Quartiles and the Interquartile Range · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Cutting a list into quarters

Mathematics · Data & Statistics · ages 15-16
Name ______________________   Date ____________

  1. Seven scores in order are 8, 11, 14, 16, 19, 23, 30. Use quartile positions (n + 1) / 4 and 3(n + 1) / 4. What is the interquartile range?

    Answer: ______________

  2. You have to put the data in order before finding the quartiles.

    Circle one:   True   False

  3. Using upper-quartile position 3(n + 1) / 4, where is the upper quartile in an ordered list of 19 values?

    • the 14th value
    • the 19th value
    • the 15th value
    • the 5th value

  4. The ordered values are 2, 4, 4, 4, 7, 9, 10, 12, 15, 18, 20. The lower quartile is the 3rd value, 4. How many observations are strictly less than the lower quartile?

    Answer: ______________

  5. Nine values in order are 4, 6, 7, 9, 12, 14, 15, 19, 22. Use lower-quartile position (n + 1) / 4, averaging adjacent values for a halfway position. What is the lower quartile?

    Answer: ______________

  6. Sam finds quartiles for 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. He removes both middle observations before splitting the list. Under the median-of-halves convention, what should he do instead?

    • Remove only 10 and keep the remaining nine values
    • Keep all ten values and form two halves of five
    • Remove 10 and 12, then average the two extremes
    • Use all ten values together to find each quartile

  7. The median can lie closer to one quartile value than to the other.

    Circle one:   True   False

  8. Append 100 to 2, 4, 7, 8, 11, 13, 14. For the new ordered list, find each quartile as the median of its four-value half. What are the new range and IQR?

    • Range 12; IQR 9
    • Range 98; IQR 9
    • Range 98; IQR 6
    • Range 98; IQR 8

  9. Can a set's interquartile range ever be bigger than its range?

    • Yes, if the largest observation is extreme
    • Yes, if the middle observations are spread out
    • No, the quartiles lie within the extremes
    • Yes, if the ordered list has an even count

  10. The interval from the minimum to Q1 is 40 to 43. The interval from Q3 to the maximum is 55 to 78. Leo says quartile cuts always create equal value widths. What do these intervals show?

    • Quartile intervals can have unequal widths
    • All four quartile intervals must have width 3
    • All four quartile intervals must have width 23
    • Opposite quartile intervals cannot be compared
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Cutting a list into quarters W1-mt_vN10d1I9dA-s1

  1. 12 · The lower quartile is the 2nd score, 11, and the upper quartile is at position 3 × 8 / 4 = 6, the 6th score, 23. So the interquartile range is 23 - 11 = 12.
  2. True · Both quartiles are found by counting to a position, and a position only means something once the values are lined up smallest to largest.
  3. the 15th value · The position is 3 × (19 + 1) / 4 = 3 × 5 = 15, so the upper quartile is the 15th value. The 5th value is the lower quartile.
  4. 1 · Only 2 is strictly less than 4. Observations equal to 4 do not count as less than it.
  5. 6.5 · The position is (9 + 1) / 4 = 2.5, which falls halfway between the 2nd and 3rd values, 6 and 7. The halfway point is (6 + 7) / 2 = 6.5.
  6. Keep all ten values and form two halves of five · An even-sized list has no single central observation to exclude. Keep all ten values. The halves are 2, 4, 6, 8, 10 and 12, 14, 16, 18, 20, giving quartiles 6 and 16.
  7. True · The median is located by ordered position, not by averaging the quartiles. For example, quartiles 10 and 21 with median 14 give unequal gaps of 4 and 7.
  8. Range 98; IQR 8 · The new halves are 2, 4, 7, 8 and 11, 13, 14, 100. Their medians are 5.5 and 13.5, giving IQR 8. The range is 100 - 2 = 98.
  9. No, the quartiles lie within the extremes · The lower quartile is at or above the smallest value and the upper quartile is at or below the largest, so the gap between them can never exceed the gap between the ends.
  10. Quartile intervals can have unequal widths · The widths are 43 - 40 = 3 and 78 - 55 = 23. They differ, so quartile cuts do not guarantee equal value widths.
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