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Question

The stem-and-leaf plot shows the heights in centimeters of Teddy Bear sunflowers grown in two different types of soil.

Soil A Soil B
5 9
5 2 1 1 6 3 9
5 1 0 7 0 2 3 6 7 8
2 1 8 3
0 9
Key: 9|6 means 69 Key: 5|8 means 58
Calculate the mean of each data set.
Calculate the mean absolute deviation (MAD) of each data set.
Which set is more variable? How do you know?

Answer 1

Mean of Soil A:

- Add up all the values in the Soil A stem-and-leaf plot: 50 + 51 + 52 + 55 + 56 + 56 + 57 + 58 + 59 + 91 = 485
- Count the total number of values in the Soil A plot: 11
- Divide the sum by the count to find the mean: 485 ÷ 11 = 44.09

Mean of Soil B:

- Add up all the values in the Soil B stem-and-leaf plot: 60 + 63 + 67 + 68 + 69 + 69 + 70 + 72 + 73 + 76 + 78 = 775
- Count the total number of values in the Soil B plot: 11
- Divide the sum by the count to find the mean: 775 ÷ 11 = 70.45

MAD of Soil A:

- Find the absolute deviation of each value from the mean. For example, the absolute deviation of 50 from the mean (44.09) is |50 - 44.09| = 5.91.
- List all of the absolute deviations: 5.91, 6.09, 7.09, 10.91, 11.91, 11.91, 12.91, 13.91, 14.91, 46.91, and 46.91.
- Find the mean of the absolute deviations: (5.91 + 6.09 + 7.09 + 10.91 + 11.91 + 11.91 + 12.91 + 13.91 + 14.91 + 46.91 + 46.91) ÷ 11 = 17.27

MAD of Soil B:

- Find the absolute deviation of each value from the mean. For example, the absolute deviation of 60 from the mean (70.45) is |60 - 70.45| = 10.45.
- List all of the absolute deviations: 10.45, 6.45, 2.45, 1.45, 0.45, 0.45, 0.45, 1.45, 2.45, 5.45, and 7.45.
- Find the mean of the absolute deviations: (10.45 + 6.45 + 2.45 + 1.45 + 0.45 + 0.45 + 0.45 + 1.45 + 2.45 + 5.45 + 7.45) ÷ 11 = 3.73

Soil A has a higher MAD (17.27) than Soil B (3.73), indicating that Soil A is more variable. The MAD is a measure of how spread out the data is, and a higher MAD means that there is more variability. We also see this in the stem-and-leaf plots, where Soil A has values ranging from 21 to 91, while Soil B has values ranging from 63 to 78, indicating that Soil A has a wider range of heights.