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A. Calculate the values for the 5-number summary of the skewed dataset:

5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16

(1 point)
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The minimum value = Response area

The maximum value = Response area

The medial (or Q2) = Response area

Quartile 1 (Q1) = Response area

Quartile 3 (Q3) = Response area

Question 2
B. Using the dataset and work ABOVE, determine the range, standard deviation, and interquartile range. Round to the nearest hundredth. (1 point)
Range =

Standard Deviation =

Interquartile Range (IQR) =

1 answer

To calculate the 5-number summary, range, standard deviation, and interquartile range (IQR) for the dataset: 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16, let's go through each step.

A. 5-Number Summary

  1. Minimum Value: The smallest number in the dataset.

    • Minimum = 5

  2. Maximum Value: The largest number in the dataset.

    • Maximum = 16

  3. Median (Q2): The middle value of the dataset when sorted. Since there are 12 values (even number), the median is the average of the 6th and 7th values.

    • Median = (8 + 8) / 2 = 8

  4. Quartile 1 (Q1): The median of the first half of the data (the lower half: 5, 6, 6, 7, 7, 8). Again, since there are 6 values, Q1 is the average of the 3rd and 4th values.

    • Q1 = (6 + 7) / 2 = 6.5

  5. Quartile 3 (Q3): The median of the second half of the data (the upper half: 8, 8, 9, 9, 14, 16). Q3 is the average of the 3rd and 4th values.

    • Q3 = (9 + 9) / 2 = 9

5-Number Summary:

  • Minimum Value = 5
  • Maximum Value = 16
  • Median (Q2) = 8
  • Quartile 1 (Q1) = 6.5
  • Quartile 3 (Q3) = 9

B. Additional calculations

  1. Range: The difference between the maximum and minimum values.

    • Range = Maximum - Minimum = 16 - 5 = 11

  2. Interquartile Range (IQR): The difference between Q3 and Q1.

    • IQR = Q3 - Q1 = 9 - 6.5 = 2.5

  3. Standard Deviation:

    • First, find the mean:

      • Mean = (5 + 6 + 6 + 7 + 7 + 8 + 8 + 8 + 9 + 9 + 14 + 16) / 12 = 8.25

    • Then calculate the variance: \[ \text{Variance} = \frac{1}{n} \sum (x_i - \text{Mean})^2 \]

    • Calculate each deviation squared:

      • (5 - 8.25)² = 10.5625
      • (6 - 8.25)² = 5.0625
      • (6 - 8.25)² = 5.0625
      • (7 - 8.25)² = 1.5625
      • (7 - 8.25)² = 1.5625
      • (8 - 8.25)² = 0.0625
      • (8 - 8.25)² = 0.0625
      • (8 - 8.25)² = 0.0625
      • (9 - 8.25)² = 0.5625
      • (9 - 8.25)² = 0.5625
      • (14 - 8.25)² = 32.0625
      • (16 - 8.25)² = 60.0625

    • Total sum of squared deviations: = 10.5625 + 5.0625 + 5.0625 + 1.5625 + 1.5625 + 0.0625 + 0.0625 + 0.0625 + 0.5625 + 0.5625 + 32.0625 + 60.0625 = 117.5

    • Variance = 117.5 / 12 ≈ 9.79

    • Standard Deviation = √Variance ≈ √9.79 ≈ 3.13

Summary of Results:

  • Range = 11
  • Standard Deviation ≈ 3.13
  • Interquartile Range (IQR) = 2.5

Final Inputs:

Minimum Value = 5
Maximum Value = 16
Median (Q2) = 8
Quartile 1 (Q1) = 6.5
Quartile 3 (Q3) = 9
Range = 11
Standard Deviation = 3.13
Interquartile Range (IQR) = 2.5