## Mean absolute deviation for ungrouped data

Let $x_i, i=1,2, \cdots , n$ be $n$ observations. The mean of $X$ is denoted by $\overline{x}$ and is given by $$\begin{eqnarray*} \overline{x}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*}$$

## Formula:

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

where,

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

## Example 1

The age (in years) of 6 randomly selected students from a class are : 22,25,24,23,24,20.

Compute mean absolute deviation about mean.

### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
22 -1 1
25 2 2
24 1 1
23 0 0
24 1 1
20 -3 3
Total 138 8

The sample mean of $X$ is

\begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{138}{6}\\ &=23 \text{ years} \end{aligned} The mean absolute deviation about mean is \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{8}{6}\\ &=\frac{8}{6}\\ &= 1.3333 \text{ years} \end{aligned}

## Example 2

The systolic blood pressure (in mmHg) of 10 randomly selected patients are :

123, 128, 136, 112, 143, 114, 104, 137, 145, 150.

Compute mean absolute deviation about mean.

### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
123 -6.2 6.2
128 -1.2 1.2
136 6.8 6.8
112 -17.2 17.2
143 13.8 13.8
114 -15.2 15.2
104 -25.2 25.2
137 7.8 7.8
145 15.8 15.8
150 20.8 20.8
Total 1292 130

The sample mean of $X$ is

\begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1292}{10}\\ &=129.2 \text{ mmHg} \end{aligned} The mean absolute deviation about mean is \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{130}{10}\\ &=\frac{130}{10}\\ &= 13 \text{ mmHg} \end{aligned}