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:
- We add up all the values in the Soil A stem-and-leaf plot:
50 + 51 + 52 + 55 + 60 + 61 + 62 + 65 + 69
- There are 9 values. Divide the sum by 9 to get the mean:
(50 + 51 + 52 + 55 + 60 + 61 + 62 + 65 + 69)/9 ≈ 58.11
Mean of Soil B:
- We add up all the values in the Soil B stem-and-leaf plot:
36 + 39 + 70 + 72 + 73 + 76 + 78 + 79
- There are 8 values. Divide the sum by 8 to get the mean:
(36 + 39 + 70 + 72 + 73 + 76 + 78 + 79)/8 = 68.125
Mean absolute deviation (MAD) of Soil A:
- First, we find the absolute deviations of each value from the mean:
|50 - 58.11| = 8.11
|51 - 58.11| = 7.11
|52 - 58.11| = 6.11
|55 - 58.11| = 3.11
|60 - 58.11| = 1.89
|61 - 58.11| = 2.89
|62 - 58.11| = 3.89
|65 - 58.11| = 6.89
|69 - 58.11| = 10.89
- Next, we find the mean of these absolute deviations:
(8.11 + 7.11 + 6.11 + 3.11 + 1.89 + 2.89 + 3.89 + 6.89 + 10.89)/9 ≈ 5.56
MAD of Soil B:
- First, we find the absolute deviations of each value from the mean:
|36 - 68.125| = 32.125
|39 - 68.125| = 29.125
|70 - 68.125| = 1.875
|72 - 68.125| = 3.875
|73 - 68.125| = 4.875
|76 - 68.125| = 7.875
|78 - 68.125| = 9.875
|79 - 68.125| = 10.875
- Next, we find the mean of these absolute deviations:
(32.125 + 29.125 + 1.875 + 3.875 + 4.875 + 7.875 + 9.875 + 10.875)/8 ≈ 12.09
Soil B is more variable because its MAD is larger. We can see from the stem-and-leaf plot that Soil B has more spread out values and outliers, while Soil A has a more concentrated set of values. The MAD confirms this, as it measures how far the values are from the mean on average.
could you explain in just words not numbers.?
?
Sure, I can explain it in words.
To find the mean of each data set, we add up all the values and divide by the number of values. The mean tells us the "average" or "typical" value in the data set. In this case, Soil A has a mean height of about 58 centimeters, while Soil B has a mean height of about 68 centimeters. This means that, on average, Teddy Bear sunflowers grown in Soil B are taller than those grown in Soil A.
To find the mean absolute deviation (MAD) of each data set, we first find the absolute deviation of each value from the mean. This means we subtract the mean from each value and take the absolute value (ignoring any negative signs). Then, we find the mean of these absolute deviations. This tells us, on average, how far each value is from the mean. In this case, Soil B has a larger MAD than Soil A, which means that its values are generally more spread out (with some larger deviations from the mean).
Therefore, we can say that Soil B is more variable than Soil A. One way to see this from the stem-and-leaf plot is to notice that Soil B has more values that are farther away from the middle (the "5" in the middle of the plot), while Soil A has more values that are clustered around the middle.
To calculate the mean of each data set, we need to find the average value.
For Soil A:
5 + 52 + 51 + 10 + 21 + 81 + 90 = 310
There are 9 values in Soil A.
Mean of Soil A = 310 / 9 = 34.4444 (rounded to four decimal places)
For Soil B:
59 + 63 + 73 + 76 + 78 + 83 + 89 = 481
There are 13 values in Soil B.
Mean of Soil B = 481 / 13 = 37 (rounded to the nearest whole number)
To calculate the mean absolute deviation (MAD), we need to find the average absolute difference between each value and the mean.
For Soil A:
|5 - 34.4444| + |2 - 34.4444| + |1 - 34.4444| + |1 - 34.4444| + |0 - 34.4444| + |1 - 34.4444| + |0 - 34.4444| + |2 - 34.4444| + |1 - 34.4444|
= 29.4444 + 32.4444 + 33.4444 + 33.4444 + 34.4444 + 33.4444 + 34.4444 + 32.4444 + 33.4444
= 296.5176 (rounded to four decimal places)
MAD of Soil A = 296.5176 / 9 = 32.9453 (rounded to four decimal places)
For Soil B:
|9 - 37| + |6 - 37| + |3 - 37| + |9 - 37| + |5 - 37| + |8 - 37| + |3 - 37| + |0 - 37| + |2 - 37| + |3 - 37| + |6 - 37| + |7 - 37| + |8 - 37|
= 28 + 31 + 34 + 28 + 32 + 29 + 34 + 37 + 35 + 34 + 31 + 30 + 29
= 392
MAD of Soil B = 392 / 13 = 30.1538 (rounded to four decimal places)
To determine which set is more variable, we can compare the mean absolute deviations (MAD) of Soil A and Soil B. The data set with a higher MAD is more variable.
In this case, Soil A has an MAD of 32.9453 while Soil B has an MAD of 30.1538. Therefore, Soil A is more variable based on the MAD.
To calculate the mean of each data set, you need to find the sum of all the values in the set, and then divide that sum by the total number of values.
For Soil A:
Sum of values = 5 + 5 + 2 + 1 + 1 + 5 + 1 + 0 + 2 + 1 + 8 + 0 + 9 = 40
Total number of values = 13
Mean = Sum of values / Total number of values = 40 / 13 ≈ 3.08
For Soil B:
Sum of values = 9 + 6 + 3 + 9 + 7 + 0 + 2 + 3 + 6 + 7 + 8 + 3 + 9 = 72
Total number of values = 13
Mean = Sum of values / Total number of values = 72 / 13 ≈ 5.54
To calculate the mean absolute deviation (MAD) of each data set, you need to find the absolute difference between each value and the mean of the set, and then find the mean of those absolute differences.
For Soil A:
Absolute differences = |5 - 3.08| + |5 - 3.08| + |2 - 3.08| + |1 - 3.08| + |1 - 3.08| + |5 - 3.08| + |1 - 3.08| + |0 - 3.08| + |2 - 3.08| + |1 - 3.08| + |8 - 3.08| + |0 - 3.08| + |9 - 3.08|
= 1.92 + 1.92 + 1.08 + 2.08 + 2.08 + 1.92 + 2.08 + 3.08 + 1.08 + 2.08 + 4.92 + 4.92 + 5.92
= 34.92
Number of values = 13
MAD = Absolute differences / Number of values = 34.92 / 13 ≈ 2.68
For Soil B:
Absolute differences = |9 - 5.54| + |6 - 5.54| + |3 - 5.54| + |9 - 5.54| + |7 - 5.54| + |0 - 5.54| + |2 - 5.54| + |3 - 5.54| + |6 - 5.54| + |7 - 5.54| + |8 - 5.54| + |3 - 5.54| + |9 - 5.54|
= 3.46 + 0.46 + 2.54 + 3.46 + 1.46 + 5.54 + 3.54 + 2.54 + 0.46 + 1.46 + 2.46 + 2.54 + 3.46
= 33.16
Number of values = 13
MAD = Absolute differences / Number of values = 33.16 / 13 ≈ 2.55
To determine which set is more variable, you compare the MAD values of the two data sets. The data set with a higher MAD is considered more variable because it indicates that the values deviate more from the mean. In this case, Soil A has a MAD of approximately 2.68, while Soil B has a MAD of approximately 2.55. Therefore, Soil A is more variable.
You can tell which set is more variable by comparing the MAD values. If one set has a higher MAD than the other, it means that the values in that set deviate more from the mean on average, indicating higher variability.