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The mean of each dataset that follows is 10. Which dataset is more spread out? Justify your answer using the mean absolute deviation of each dataset. Round your answers to the nearest tenth.

Dataset #1: {5,8,9,10,12,12,14}
dataset #2: {5,6,8,9,10,10,22}

(2 points)
Dataset #__ is more spread out because it has a mean absolute deviation of while Dataset #__ has a mean absolute deviation of __.

1 answer

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

Step 1: Calculate the Mean Absolute Deviation (MAD) for Dataset #1

Dataset #1: {5, 8, 9, 10, 12, 12, 14}

  1. Find the mean (already given as 10, confirming our calculations).

  2. Calculate the absolute deviations from the mean:

    • |5 - 10| = 5
    • |8 - 10| = 2
    • |9 - 10| = 1
    • |10 - 10| = 0
    • |12 - 10| = 2
    • |12 - 10| = 2
    • |14 - 10| = 4

  3. Sum absolute deviations:

    \[ 5 + 2 + 1 + 0 + 2 + 2 + 4 = 16 \]

  4. Calculate MAD:

    \[ MAD = \frac{16}{7} \approx 2.3 \]

Step 2: Calculate the Mean Absolute Deviation (MAD) for Dataset #2

Dataset #2: {5, 6, 8, 9, 10, 10, 22}

  1. Find the mean (already given as 10).

  2. Calculate the absolute deviations from the mean:

    • |5 - 10| = 5
    • |6 - 10| = 4
    • |8 - 10| = 2
    • |9 - 10| = 1
    • |10 - 10| = 0
    • |10 - 10| = 0
    • |22 - 10| = 12

  3. Sum absolute deviations:

    \[ 5 + 4 + 2 + 1 + 0 + 0 + 12 = 24 \]

  4. Calculate MAD:

    \[ MAD = \frac{24}{7} \approx 3.4 \]

Step 3: Compare the Results

  • Dataset #1: MAD ≈ 2.3
  • Dataset #2: MAD ≈ 3.4

Conclusion

Dataset #2 is more spread out because it has a mean absolute deviation of 3.4, while Dataset #1 has a mean absolute deviation of 2.3.

So, the final answer is: Dataset #2 is more spread out because it has a mean absolute deviation of 3.4 while Dataset #1 has a mean absolute deviation of 2.3.