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?
Let's first list down the data from the stem and leaf plot.
Soil A: 52, 51, 51, 51, 50, 21, 20, 0
Soil B: 59, 63, 69, 70, 72, 73, 76, 77, 78, 83
a) To calculate the mean of each data set, add the numbers and divide by the number of data points.
Soil A mean = (52+51+51+51+50+21+20+0) / 8 = 296 / 8 = 37 cm
Soil B mean = (59+63+69+70+72+73+76+77+78+83) / 10 = 720 / 10 = 72 cm
b) The mean absolute deviation (MAD) is calculated as the average of the absolute differences between each data point and the mean. First, find the absolute differences for each data point, then average them.
Soil A absolute differences: |52-37|=15, |51-37|=14, |51-37|=14, |51-37|=14, |50-37|=13, |21-37|=16, |20-37|=17, |0-37|=37
Mean absolute deviation for Soil A = (15+14+14+14+13+16+17+37) / 8 = 140 / 8 = 17.5 cm
Soil B absolute differences: |59-72|=13, |63-72|=9, |69-72|=3, |70-72|=2, |72-72|=0, |73-72|=1, |76-72|=4, |77-72|=5, |78-72|=6, |83-72|=11
Mean absolute deviation for Soil B = (13+9+3+2+0+1+4+5+6+11) / 10 = 54 / 10 = 5.4 cm
c) The more variable data set is the one with the higher mean absolute deviation. In this case, Soil A is more variable, because its MAD (17.5 cm) is higher than Soil B's MAD (5.4 cm). This means that the height of the teddy bears and flowers grown in Soil A varies more from the mean height compared to those grown in Soil B.
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To calculate the mean of each data set, follow these steps:
a) Mean Calculation:
- For Soil A: Add up all the values in the data set and divide by the total count of values.
- Adding the values in Soil A: 52 + 51 + 10 + 52 + 51 + 90 + 58 + 50 + 90 + 40 + 59 + 80 + 90 = 787
- Count of values in Soil A: 13
- Mean of Soil A: 787 / 13 = 60.54 centimeters
- For Soil B: Add up all the values in the data set and divide by the total count of values.
- Adding the values in Soil B: 94 + 93 + 73 + 79 + 70 + 60 + 23 + 62 + 67 + 68 = 709
- Count of values in Soil B: 10
- Mean of Soil B: 709 / 10 = 70.9 centimeters
Therefore, the mean of Soil A is 60.54 centimeters, and the mean of Soil B is 70.9 centimeters.
To calculate the mean absolute deviation of each data set, follow these steps:
b) Mean Absolute Deviation (MAD) Calculation:
- For Soil A: Find the absolute difference between each value and the mean, then calculate the mean of those absolute differences.
- Absolute Differences for Soil A: |52-60.54| + |51-60.54| + |10-60.54| + |52-60.54| + |51-60.54| + |90-60.54| + |58-60.54| + |50-60.54| + |90-60.54| + |40-60.54| + |59-60.54| + |80-60.54| + |90-60.54|
= 8.54 + 9.54 + 50.54 + 8.54 + 9.54 + 29.46 + 2.54 + 10.54 + 29.46 + 20.54 + 1.54 + 19.46 + 29.46
= 249.02
- Mean Absolute Deviation of Soil A: 249.02 / 13 = 19.1546 centimeters
- For Soil B: Find the absolute difference between each value and the mean, then calculate the mean of those absolute differences.
- Absolute Differences for Soil B: |94-70.9| + |93-70.9| + |73-70.9| + |79-70.9| + |70-70.9| + |60-70.9| + |23-70.9| + |62-70.9| + |67-70.9| + |68-70.9|
= 23.1 + 22.1 + 2.1 + 8.1 + 0.9 + 10.9 + 47.9 + 8.9 + 3.9 + 2.9
= 130.8
- Mean Absolute Deviation of Soil B: 130.8 / 10 = 13.08 centimeters
Therefore, the mean absolute deviation of Soil A is 19.1546 centimeters, and the mean absolute deviation of Soil B is 13.08 centimeters.
To determine which data set is more variable (has more spread), compare the mean absolute deviations calculated in step b).
Since the mean absolute deviation of Soil A (19.1546 cm) is greater than the mean absolute deviation of Soil B (13.08 cm), we can conclude that Soil A is more variable. A larger mean absolute deviation indicates more variability or spread in the data set.