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