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 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.
To calculate the mean of each data set, we need to sum up all the values and then divide by the total number of values.
For Soil A:
Sum = 5 + 5 + 5 + 2 + 1 + 1 + 1 + 6 + 5 + 2 + 1 + 0 = 34
Total number of values = 12
Mean of Soil A = Sum / Total number of values = 34 / 12 ≈ 2.83
For Soil B:
Sum = 9 + 9 + 8 + 7 + 7 + 6 + 6 + 6 + 3 + 6 + 0 + 2 + 3 + 6 + 7 + 8 + 3 + 9 = 106
Total number of values = 18
Mean of Soil B = Sum / Total number of values = 106 / 18 ≈ 5.89
Now, let's calculate the mean absolute deviation (MAD) of each data set. To find the MAD, we need to calculate the absolute difference between each value and the mean, sum up these differences, and then divide by the total number of values.
For Soil A:
Absolute differences from mean: |5-2.83|, |5-2.83|, |5-2.83|, |2-2.83|, |1-2.83|, |1-2.83|, |1-2.83|, |6-2.83|, |5-2.83|, |2-2.83|, |1-2.83|, |0-2.83|
Sum of absolute differences = 2.83 + 2.83 + 2.83 + 0.83 + 1.83 + 1.83 + 1.83 + 3.17 + 2.83 + 0.83 + 1.83 + 2.83 ≈ 25.68
MAD of Soil A = Sum of absolute differences / Total number of values = 25.68 / 12 ≈ 2.14
For Soil B:
Absolute differences from mean: |9-5.89|, |9-5.89|, |8-5.89|, |7-5.89|, |7-5.89|, |6-5.89|, |6-5.89|, |6-5.89|, |3-5.89|, |6-5.89|, |0-5.89|, |2-5.89|, |3-5.89|, |6-5.89|, |7-5.89|, |8-5.89|, |3-5.89|, |9-5.89|
Sum of absolute differences = 3.11 + 3.11 + 2.11 + 1.11 + 1.11 + 0.11 + 0.11 + 0.11 + 2.89 + 0.11 + 5.89 + 3.89 + 2.89 + 0.11 + 1.11 + 2.11 + 2.89 + 3.11 ≈ 36.71
MAD of Soil B = Sum of absolute differences / Total number of values = 36.71 / 18 ≈ 2.04
To determine which set is more variable, we compare the MAD values. The higher the MAD value, the more variable the data set. In this case, Soil A has an MAD of approximately 2.14, while Soil B has an MAD of approximately 2.04. Therefore, Soil A is slightly more variable than Soil B.
To calculate the mean of each dataset, you need to sum up all the values and divide by the total number of values. Let's start with Soil A:
Soil A:
Sum of heights: 5 + 5 + 2 + 1 + 1 + 5 + 2 + 1 + 0 = 22
Total values: 9 (counting the individual values)
Mean = Sum of heights / Total values = 22 / 9 ≈ 2.44
Now let's calculate the mean for Soil B:
Soil B:
Sum of heights: 9 + 6 + 5 + 3 + 9 + 7 + 0 + 2 + 3 + 6 + 7 + 8 + 3 = 68
Total values: 13
Mean = Sum of heights / Total values = 68 / 13 ≈ 5.23
Next, let's calculate the mean absolute deviation (MAD) for each data set. To calculate the MAD, you need to find the absolute difference between each value and the mean, then calculate the average of those differences.
For Soil A, we can calculate the MAD as follows:
|5 - 2.44| + |5 - 2.44| + |2 - 2.44| + |1 - 2.44| + |1 - 2.44| + |5 - 2.44| + |2 - 2.44| + |1 - 2.44| + |0 - 2.44| ≈ 8.44
MAD for Soil A ≈ 8.44 / 9 ≈ 0.94
For Soil B, the calculation would be:
|9 - 5.23| + |6 - 5.23| + |5 - 5.23| + |3 - 5.23| + |9 - 5.23| + |7 - 5.23| + |0 - 5.23| + |2 - 5.23| + |3 - 5.23| + |6 - 5.23| + |7 - 5.23| + |8 - 5.23| + |3 - 5.23| ≈ 30.22
MAD for Soil B ≈ 30.22 / 13 ≈ 2.32
To determine which data set is more variable, we compare the MAD values. The data set with the higher MAD is considered more variable. In this case, Soil B has a higher MAD (2.32) compared to Soil A (0.94). Therefore, Soil B is more variable.
You can identify which set is more variable by comparing the MAD values. The higher the MAD, the greater the variability in the data set.