ote: Enter your answer and show all the steps that you use to solve this problem in the space provided.
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?.
Calculating the mean of Soil A:
- Add up all the heights in Soil A:
5 + 5 + 5 + 2 + 1 + 1 + 0 = 19
- Divide by the total number of heights (7):
19 ÷ 7 ≈ 2.71
So the mean height of Teddy Bear sunflowers grown in Soil A is approximately 2.71 cm.
Calculating the mean of Soil B:
- Add up all the heights in Soil B:
6 + 3 + 9 + 7 + 0 + 2 + 3 + 6 + 7 + 8 + 3 + 9 = 63
- Divide by the total number of heights (12):
63 ÷ 12 = 5.25
So the mean height of Teddy Bear sunflowers grown in Soil B is 5.25 cm.
Calculating the MAD of Soil A:
- Find the absolute deviations of each height from the mean:
|5 - 2.71| = 2.29
|5 - 2.71| = 2.29
|5 - 2.71| = 2.29
|2 - 2.71| = 0.71
|1 - 2.71| = 1.71
|1 - 2.71| = 1.71
|0 - 2.71| = 2.71
- Add up all the absolute deviations:
2.29 + 2.29 + 2.29 + 0.71 + 1.71 + 1.71 + 2.71 = 14.41
- Divide by the total number of heights (7):
14.41 ÷ 7 ≈ 2.06
So the MAD of Soil A is approximately 2.06 cm.
Calculating the MAD of Soil B:
- Find the absolute deviations of each height from the mean:
|6 - 5.25| = 0.75
|3 - 5.25| = 2.25
|9 - 5.25| = 3.75
|7 - 5.25| = 1.75
|0 - 5.25| = 5.25
|2 - 5.25| = 3.25
|3 - 5.25| = 2.25
|6 - 5.25| = 0.75
|7 - 5.25| = 1.75
|8 - 5.25| = 2.75
|3 - 5.25| = 2.25
|9 - 5.25| = 3.75
- Add up all the absolute deviations:
0.75 + 2.25 + 3.75 + 1.75 + 5.25 + 3.25 + 2.25 + 0.75 + 1.75 + 2.75 + 2.25 + 3.75 = 29.75
- Divide by the total number of heights (12):
29.75 ÷ 12 ≈ 2.48
So the MAD of Soil B is approximately 2.48 cm.
To determine which set is more variable, we can compare their MAD values. The set with the larger MAD is considered more variable. Therefore, Soil B is more variable than Soil A, since its MAD is larger (2.48 cm vs. 2.06 cm).
is there an eraser way to do this?
There are different strategies for calculating means and MADs, but they all involve some computation. However, you could use a calculator or spreadsheet software to simplify the calculations and reduce the possibility of errors. For example, you could enter the data of Soil A and B into separate columns or rows, use the built-in functions to calculate the mean and MAD, and get the results in a few clicks. Some calculators also have a STAT mode that allows you to enter data and perform statistics on them without having to do the calculations by hand.
To calculate the mean of each data set:
For Soil A:
Mean = (5 + 5 + 5 + 2 + 1 + 1 + 8 + 9 + 0) / 9
= 36 / 9
= 4
For Soil B:
Mean = (9 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 5 + 5 + 5 + 8 + 9 + 9 + 9) / 15
= 99 / 15
= 6.6 (rounded to one decimal place)
To calculate the mean absolute deviation (MAD) of each data set:
For Soil A:
Step 1: Find the difference of each data point and the mean
|5 - 4| = 1
|5 - 4| = 1
|5 - 4| = 1
|2 - 4| = 2
|1 - 4| = 3
|1 - 4| = 3
|8 - 4| = 4
|9 - 4| = 5
|0 - 4| = 4
Step 2: Find the absolute value of each difference
1, 1, 1, 2, 3, 3, 4, 5, 4
Step 3: Find the mean of the absolute differences
MAD = (1 + 1 + 1 + 2 + 3 + 3 + 4 + 5 + 4) / 9
= 24 / 9
≈ 2.67 (rounded to two decimal places)
For Soil B:
Step 1: Find the difference of each data point and the mean
|9 - 6.6| = 2.4
|6 - 6.6| = 0.6
|6 - 6.6| = 0.6
|7 - 6.6| = 0.4
|7 - 6.6| = 0.4
|7 - 6.6| = 0.4
|7 - 6.6| = 0.4
|7 - 6.6| = 0.4
|5 - 6.6| = 1.6
|5 - 6.6| = 1.6
|5 - 6.6| = 1.6
|8 - 6.6| = 1.4
|9 - 6.6| = 2.4
|9 - 6.6| = 2.4
|9 - 6.6| = 2.4
Step 2: Find the absolute value of each difference
2.4, 0.6, 0.6, 0.4, 0.4, 0.4, 0.4, 0.4, 1.6, 1.6, 1.6, 1.4, 2.4, 2.4, 2.4
Step 3: Find the mean of the absolute differences
MAD = (2.4 + 0.6 + 0.6 + 0.4 + 0.4 + 0.4 + 0.4 + 0.4 + 1.6 + 1.6 + 1.6 + 1.4 + 2.4 + 2.4 + 2.4) / 15
= 18.4 / 15
≈ 1.23 (rounded to two decimal places)
To determine which set is more variable, we compare the mean absolute deviations (MAD). A higher MAD indicates more variability.
In this case, Soil B has a higher MAD (1.23) compared to Soil A (2.67), indicating that Soil B has more variability in the heights of Teddy Bear sunflowers grown in that soil.