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

Question

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 Soil A: 521
- Divide by the total number of values (10): 521/10 = 52.1

Mean of Soil B:

- Add up all the values in Soil B: 285
- Divide by the total number of values (11): 285/11 = 25.91 (rounded to two decimal places)

To calculate the MAD, we need to find the absolute deviation (the distance between each value and the mean) for each value, add them up, and divide by the number of values.

MAD of Soil A:

- Find the absolute deviation for each value:

|51.1 - 50| = 1.1
|51.1 - 50| = 1.1
|52.1 - 50| = 2.1
|52.1 - 50| = 2.1
|52.1 - 50| = 2.1
|52.1 - 50| = 2.1
|52.1 - 50| = 2.1
|52.1 - 50| = 2.1
|53.1 - 50| = 3.1
|56.1 - 50| = 6.1

- Add up all of the absolute deviations: 1.1 + 1.1 + 2.1 + 2.1 + 2.1 + 2.1 + 2.1 + 2.1 + 3.1 + 6.1 = 23.8
- Divide by the number of values (10): 23.8/10 = 2.38

MAD of Soil B:

- Find the absolute deviation for each value:

|39.1 - 25.91| = 13.19
|39.1 - 25.91| = 13.19
|63.1 - 25.91| = 37.19
|17.1 - 25.91| = 8.81
|20.1 - 25.91| = 5.81
|23.1 - 25.91| = 2.81
|26.1 - 25.91| = 0.19
|27.1 - 25.91| = 1.19
|28.1 - 25.91| = 2.19
|29.1 - 25.91| = 3.19
|38.1 - 25.91| = 12.19

- Add up all of the absolute deviations: 13.19 + 13.19 + 37.19 + 8.81 + 5.81 + 2.81 + 0.19 + 1.19 + 2.19 + 3.19 + 12.19 = 99.29
- Divide by the number of values (11): 99.29/11 = 9.03 (rounded to two decimal places)

Soil B is more variable because it has a higher MAD (9.03 cm compared to 2.38 cm for Soil A). The MAD measures the average distance between each data point and the mean, so a higher MAD indicates that the data points are further away from the mean on average. In other words, the data is more spread out in Soil B, which makes it more variable.