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, we need to calculate the mean absolute deviation (MAD) for both datasets.
**Mean Absolute Deviation (MAD)** is calculated using the formula:
\[ \text{MAD} = \frac{1}{n} \sum |x_i - \bar{x}| \]
where \( n \) is the number of data points, \( x_i \) is each individual data point, and \( \bar{x} \) is the mean of the dataset.
Given that the mean for both datasets is 62, we can calculate the MAD for each dataset.
**Dataset #1: {51, 53, 56, 60, 72, 80}**
1. Deviation from the mean:
- |51 - 62| = 11
- |53 - 62| = 9
- |56 - 62| = 6
- |60 - 62| = 2
- |72 - 62| = 10
- |80 - 62| = 18
2. Sum of absolute deviations:
- 11 + 9 + 6 + 2 + 10 + 18 = 56
3. Mean Absolute Deviation:
- MAD = 56 / 6 ≈ 9.3
**Dataset #2: {49, 55, 61, 63, 70, 74}**
1. Deviation from the mean:
- |49 - 62| = 13
- |55 - 62| = 7
- |61 - 62| = 1
- |63 - 62| = 1
- |70 - 62| = 8
- |74 - 62| = 12
2. Sum of absolute deviations:
- 13 + 7 + 1 + 1 + 8 + 12 = 42
3. Mean Absolute Deviation:
- MAD = 42 / 6 = 7.0
Now comparing the MAD:
- Dataset #1: MAD ≈ 9.3
- Dataset #2: MAD = 7.0
Thus, **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.**
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.**
**Mean Absolute Deviation (MAD)** is calculated using the formula:
\[ \text{MAD} = \frac{1}{n} \sum |x_i - \bar{x}| \]
where \( n \) is the number of data points, \( x_i \) is each individual data point, and \( \bar{x} \) is the mean of the dataset.
Given that the mean for both datasets is 62, we can calculate the MAD for each dataset.
**Dataset #1: {51, 53, 56, 60, 72, 80}**
1. Deviation from the mean:
- |51 - 62| = 11
- |53 - 62| = 9
- |56 - 62| = 6
- |60 - 62| = 2
- |72 - 62| = 10
- |80 - 62| = 18
2. Sum of absolute deviations:
- 11 + 9 + 6 + 2 + 10 + 18 = 56
3. Mean Absolute Deviation:
- MAD = 56 / 6 ≈ 9.3
**Dataset #2: {49, 55, 61, 63, 70, 74}**
1. Deviation from the mean:
- |49 - 62| = 13
- |55 - 62| = 7
- |61 - 62| = 1
- |63 - 62| = 1
- |70 - 62| = 8
- |74 - 62| = 12
2. Sum of absolute deviations:
- 13 + 7 + 1 + 1 + 8 + 12 = 42
3. Mean Absolute Deviation:
- MAD = 42 / 6 = 7.0
Now comparing the MAD:
- Dataset #1: MAD ≈ 9.3
- Dataset #2: MAD = 7.0
Thus, **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.**
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.**