## 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
```
$$
\begin{aligned}
MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\
&=\frac{130}{10}\\
&=\frac{130}{10}\\
&= 13 \text{ mmHg}
\end{aligned}
$$
```