Question

stefano calculated the mean absolute deviation for the data set 32, 4, 12, 40, 20, and 24. his work is shown below. step 1: find the mean. mean = (32 + 4 + 12 + 40 + 20 + 24) / 6 = 22 step 2: find each absolute deviation. 10, 18, 10, 18, 2, 2 step 3: find the mean absolute deviation. mad = (10 + 18 + 10 + 18 + 2 + 2) / 4 = 15 what is stefanos error?

Explanation:

Step1: Recall MAD formula

Mean Absolute Deviation (MAD) is calculated as the mean of the absolute deviations from the mean. So, first, find the absolute difference between each data point and the mean, then take the mean of those absolute differences.

Step2: Analyze Stefano's Step 2

The data set is \( 32, 4, 12, 40, 20, 24 \) (6 data points), mean is \( 22 \) (from Step 1: \( \frac{32 + 4+12 + 40+20 + 24}{6}=\frac{132}{6} = 22 \)).

For each data point, absolute deviation from mean:

• \( |32 - 22|=10 \)
• \( |4 - 22| = 18 \)
• \( |12 - 22|=10 \)
• \( |40 - 22| = 18 \)
• \( |20 - 22|=2 \)
• \( |24 - 22|=2 \)

Step3: Analyze Stefano's Step 3

Stefano used \( 4 \) as the number of data points in the denominator, but there are \( 6 \) data points. Also, when calculating MAD, we take the mean of these absolute deviations, so the denominator should be the number of data points (6), not 4. Also, his absolute deviations are correct, but the count of data points is wrong.

Answer:

Stefano used 4 (incorrect number of data points, should be 6) in the denominator when calculating MAD. Also, he misapplied the number of data points (there are 6 data points, not 4) while computing the mean of absolute deviations.