NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Comparing data distributions

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A school in Melbourne orders two types of science lab equipment for Year 9 practicals. The delivery times (in days) for 12 orders of Type X equipment are: 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28.

The delivery times (in days) for 12 orders of Type Y equipment are: 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 65. Both data sets are ordered.

Calculate the exact interquartile range (IQR) for the Type Y delivery times. The IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1).

For a data set with 12 values, Q1 is the average of the 3rd and 4th values, and Q3 is the average of the 9th and 10th values. Show all steps and give the exact result as a single number or fraction.

Concept illustrations

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A five-number summary. Illustrative five-number summary: minimum 1, Q1 4, median 7, Q3 10, maximum 13. IQR = 10 − 4 = 6. Whiskers use the minimum/maximum convention.1471013IQR = Q3 − Q1 = 6
Illustrative five-number summary: minimum 1, Q1 4, median 7, Q3 10, maximum 13. IQR = 10 − 4 = 6. Whiskers use the minimum/maximum convention.
Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The data set for Type Y delivery times is given as: 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 65. There are 12 values, so n = 12.
  2. To find the lower quartile (Q1), use the formula for the position of the 1st quartile in an ordered data set: (n + 1)/4. For n = 12, this gives (12 + 1)/4 = 13/4 = 3.25. This means Q1 lies between the 3rd and 4th values.
  3. The 3rd value is 17 and the 4th value is 18. Q1 is the average of these two: (17 + 18)/2 = 35/2 = 17.5.
  4. To find the upper quartile (Q3), use the formula for the 3rd quartile: 3(n + 1)/4. For n = 12, this gives 3(13)/4 = 39/4 = 9.75. This means Q3 lies between the 9th and 10th values.
  5. The 9th value is 23 and the 10th value is 24. Q3 is the average of these two: (23 + 24)/2 = 47/2 = 23.5.
  6. The interquartile range (IQR) is calculated as IQR = Q3 - Q1. Substitute the values: IQR = 23.5 - 17.5.
  7. Perform the subtraction: 23.5 - 17.5 = 6.0.
  8. The result is a whole number, so no further simplification is needed. The IQR is exactly 6.
  9. Verify the positions: for Q1, 3rd and 4th values are 17 and 18; average is 17.5. For Q3, 9th and 10th values are 23 and 24; average is 23.5. Correct.
  10. Check for outliers: the value 65 is much higher than the rest, but since the IQR is based only on Q1 and Q3, and the question asks for the IQR, outlier presence does not affect the calculation.
  11. The IQR is 6 days, which represents the spread of the middle 50% of the Type Y delivery times.
  12. The final answer is 6 days.

Reference answer: 6

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