 2 19 -1.59 1.59 3.18
9.75-10.25 10 5 50 -1.09 1.09 5.46
10.25-10.75 10.5 12 126 -0.59 0.59 7.1
10.75-11.25 11 17 187 -0.09 0.09 1.56
11.25-11.75 11.5 14 161 0.41 0.41 5.72
11.75-12.25 12 6 72 0.91 0.91 5.45
12.25-12.75 12.5 3 37.5 1.41 1.41 4.22
12.75-13.25 13 1 13 1.91 1.91 1.91
Total 60 665.5 34.6

The mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{665.5}{60}\\ &=11.09 \text{ tons} \end{aligned} 

The mean absolute deviation about mean is

 \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\\ &= \frac{34.6}{60}\\ &= 0.58 \text{ tons} \end{aligned} `

## Conclusion

In this tutorial, you learned about formula for mean absolute deviation (MAD) for grouped data and how to calculate mean absolute deviation for grouped data. You also learned about how to solve numerical problems based on mean absolute deviation for grouped data.