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
Calculate the mean absolute deviation (MAD) of each data set.
Which set is more variable? How do you know?
could you explain in words instead numbers
?
Certainly! The stem-and-leaf plot is a way to organize data. The numbers on the left side (the "stem") represent the tens digit, and the numbers on the right side (the "leaves") represent the ones digit. For example, in Soil A, there is one sunflower that is 50 centimeters tall (the 5 is the stem and the 0 is the leaf), two sunflowers that are 52 centimeters tall (the 5 is the stem and the 2's are the leaves), and so on.
Mean absolute deviation (MAD) measures how much the data deviates, or varies, from the mean or average. To find the MAD, you first find the mean of the data set. Then, you subtract the mean from each data point and take the absolute value (to eliminate negatives), and finally find the mean of those absolute differences.
In this problem, you need to calculate the MAD for both Soil A and Soil B. The data set that has a larger MAD is more variable, meaning it has more spread out data points.
To calculate the mean absolute deviation (MAD) of each data set using the stem-and-leaf plot, we need to find the average deviation of each value from the mean.
For Soil A:
1. Find the mean by adding up all the values in Soil A and dividing by the total count. In this case, the mean of Soil A is 5.
2. Find the deviation of each value in Soil A from the mean. Subtract the mean from each value.
3. Take the absolute value of each deviation. This ensures that we are considering the magnitude of the deviation, regardless of whether it is positive or negative.
4. Find the sum of all the absolute deviations.
5. Divide the sum of absolute deviations by the total count to find the mean absolute deviation.
Repeat the same steps for Soil B.
After calculating the MAD of each data set, we can compare the values to determine which set is more variable. The data set with a higher MAD is considered more variable because it has larger absolute deviations from the mean.
To find the mean absolute deviation (MAD) of each data set, you need to follow these steps:
1. Calculate the mean (average) of each data set: To do this, you add up all the values in the data set and then divide by the number of values.
2. Find the absolute deviation for each data point: To do this, you subtract each individual data point from the mean and take the absolute value (ignore the negative sign).
3. Calculate the average of the absolute deviations: Add up all the absolute deviations and divide by the total number of values.
4. Compare the MAD of each data set: The data set with the larger MAD is more variable because it has larger deviations from the mean.
Now, let's apply these steps to the given stem-and-leaf plot:
For Soil A:
1. Calculate the mean: Add up the values in Soil A (5, 5, 5, 2, 1, 1, 0) and divide by the total number of values (7). The mean is (5+5+5+2+1+1+0) / 7 = 19 / 7.
2. Find the absolute deviation for each data point: Subtract the mean from each data point and ignore the negative sign (taking the absolute value). This will give us the deviations: 0, 0, 0, 3, 4, 4, 5.
3. Calculate the average of the absolute deviations: Add up all the absolute deviations (0+0+0+3+4+4+5) and divide by the total number of values (7). The average is (0+0+0+3+4+4+5) / 7 = 16 / 7.
For Soil B:
1. Calculate the mean: Add up the values in Soil B (9, 6, 3, 9, 0, 2, 3, 6, 7, 8) and divide by the total number of values (10). The mean is (9+6+3+9+0+2+3+6+7+8) / 10 = 53 / 10.
2. Find the absolute deviation for each data point: Subtract the mean from each data point and ignore the negative sign (taking the absolute value). This will give us the deviations: 4, 3, 2, 4, 5, 3, 0, 3, 4, 5.
3. Calculate the average of the absolute deviations: Add up all the absolute deviations (4+3+2+4+5+3+0+3+4+5) and divide by the total number of values (10). The average is (4+3+2+4+5+3+0+3+4+5) / 10 = 33 / 10.
To determine which set is more variable, compare the MAD of each data set. In this case, Soil B has a larger MAD (33/10) compared to Soil A (16/7). Therefore, Soil B is more variable, indicating that the heights of Teddy Bear sunflowers grown in Soil B are more spread out from the mean compared to Soil A.