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