Question

A beekeeper reported the number of bees that escaped from each of the 7 hives she maintained. She reported:
8 bees1 bee8 bees1 bee6 bees2 bees9 bees
What was the mean absolute deviation of the numbers of escaped bees?

If the answer is a decimal, round it to the nearest tenth.

mean absolute deviation (MAD):
bees

Answer 1

To find the mean absolute deviation (MAD) of the numbers of escaped bees, we will first gather the data, calculate the mean of the dataset, and then calculate the absolute deviations from the mean. Finally, we will find the average of those absolute deviations.

1. **Collect the data:**
The number of escaped bees reported is:
- 8, 1, 8, 1, 6, 2, 9

2. **Calculate the mean:**
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{8 + 1 + 8 + 1 + 6 + 2 + 9}{7}
\]
\[
\text{Sum} = 8 + 1 + 8 + 1 + 6 + 2 + 9 = 35
\]
\[
\text{Mean} = \frac{35}{7} = 5
\]

3. **Calculate the absolute deviations from the mean:**
- For 8: \( |8 - 5| = 3 \)
- For 1: \( |1 - 5| = 4 \)
- For 8: \( |8 - 5| = 3 \)
- For 1: \( |1 - 5| = 4 \)
- For 6: \( |6 - 5| = 1 \)
- For 2: \( |2 - 5| = 3 \)
- For 9: \( |9 - 5| = 4 \)

The absolute deviations are:
- 3, 4, 3, 4, 1, 3, 4

4. **Calculate the mean of the absolute deviations:**
\[
\text{Mean absolute deviation} = \frac{\text{Sum of absolute deviations}}{\text{Number of values}} = \frac{3 + 4 + 3 + 4 + 1 + 3 + 4}{7}
\]
\[
\text{Sum of absolute deviations} = 3 + 4 + 3 + 4 + 1 + 3 + 4 = 22
\]
\[
\text{Mean absolute deviation} = \frac{22}{7} \approx 3.142857143
\]

5. **Round to the nearest tenth:**
The mean absolute deviation rounded to the nearest tenth is approximately **3.1**.

Thus, the mean absolute deviation of the numbers of escaped bees is:
\[
\text{MAD} = 3.1 \text{ bees}
\]