Question
The mean of each dataset that follows is 62. Which dataset is less spread out? Justify your answer using the mean absolute deviation of each dataset. Round your answers to the nearest tenth.
Dataset #1: {51, 53, 56, 60, 72, 80}
Dataset #2: {49, 55, 61, 63, 70, 74}
(1 point)
Dataset #1 and Dataset #2 are equally spread because they both have a mean absolute deviation of 0.
Dataset #2 is less spread out because it has a mean absolute deviation of 7 whereas Dataset #1 has a mean absolute deviation of approximately 9.3.
Dataset #1 is less spread out because it has a mean absolute deviation of 9.3 whereas Dataset #2 has a mean absolute deviation of 7
Dataset #1 and Dataset #2 are equally spread because they both have a mean of 62.
Dataset #1: {51, 53, 56, 60, 72, 80}
Dataset #2: {49, 55, 61, 63, 70, 74}
(1 point)
Dataset #1 and Dataset #2 are equally spread because they both have a mean absolute deviation of 0.
Dataset #2 is less spread out because it has a mean absolute deviation of 7 whereas Dataset #1 has a mean absolute deviation of approximately 9.3.
Dataset #1 is less spread out because it has a mean absolute deviation of 9.3 whereas Dataset #2 has a mean absolute deviation of 7
Dataset #1 and Dataset #2 are equally spread because they both have a mean of 62.
Answers
GPT-4o mini
To determine which dataset is less spread out using the mean absolute deviation (MAD), we first calculate the MAD for each dataset.
**Dataset #1: {51, 53, 56, 60, 72, 80}**
1. Calculate the absolute deviations from the mean (62):
- |51 - 62| = 11
- |53 - 62| = 9
- |56 - 62| = 6
- |60 - 62| = 2
- |72 - 62| = 10
- |80 - 62| = 18
2. Calculate the mean of these absolute deviations:
\[
\text{MAD}_1 = \frac{11 + 9 + 6 + 2 + 10 + 18}{6} = \frac{56}{6} \approx 9.3
\]
**Dataset #2: {49, 55, 61, 63, 70, 74}**
1. Calculate the absolute deviations from the mean (62):
- |49 - 62| = 13
- |55 - 62| = 7
- |61 - 62| = 1
- |63 - 62| = 1
- |70 - 62| = 8
- |74 - 62| = 12
2. Calculate the mean of these absolute deviations:
\[
\text{MAD}_2 = \frac{13 + 7 + 1 + 1 + 8 + 12}{6} = \frac{42}{6} = 7
\]
Now that we have both MADs, we can compare them:
- MAD for Dataset #1: approximately **9.3**
- MAD for Dataset #2: **7**
Since a lower mean absolute deviation indicates that the dataset is less spread out, we conclude that:
**Dataset #2 is less spread out because it has a mean absolute deviation of 7 whereas Dataset #1 has a mean absolute deviation of approximately 9.3.**
So the correct answer is:
**Dataset #2 is less spread out because it has a mean absolute deviation of 7 whereas Dataset #1 has a mean absolute deviation of approximately 9.3.**
**Dataset #1: {51, 53, 56, 60, 72, 80}**
1. Calculate the absolute deviations from the mean (62):
- |51 - 62| = 11
- |53 - 62| = 9
- |56 - 62| = 6
- |60 - 62| = 2
- |72 - 62| = 10
- |80 - 62| = 18
2. Calculate the mean of these absolute deviations:
\[
\text{MAD}_1 = \frac{11 + 9 + 6 + 2 + 10 + 18}{6} = \frac{56}{6} \approx 9.3
\]
**Dataset #2: {49, 55, 61, 63, 70, 74}**
1. Calculate the absolute deviations from the mean (62):
- |49 - 62| = 13
- |55 - 62| = 7
- |61 - 62| = 1
- |63 - 62| = 1
- |70 - 62| = 8
- |74 - 62| = 12
2. Calculate the mean of these absolute deviations:
\[
\text{MAD}_2 = \frac{13 + 7 + 1 + 1 + 8 + 12}{6} = \frac{42}{6} = 7
\]
Now that we have both MADs, we can compare them:
- MAD for Dataset #1: approximately **9.3**
- MAD for Dataset #2: **7**
Since a lower mean absolute deviation indicates that the dataset is less spread out, we conclude that:
**Dataset #2 is less spread out because it has a mean absolute deviation of 7 whereas Dataset #1 has a mean absolute deviation of approximately 9.3.**
So the correct answer is:
**Dataset #2 is less spread out because it has a mean absolute deviation of 7 whereas Dataset #1 has a mean absolute deviation of approximately 9.3.**