Comparing box plots

When you reach for it

You have measured the same thing in two groups, such as the reaction times of two classes, and want to say how the groups differ. Two box plots drawn one above the other on one number line show at a glance where each group sits, how spread out it is, and which way it leans.

As a reminder, a box plot draws a data set’s five-number summary: a box from the lower quartile Q_1 to the upper quartile Q_3 with a line at the median, and whiskers out to the minimum and maximum. Each of the four pieces holds about a quarter of the values, and the box’s width is the interquartile range, \text{IQR} = Q_3 - Q_1. Quartiles here are the medians of the lower and upper halves of the ordered data, with the median left out of both halves when the count is odd.

The procedure

Day 1: two box plots on one scale

In the ruler-drop test one person lets go of a ruler and the other catches it; how far it fell gives the reaction time. Two classes of 12 took the test, in milliseconds (ms), so smaller is quicker.

Morning class: 165, 172, 180, 188, 194, 198, 204, 210, 218, 226, 248, 270.

Afternoon class: 172, 196, 208, 216, 222, 226, 230, 234, 238, 244, 250, 256.

With 12 ordered values, the median is the mean of the 6th and 7th, and each quartile the mean of the 3rd and 4th values of its half.

One scale. Both plots hang from the same number line, so a given time sits at the same place in both. Position is the whole comparison; plots on separate scales can look alike when the data are not.

Center: compare the medians. The medians are 201 ms and 228 ms. Said in context, with units: the afternoon class was typically slower, by about 228 - 201 = 27 ms.

Spread: compare the boxes, then the whole plots. The box’s width is the spread of the middle half:

The ranges are 270 - 165 = 105 ms and 256 - 172 = 84 ms. By both measures the morning class is more spread out. The IQR is usually the fairer measure, because it ignores the outer quarter at each end: the morning range is stretched by one student at 270 ms, whom the box does not notice.

Shape: compare the two halves of each plot. The lower half of a group lies between the minimum and the median, the upper half between the median and the maximum. The halves hold about the same number of values, so a longer half does not hold more values; its values are spread more thinly. In the morning class the lower half covers 201 - 165 = 36 ms and the upper half 270 - 201 = 69 ms; the times crowd at the quick end and trail off toward slow ones. A distribution whose long tail points toward larger values is skewed right. The afternoon class is the mirror image, 228 - 172 = 56 ms below the median and 256 - 228 = 28 ms above, so it is skewed left. Halves of about equal length mean a roughly symmetric distribution.

Shares: place a value among the quarter marks. About a quarter of a group lies above its Q_3, about half above its median and about three quarters above its Q_1. To say what share lies above a value, find which piece of the group’s plot it falls in.

What share of the afternoon class was slower than the morning median, 201 ms? That value is below the afternoon Q_1 = 212, so at least about three quarters took longer; the data give 10 of the 12. What share of the morning class was slower than the afternoon median, 228 ms? That value lies in the morning right whisker, beyond Q_3 = 222, so fewer than about a quarter; in fact 2 of the 12. Because box plots compare shares, not counts, groups of different sizes can be compared.

Day 2: drawing two box plots and judging the difference

Drawing two on one scale. Find each group’s five-number summary from its own ordered data, then choose one evenly ticked number line running from at least the smaller minimum to the larger maximum, and draw both labeled plots above it. Example 4 draws a pair.

Is the difference real? Two groups almost never have exactly the same median, even when nothing separates them. Florentin grew two trays of seedlings in exactly the same way and measured ten from each: the medians were 9.3 cm and 9.0 cm, with boxes 1.8 cm and 2.3 cm wide. The 0.3 cm between the medians came only from which seedlings happened to be measured (Sampling variability and simulating it), and it is small next to the boxes.

So, as in comparing two distributions, compare the shift between the centers with the spread within each group. On box plots the centers are the medians and the spread is the IQR, the width of each box, which shows how much members of one group already differ. A gap of 5 ms is large when each box is 2 ms wide and tiny when each is 100 ms wide. A rough guide, not a law:

• A shift small compared with the boxes, each median inside the other group’s box: another pair of samples could shrink or reverse the gap, so no conclusion is justified.
• A shift about as wide as the boxes or wider, each median outside the other group’s box: the bulk of one group sits above the bulk of the other, which luck seldom does, so it is reasonable to say the groups differ in center.

Between the two lies a gray zone, and more values in each group make a gap of a given size more convincing. A real difference says that the groups differ, not why: the seedlings given plant food in Example 4 grew taller because of the food only if chance decided which ones got it, as in Random assignment and confounding.

Worked examples

Boundaries

Same quantity, same units, one scale. Plots in milliseconds and in seconds, or on two number lines, cannot be read against each other.

Shares are approximate. The pieces hold exactly a quarter each only when the count divides evenly and no values tie. Exact counts need the number of values and the quartile rule, and rules differ: some calculators keep the median in both halves when the count is odd. Compare box plots made by the same rule.

Five numbers, nothing else. No single values, gaps, clusters or counts, and different data sets can share a box plot.

One extreme value stretches a whisker, making a group look more spread out than most of it is. Box plots that show outliers draw such a value on its own.

Deciding is a judgment, not a proof. The guide is rough, small groups mislead, and a difference says nothing about cause unless chance decided who went into which group.

Numbers only. Categories, such as favorite color, have no order, so no median and no box plot.

Where this goes next

Comparing two distributions widens this to dot plots and histograms, which keep the gaps and clusters a box plot hides. Comparing populations with the MAD turns “shift against spread” into a number, measuring the gap between two means in mean absolute deviations.

Exercises

Day 1: reading two box plots

The box plots show ruler-drop reaction times, in milliseconds, for Kirsi’s class and Jareth’s class.

1. Find each class’s median. Which class was typically quicker, and by how much?
2. Find each class’s interquartile range and range. Which class is more spread out, and does the answer depend on the measure?
3. Describe the shape of each plot, using the lengths of its two halves.
4. About what share of Kirsi’s class took longer than 200 ms?
5. About what share of Jareth’s class took longer than 200 ms?
6. About what share of Jareth’s class was quicker than the median of Kirsi’s class?

Day 1: reading five-number summaries

7. Twenty trips on each of two bus routes took these times in minutes. Route 5: 18, 21, 24, 26, 30. Route 6: 15, 22, 24, 31, 44. Compare their centers and spreads. Which route is more predictable?
8. Describe the shape of Route 6’s times, and say what it means for a passenger.
9. Route 6 has a lower quartile of 22 minutes over 20 trips. At least how many of its trips took 22 minutes or longer?
10. Rainfall in millimeters on the same 30 days in two towns. East: 0, 1, 4, 9, 22. West: 0, 0, 1, 3, 25. True or false, with a reason: (a) West had the single wettest day. (b) West was wetter on a typical day. (c) West was dry on at least 8 days. (d) The middle half of East’s days was more spread out than West’s.

Day 2: drawing and comparing

11. Lifetimes in hours of 10 Brand C bulbs: 1150, 1210, 1240, 1260, 1300, 1320, 1350, 1380, 1420, 1500. Of 9 Brand D bulbs: 980, 1100, 1180, 1260, 1330, 1350, 1410, 1480, 1620. Find both five-number summaries.
12. Choose one scale for both brands in Exercise 11, and describe both box plots drawn on it.
13. Do the brands in Exercise 11 differ in typical lifetime? In spread?
14. Hugo-James’s class had these ruler-drop times in milliseconds: 188, 204, 176, 230, 196, 212, 184. Find the five-number summary and the IQR.
15. Gwenllian’s class had these times in milliseconds: 214, 236, 198, 252, 228, 240, 222, 206, 246. Find the five-number summary and the IQR.
16. Using Exercises 14 and 15, decide whether the classes differ in typical reaction time, and say what share of Hugo-James’s class was quicker than every student in Gwenllian’s.

Day 2: judging and the limits

17. Write two different sets of six journey times, in minutes, that both have the five-number summary 10, 12, 15, 20, 24.
18. Iona’s box plot of rainfall on 30 days shows a median of 5 mm. Must some day have had exactly 5 mm?
19. A swimming coach timed two squads over 50 m, in seconds. Squad P, old warm-up: Q_1 = 38.6, median 40.2, Q_3 = 41.4. Squad Q, new warm-up: Q_1 = 38.2, median 39.8, Q_3 = 41.0. She says the new warm-up made swimmers faster. Do the box plots support her?

Reasoning

20. Explain why two box plots have to share one number line to be compared.
21. One class has 30 students and another 20. Explain why their box plots can still be compared.
22. In a box plot, the right part of the box is three times as long as the left part. Explain why that does not mean more values lie in the right part.
23. Explain why a gap between two medians is judged against the widths of the boxes, not on its own.
24. Name two things a pair of dot plots shows that a pair of box plots cannot, and say why the box plots lose them.