Box Plots and Comparing Distributions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Five Numbers, One Picture

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

  1. Two box plots of race times sit on the same scale. Group A's median is at 46 minutes and Group B's is at 39. Which group was faster overall?

    • Group A, because its median is the larger number
    • Group B, because a shorter time is a faster run
    • Neither, medians cannot compare two groups
    • It depends on which box is wider

  2. A box plot has its box from 18 to 30, with whiskers reaching 10 and 44. What is the interquartile range?

    Answer: ______________

  3. A five-figure summary reads 6, 11, 14, 22 and 40. Where does the line inside the box go?

    • at 14, the median
    • at 23, halfway between the whisker tips
    • at 22, the upper quartile
    • at 11, the lower quartile

  4. Which set of numbers makes up a five-figure summary?

    • the smallest value, both quartiles, the median and the largest value
    • the mean, the median, the mode, the range and the total
    • five values picked evenly from the middle of the data
    • the five values that came up most often

  5. Two box plots of the same test have equal medians. Group A's box is 15 wide and Group B's is 4 wide. What is the honest sentence to write?

    • Group A scored higher than Group B
    • Group B had fewer students than Group A
    • Both groups scored about the same typically, but B's middle half was packed much closer together
    • Group A's test was harder

  6. Two box plots that look identical must have come from the same number of values.

    Circle one:   True   False

  7. A box runs from 60 to 80 with the median line at 65. What shape is the data?

    • Symmetric, since the median is inside the box
    • Skewed to the left, because the median is a low number
    • Skewed to the right, because the quarter above the median is spread over 15 while the quarter below it covers 5
    • Impossible to say from a box plot

  8. Leo draws a box plot by putting the box between the smallest and largest values and marking the quartiles as short lines inside it. What is wrong?

    • The quartiles should be the box ends and the extremes should be the whisker tips
    • Nothing, that is another correct way to draw one
    • He should have used the mean rather than the median
    • The box should always be drawn to the same width

  9. Two box plots of race times are identical. Race A had 14 finishers and race B had 350. Which comparison is still safe?

    • The two fields had the same typical time and the same middle-half spread
    • The two races were equally popular
    • The same number of runners beat 30 minutes in each race
    • Race B was better organised

  10. Sam compares two box plots and writes: Group A has more people scoring highly, because A's right whisker is much longer. Where is he wrong?

    • He should have compared the left whiskers instead
    • Right whiskers cannot be compared between plots
    • A longer whisker means A's top quarter is spread wider, and both groups still have exactly a quarter up there
    • Nothing, a longer whisker does mean more high scores
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Answer key

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

Five Numbers, One Picture W1-mt_G_SnfEllzQ-s1

  1. Group B, because a shorter time is a faster run · The median is the typical time, and in a race a smaller time is better. Group B's typical runner came home seven minutes sooner.
  2. 12 · The box ends are the quartiles, so the interquartile range is 30 - 18 = 12. The whisker tips are the smallest and largest values and belong to the range instead.
  3. at 14, the median · The line inside the box marks the median, which is the middle figure of the five, 14. Halfway between the whisker tips is 23, and that is not what the line shows.
  4. the smallest value, both quartiles, the median and the largest value · The five are the smallest value, the lower quartile, the median, the upper quartile and the largest value. Those are exactly what a box plot draws.
  5. Both groups scored about the same typically, but B's middle half was packed much closer together · Equal medians means neither group is typically higher. The box widths are the interquartile ranges, so they compare spread and nothing else. Numbers of students and test difficulty are not on the plot.
  6. False · A box plot draws five numbers. Any two sets that share those five figures draw the same picture, whether one holds 14 values and the other holds 350.
  7. Skewed to the right, because the quarter above the median is spread over 15 while the quarter below it covers 5 · The median sits 5 above the lower quartile and 15 below the upper one. The upper middle quarter is stretched over three times as much ground, so the data trails off towards the higher values.
  8. The quartiles should be the box ends and the extremes should be the whisker tips · Leo's drawing puts the whole range in the box, so the box no longer marks the middle half. A reader would take his box width as the interquartile range and be badly out.
  9. The two fields had the same typical time and the same middle-half spread · The five figures match, so the medians and interquartile ranges match. Anything about how many runners did something needs the counts, and the plots never carried them.
  10. A longer whisker means A's top quarter is spread wider, and both groups still have exactly a quarter up there · Every box plot puts a quarter of its values in each whisker by construction. Length is about how far those values reach, so A's top quarter is more spread out and not more numerous.
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