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

Cutting a List Into Quarters

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

  1. Where is the upper quartile in an ordered list of 19 values?

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

  2. A set of times has a lower quartile of 14 minutes and an upper quartile of 26 minutes. Which calculation gives the interquartile range?

    • 26 - 14
    • 26 + 14
    • (26 + 14) / 2
    • 26 - 20, using the median

  3. The interquartile range is the largest value minus the smallest value.

    Circle one:   True   False

  4. Seven scores in order are 12, 15, 18, 20, 25, 28, 33. What is the lower quartile?

    Answer: ______________

  5. Fifteen times in order are 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 55. What is the upper quartile?

    Answer: ______________

  6. Changing the largest value in a data set always changes the interquartile range.

    Circle one:   True   False

  7. Dee works out the upper quartile as 47 and the median as 41, then reports an interquartile range of 6. What has gone wrong?

    • She has subtracted in the wrong order
    • Nothing, because the interquartile range does use the median
    • She should have added the two numbers instead of subtracting
    • She has subtracted the median instead of the lower quartile

  8. Seven times in order are 5, 6, 7, 8, 9, 10 and 40. The 40 is changed to 100. What happens to the range and the interquartile range?

    • Both grow by 60
    • The range grows by 60 and the interquartile range stays at 4
    • Neither changes, since only one value moved
    • The interquartile range grows and the range stays the same

  9. A report says one runner's time was mistyped and is far too large. Which summary should the organisers quote while they check?

    • the range, since it uses every value
    • the mean, since it uses every value
    • the largest value, since that is the one in doubt
    • the interquartile range, since one wrong entry at the end cannot move it

  10. In one set the bottom quarter runs from 40 to 43 and the top quarter runs from 55 to 78. Leo says the three quartile cuts split a set into four groups covering equal stretches of value. What do those two figures show?

    • The stretches are 3 and 23, so the groups hold equal numbers of values rather than equal stretches
    • The stretches are 3 and 23, which is what equal groups look like
    • Nothing, because the two quarters are at opposite ends of the set
    • That the set must have been sorted wrongly
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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. 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.
  2. 26 - 14 · The interquartile range is the distance between the two outer cuts, so it is 26 - 14 = 12 minutes. Adding them or halving them describes a position rather than a width.
  3. False · That is the range. The interquartile range uses the two quartiles instead, so it describes the middle half rather than the whole set.
  4. 15 · The position is (7 + 1) / 4 = 2, so the lower quartile is the 2nd score, which is 15.
  5. 47 · The position is 3 × (15 + 1) / 4 = 3 × 4 = 12, so the upper quartile is the 12th time, which is 47.
  6. False · The quartiles are found by counting to a position, and the largest value stays in last place however far it moves. Unless the change reorders the list, both quartiles are untouched.
  7. She has subtracted the median instead of the lower quartile · The interquartile range is upper quartile minus lower quartile. Using the median instead measures only the upper part of the middle half, so her figure is too small.
  8. The range grows by 60 and the interquartile range stays at 4 · The range goes from 40 - 5 = 35 to 100 - 5 = 95. The quartiles are the 2nd and 6th values, 6 and 10, and both are untouched, so the interquartile range stays 10 - 6 = 4.
  9. the interquartile range, since one wrong entry at the end cannot move it · Both the range and the mean take the suspect value straight into the answer. The quartiles are set by position, so a single wrong entry at the end leaves the interquartile range exactly where it was.
  10. The stretches are 3 and 23, so the groups hold equal numbers of values rather than equal stretches · The bottom quarter covers 3 units and the top quarter covers 23, nearly eight times as wide. The cuts are made by counting values, so the groups match in size and not in the stretch of value they cover.
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