## Mean absolute deviation for grouped data

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

## Formula

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

## Example 1

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

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}