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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?
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.