To find the mean absolute deviation (MAD) of the given data set, follow these steps:
Step 1: Calculate the Mean of the Data Set
The data set comprises the number of boxes collected by six volunteers: [tex]\( 32, 35, 20, 16, 28, 13 \)[/tex].
First, sum the values:
[tex]\[ 32 + 35 + 20 + 16 + 28 + 13 = 144 \][/tex]
Next, divide the sum by the number of data points (which is 6):
[tex]\[ \text{Mean} = \frac{144}{6} = 24 \][/tex]
Therefore, the mean number of boxes collected is [tex]\( 24 \)[/tex].
Step 2: Calculate the Absolute Deviations from the Mean
For each data point, calculate the absolute deviation by subtracting the mean and taking the absolute value:
[tex]\[
\begin{align*}
|32 - 24| & = 8 \\
|35 - 24| & = 11 \\
|20 - 24| & = 4 \\
|16 - 24| & = 8 \\
|28 - 24| & = 4 \\
|13 - 24| & = 11 \\
\end{align*}
\][/tex]
This gives the absolute deviations: [tex]\( 8, 11, 4, 8, 4, 11 \)[/tex].
Step 3: Calculate the Mean of the Absolute Deviations
Sum the absolute deviations:
[tex]\[ 8 + 11 + 4 + 8 + 4 + 11 = 46 \][/tex]
Next, divide this sum by the number of data points (which is 6):
[tex]\[ \text{Mean Absolute Deviation} = \frac{46}{6} \approx 7.666666666666667 \][/tex]
Step 4: Round to the Nearest Tenth
Finally, round the mean absolute deviation to the nearest tenth:
[tex]\[ 7.666666666666667 \approx 7.7 \][/tex]
Therefore, the mean absolute deviation (rounded to the nearest tenth) is [tex]\( 7.7 \)[/tex].
