Question

Find and interpret the mean absolute deviation of the data.
5, 8, 8, 10, 13, 14, 16, 22

Answer 1

To calculate the Mean Absolute Deviation (MAD) of the given data set \( 5, 8, 8, 10, 13, 14, 16, 22 \), we can follow these steps:

1. **Calculate the Mean of the Data:**

First, we find the mean (average) of the data set.

\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]

\[
\text{Mean} = \frac{5 + 8 + 8 + 10 + 13 + 14 + 16 + 22}{8} = \frac{ 96 }{8} = 12
\]

2. **Calculate the Absolute Deviations:**

Next, we calculate the absolute deviation of each data point from the mean:

- \(|5 - 12| = 7\)
- \(|8 - 12| = 4\)
- \(|8 - 12| = 4\)
- \(|10 - 12| = 2\)
- \(|13 - 12| = 1\)
- \(|14 - 12| = 2\)
- \(|16 - 12| = 4\)
- \(|22 - 12| = 10\)

The absolute deviations are:

\[
7, 4, 4, 2, 1, 2, 4, 10
\]

3. **Calculate the Mean of the Absolute Deviations:**

Now we find the mean of these absolute deviations.

\[
\text{Mean Absolute Deviation (MAD)} = \frac{\text{Sum of absolute deviations}}{\text{Number of data points}}
\]

\[
\text{Mean Absolute Deviation (MAD)} = \frac{7 + 4 + 4 + 2 + 1 + 2 + 4 + 10}{8} = \frac{34}{8} = 4.25
\]

### Interpretation:

The Mean Absolute Deviation (MAD) of the data set is \(4.25\). This value represents the average distance between each data point and the mean of the data set. In practical terms, it gives an idea of how spread out the data points are around the mean. A MAD of \(4.25\) indicates that, on average, the data points differ from the mean by \(4.25\) units.