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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}
{
51
,
53
,
56
,
60
,
72
,
80
}

Dataset #2: {49,55,61,63,70,74}
{
49
,
55
,
61
,
63
,
70
,
74
}


(1 point)
Responses

Dataset #1 and Dataset #2 are equally spread because they both have a mean of 62.
Dataset #1 and Dataset #2 are equally spread because they both have a mean of 62.

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 #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 and Dataset #2 are equally spread because they both have a mean absolute deviation of 0.
Dataset #1 and Dataset #2 are equally spread because they both have a mean absolute deviation of 0.

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 is less spread out because it has a mean absolute deviation of 9.3 whereas Dataset #2 has a mean absolute deviation of 7.

1 answer

To determine which dataset is less spread out, we need to calculate the mean absolute deviation (MAD) for each dataset.

Dataset #1: {51, 53, 56, 60, 72, 80}

  1. Compute the absolute deviations from the mean (which is 62):

    • |51 - 62| = 11
    • |53 - 62| = 9
    • |56 - 62| = 6
    • |60 - 62| = 2
    • |72 - 62| = 10
    • |80 - 62| = 18

  2. Sum the absolute deviations:

    • 11 + 9 + 6 + 2 + 10 + 18 = 56

  3. Divide by the number of data points (6) to get the mean absolute deviation:

    • MAD = 56 / 6 ≈ 9.3

Dataset #2: {49, 55, 61, 63, 70, 74}

  1. Compute the absolute deviations from the mean (which is 62):

    • |49 - 62| = 13
    • |55 - 62| = 7
    • |61 - 62| = 1
    • |63 - 62| = 1
    • |70 - 62| = 8
    • |74 - 62| = 12

  2. Sum the absolute deviations:

    • 13 + 7 + 1 + 1 + 8 + 12 = 42

  3. Divide by the number of data points (6) to get the mean absolute deviation:

    • MAD = 42 / 6 = 7.0

Comparison:

  • Mean Absolute Deviation for Dataset #1: 9.3
  • Mean Absolute Deviation for Dataset #2: 7.0

Since Dataset #2 has a lower mean absolute deviation (7.0) compared to Dataset #1 (9.3), this indicates that Dataset #2 is less spread out.

Therefore, the correct response 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.