Exercise 1 Page 712

The mean absolute deviation (MAD) is the average of the absolute values of the differences between the mean and each value in the data set. We will start by calculating the mean of the given set of numbers.

Mean

The mean of a data set or $\overline{x},$ is calculated by finding the sum of all of the values in the set and then dividing by the number of values in the set. In this case, there are $10$ values. Let's now look at the table with given values.

Amount of Caffeine in Tea (milligrams)
$9$	$46$	$18$	$35$	$30$
$12$	$56$	$24$	$38$	$32$

Now, we will find the sum of the given values. The mean of the set is $30.$

Mean Absolute Deviation

As previously stated, the MAD of a set of data is the average of the absolute values of the differences between the mean and each value in the data set. In this formula, $x_1,\dots ,x_n$ are the values in the set of data, $\overline{x}$ is the mean, and $n$ is the number of values. We already know that $\overline{x}=30$ and $n=10.$ Let's use a table to find the sum of the absolute values of the differences.

$x_n$	$\overline{x}-x_n$	$\left|\overline{x}-x_n\right|$
$9$	$30-9=21$	$|21|=21$
$46$	$30-46=\text{-}16$	$|\text{-}16|=16$
$18$	$30-18=12$	$|12|=12$
$35$	$30-35=\text{-}5$	$|\text{-}5|=5$
$30$	$30-30=0$	$|0|=0$
$12$	$30-12=18$	$|18|=18$
$56$	$30-56=\text{-}26$	$|\text{-}26|=26$
$24$	$30-24=6$	$|6|=6$
$38$	$30-38=\text{-}8$	$|\text{-}8|=8$
$32$	$30-32=\text{-}2$	$|\text{-}2|=2$
Sum of Values		$114$

Finally, we need to divide by $10.$ A MAD of $11.4$ indicates that the data, on average, are $11.4$ units away from the mean. In the context of the problem, this means that the amount of caffeine in milligrams in certain types of tea is, on average, $11.4$ units away from the mean of $30$ milligrams. There are no outliers that influence this value.