The stem and leaf plot shows the height in centimeters of teddy bears and flowers 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
a) calculate the mean of each data set
b) calculate the mean, absolute deviation of each data set
c) which state is more variable? How do you know?
a) To calculate the mean of each data set, we need to first find the total height in each data set and divide by the number of data points.
For Soil A:
Total height: 59 + 55 + 51 + 50 + 56 + 57 + 52 + 58 + 53 + 61 + 60 + 63 + 67 + 68 = 810
Number of data points: 14
Mean of Soil A: 810/14 = 57.86 cm
For Soil B:
Total height: 69 + 52 + 51 + 56 + 63 + 59 + 50 + 52 + 53 + 56 + 57 + 58 = 676
Number of data points: 12
Mean of Soil B: 676/12 = 56.33 cm
b) To calculate the mean absolute deviation of each data set, we first need to find the absolute differences between each data point and its data set's mean, then find the mean of these absolute differences.
Mean Absolute Deviation of Soil A:
|59-57.86|+|55-57.86|+|51-57.86|+|50-57.86|+|56-57.86|+|57-57.86|+|52-57.86|+|58-57.86|+|53-57.86|+|61-57.86|+|60-57.86|+|63-57.86|+|67-57.86|+|68-57.86|=1.14+2.86+6.86+7.86+1.86+0.86+5.86+0.14+4.86+3.14+2.14+5.14+9.14+10.14=70.08
Mean Absolute Deviation of Soil A: 70.08/14 = 5.006
Mean Absolute Deviation of Soil B:
|69-56.33|+|52-56.33|+|51-56.33|+|56-56.33|+|63-56.33|+|59-56.33|+|50-56.33|+|52-56.33|+|53-56.33|+|56-56.33|+|57-56.33|+|58-56.33|=12.67+4.33+5.33+0.33+6.67+2.67+6.33+4.33+3.33+0.33+0.67+1.67=49.67
Mean Absolute Deviation of Soil B: 49.67/12 = 4.139
c) Soil A is more variable than Soil B because it has a larger mean absolute deviation (5.006) compared to Soil B (4.139). A larger mean absolute deviation indicates that the data points in Soil A deviate more from the mean than those in Soil B, hence making it more variable.