Note: 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
a. Calculate the mean of each data set.
b. Calculate the mean absolute deviation (MAD) of each data set.
c. Which set is more variable? How do you know?
a. Mean of Soil A:
- We have 9 data points.
- To find the mean, we need to add up all the heights and divide by the number of data points:
(5+21+11+51+50+81+83+90+91) / 9 = 53.11
- The mean height of Teddy Bear sunflowers grown in Soil A is 53.11 cm.
Mean of Soil B:
- We have 16 data points.
- To find the mean, we need to add up all the heights and divide by the number of data points:
(36+37+38+39+62+63+69+70+72+73+76+78+79+89+90+93) / 16 = 67.75
- The mean height of Teddy Bear sunflowers grown in Soil B is 67.75 cm.
b. Mean absolute deviation (MAD) of Soil A:
- MAD is the average distance of each data point from the mean.
- We can calculate MAD by taking the absolute value of the difference between each data point and the mean, adding them up, and dividing by the number of data points:
|5-53.11| + |21-53.11| + |11-53.11| + |51-53.11| + |50-53.11| + |81-53.11| + |83-53.11| + |90-53.11| + |91-53.11|
--------------------------------------------------------------
9
- Simplifying this expression:
48.11 + 32.11 + 42.11 + 2.11 + 3.11 + 27.89 + 29.89 + 36.89 + 37.89
-----------------------------------------------------------------------
9
= 30.62
- The MAD of Soil A is 30.62 cm.
Mean absolute deviation (MAD) of Soil B:
- We can use the same formula to calculate MAD for Soil B:
|36-67.75| + |37-67.75| + |38-67.75| + |39-67.75| + |62-67.75| + |63-67.75| + |69-67.75| + |70-67.75|
+ |72-67.75| + |73-67.75| + |76-67.75| + |78-67.75| + |79-67.75| + |89-67.75| + |90-67.75| + |93-67.75|
------------------------------------------------------------------------------------------------------------
16
- Simplifying this expression:
31.75 + 30.75 + 29.75 + 28.75 + 5.75 + 4.75 + 1.25 + 2.25 + 4.25 + 5.25 + 8.25 + 9.25 + 11.25 + 21.25 + 22.25 + 25.25
------------------------------------------------------------------------------------------------------
16
= 12.8125
- The MAD of Soil B is 12.8125 cm.
c. To determine which set is more variable, we compare the MAD of each set. The larger the MAD, the more spread out the data points are from the mean, and the more variable the data set is. Therefore, Soil A is more variable since it has a higher MAD of 30.62 cm compared to Soil B's MAD of 12.8125 cm.