Question

Dora calculated the mean absolute deviation for the data set 35, 16, 23, 42, and 19. Her work is shown below.

Step 1: Find the mean.

mean = StartFraction 35 + 16 + 23 + 42 + 19 Over 5 EndFraction = 27

Step 2: Find each absolute deviation.

8, 11, 4, 15, 8

Step 3: Find the mean absolute deviation.

M A D = StartFraction 8 + 11 + 4 + 15 Over 5 EndFraction = 9.5

What is Dora’s error?
Dora should have divided by 4 when finding the mean.
Dora found the absolute deviation of 35 incorrectly.
Dora used only four numbers in finding the mean.
Dora used only four numbers in finding the mean absolute deviation.

Answer 1

Dora's error lies in the calculation of the mean absolute deviation. Let's review her steps:

1. **Mean Calculation**:
\[
\text{mean} = \frac{35 + 16 + 23 + 42 + 19}{5} = \frac{135}{5} = 27
\]
This is correct.

2. **Finding Absolute Deviations**:
- For 35: \( |35 - 27| = 8 \)
- For 16: \( |16 - 27| = 11 \)
- For 23: \( |23 - 27| = 4 \)
- For 42: \( |42 - 27| = 15 \)
- For 19: \( |19 - 27| = 8 \)

**Absolute deviations** are \( 8, 11, 4, 15, 8 \). This also is correct.

3. **Finding the Mean Absolute Deviation (MAD)**:
To find the MAD, she must take the average of all absolute deviations. The sum of the absolute deviations is:
\[
8 + 11 + 4 + 15 + 8 = 46
\]
Since there are **five** deviations, Dora should divide by **5**:
\[
\text{MAD} = \frac{46}{5} = 9.2
\]

Dora incorrectly calculated the mean absolute deviation by dividing the sum of the absolute deviations by **4** instead of **5**. Therefore, Dora’s error is:

**Dora used only four numbers in finding the mean absolute deviation.**