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?
To calculate the mean of each data set, we need to first convert the stem-and-leaf plot into a list of all the individual heights:
Soil A: 110, 111, 115, 121, 125, 150, 152, 159
Soil B: 230, 231, 233, 236, 237, 238, 250, 263, 269
To find the mean of each data set, we add up all the heights and divide by the number of heights:
Mean of Soil A = (110 + 111 + 115 + 121 + 125 + 150 + 152 + 159) / 8 = 132.875
Mean of Soil B = (230 + 231 + 233 + 236 + 237 + 238 + 250 + 263 + 269) / 9 = 241.111
To calculate the mean absolute deviation (MAD) of each data set, we first find the absolute deviation of each height from the mean. For Soil A:
110 - 132.875 = 22.875
111 - 132.875 = 21.875
115 - 132.875 = 17.875
121 - 132.875 = 11.875
125 - 132.875 = 7.875
150 - 132.875 = 17.125
152 - 132.875 = 19.125
159 - 132.875 = 26.125
We take the absolute value of each deviation, because we only want to know how far each height is from the mean, regardless of whether it's higher or lower. Then we find the average of these absolute deviations:
MAD of Soil A = (22.875 + 21.875 + 17.875 + 11.875 + 7.875 + 17.125 + 19.125 + 26.125) / 8 = 17.375
We do the same thing for Soil B:
230 - 241.111 = 11.111
231 - 241.111 = 10.111
233 - 241.111 = 8.111
236 - 241.111 = 5.111
237 - 241.111 = 4.111
238 - 241.111 = 3.111
250 - 241.111 = 8.889
263 - 241.111 = 21.889
269 - 241.111 = 27.889
MAD of Soil B = (11.111 + 10.111 + 8.111 + 5.111 + 4.111 + 3.111 + 8.889 + 21.889 + 27.889) / 9 = 11.778
To determine which set is more variable, we compare the MADs. A higher MAD means that the data set has more variability, because the heights are farther away from the mean. In this case, Soil A has a MAD of 17.375, while Soil B has a MAD of 11.778. Therefore, Soil A is more variable.