Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 4) Find the IQR when Q1 = 14.5 and Q3 = 22.
Answer
7.5. Q3 − Q1 = 22 − 14.5. - (Lesson 2) Find the mean of 6, 8, 11 and 15.
Answer
10. The total is 40 and 40 ÷ 4 = 10. - (Year 7, Unit 6, lesson 7) A bar chart's vertical axis starts at 50 instead of 0. What does that do to how the bars look?
Answer
It makes small differences look large, because the bars are cut off at the bottom.
Words used in this lesson
- box plot
- a picture of the five-number summary on a number line. For example: box from Q1 to Q3, line at the median.
- whisker
- the line from the box out to the smallest or largest value. For example: the line from Q3 to the maximum.
- five-number summary
- smallest, Q1, median, Q3, largest. For example: 2, 5, 8, 15, 20 for 2, 5, 7, 8, 12, 15, 20.
- consistent
- close together, with little spread. For example: a bus that is always 2 or 3 minutes late is consistent.
- compare
- say what is the same and what is different. For example: say what two groups share and where they differ.
- scale
- the numbers marked along an axis. For example: 0, 10, 20, … minutes.
Five numbers, one picture
A list of 9 waiting times is hard to hold in your head, and five numbers can say most of it: the smallest, the lower quartile, the median, the upper quartile and the largest. A box plot draws those five numbers on a number line. Predict first: out of nine values, about how many will end up inside the box once it is drawn? Then build it, one step at a time.
The data: the waiting times of nine patients, in minutes, as nine dots on a number line.
Show me the steps
- Step 0, the data: nine waiting times, 14, 22, 9, 18, 25, 11, 30, 16 and 20 minutes, as dots on a number line, not yet in order.
- Step 1, the ends: put them in order, 9, 11, 14, 16, 18, 20, 22, 25, 30. The smallest is 9 and the largest 30, so the range is 21.
- Step 2, the median: nine values, so the median is the 5th, 18. It goes in neither half.
- Step 3, the quartiles: the lower half 9, 11, 14, 16 has Q1 = (11 + 14) ÷ 2 = 12.5 and the upper half 20, 22, 25, 30 has Q3 = (22 + 25) ÷ 2 = 23.5. The IQR is 11.
- Step 4, the box plot: a box from 12.5 to 23.5, a line across it at 18, and whiskers from the box out to 9 and to 30.
A box plot (box-and-whisker plot) shows the five-number summary. The box runs from Q1 to Q3 with a line at the median, and the whiskers run out to the smallest and largest values. The box holds the middle half of the data, and each whisker holds a quarter.
To compare two groups, draw both on the same scale and compare the middles (the medians) and the spreads (the IQRs and the ranges), then say what the numbers mean in the context.
Why the same scale: a box that looks longer on a different scale may be shorter in numbers. Only a shared axis makes the picture an honest comparison.
Comparing two villages
People in two villages time how long it takes to fetch water, 11 people in each. In one comparison the medians are the same, 18 minutes. Predict first: if both medians are 18, will the two box plots look alike? Then press the first button to test it, and try the other two.
Village B's median is 26 against 18 for village A. The IQRs are equal, 6 and 6, so the two boxes are the same width and B's is shifted 8 minutes up.
Show me the steps
- Same middle, different spread: A is 12, 15, 18, 21, 24 and B is 6, 12, 18, 25, 32. Both medians are 18. A's IQR is 6 and B's is 13, and the ranges are 12 and 26. B's times are much less predictable.
- Higher middle, same spread: B is 20, 23, 26, 29, 32. B's median is 26, 8 minutes above A's 18. Both IQRs are 6, so B's box is the same width, just shifted up.
- Same box, one far value: B is 12, 15, 18, 21, 60. The box and median are exactly A's. Only the upper whisker is different: the largest time is 60, so the range is 48 against 12.
A common mistake: reading a long box or whisker as “more values”. A box plot shows where the quarters of the data lie, not how many values there are. A long whisker means the top quarter of the values is spread out over a wide range, not that there are more of them: in the last comparison the whisker is long because of a single time of 60.
Worked examples
The scores of 12 students: 34, 41, 28, 39, 45, 31, 50, 37, 42, 36, 47, 33. Find the five-number summary and describe the box plot.
- In order: 28, 31, 33, 34, 36, 37, 39, 41, 42, 45, 47, 50.
- 12 values: the median is (37 + 39) ÷ 2 = 38.
- Lower half 28, 31, 33, 34, 36, 37: Q1 = (33 + 34) ÷ 2 = 33.5. Upper half 39, 41, 42, 45, 47, 50: Q3 = (42 + 45) ÷ 2 = 43.5.
- Summary: 28, 33.5, 38, 43.5, 50. The box runs from 33.5 to 43.5 with a line at 38, and the whiskers go out to 28 and 50. IQR = 10, range = 22.
Phone battery life, in hours. Brand P: 8, 11, 13, 16, 19. Brand Q: 9, 12, 13, 14, 16 (smallest, Q1, median, Q3, largest). Compare the two brands.
- Medians: both are 13 hours, so a typical phone of each brand lasts equally long.
- IQRs: P is 16 − 11 = 5 and Q is 14 − 12 = 2. Ranges: P is 11 and Q is 7.
- Conclusion: the two brands have the same typical battery life, but Q is more consistent, since its middle half is narrower. P's best phones last longer (19 hours), and its worst last less (8).
Your go. The first step is given. Two groups' reaction times in milliseconds. Group M: 20, 30, 36, 44, 52. Group N: 25, 33, 38, 42, 47 (smallest, Q1, median, Q3, largest). Which group is faster, and which is more consistent?
- The medians are 36 for M and 38 for N, so M's typical time is 2 ms lower, and M is faster (a lower time is faster).
- Now find both IQRs and both ranges and decide which is more consistent, then check.
Check the answer
IQRs: M is 44 − 30 = 14, N is 42 − 33 = 9. Ranges: M is 32, N is 22. N is more consistent. M is slightly faster in the middle, but its times are more spread.
Check before you go on
1. A box plot summarizes the heights of 40 plants. About how many of the plants have a height inside the box, between Q1 and Q3?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A box plot of reaction times has whiskers ending at 6 and 30, a box from 12 to 22, and a line in the box at 15. What is the range?
3. The same box plot has a box from 12 to 22 with the line at 15. What is the interquartile range?
Level 2 · Building
4. Class P's marks have median 62 and IQR 8. Class Q's marks have median 58 and IQR 15. By how many marks is the median of class P higher than that of class Q?
5. Waiting times in minutes (smallest, Q1, median, Q3, largest). Hospital R: 5, 20, 30, 40, 95. Hospital S: 15, 25, 30, 35, 50. Which statement is best supported?
6. Nine values in order are 8, 10, 11, 13, 14, 16, 17, 19 and 21. The 21 is replaced by 70. What is the range of the new data?
Level 3 · Stretch
7. A team's times in minutes have smallest 20, Q1 24, median 30, Q3 36 and largest 60. Over how many minutes is the slowest quarter of the times spread, from Q3 to the largest?
8. A box plot summarizes 20 values. About how many of them lie between the median and Q3?
Something to think about
Two box plots have the same median, the same Q1 and the same Q3, but one has a much longer upper whisker. Name one real situation where that long whisker is the most important thing to know.
One way to think about it
For example, the time an ambulance takes to arrive: the middle half of the times can be identical for two towns, but a town whose slowest ambulance takes 60 minutes has a problem the box does not show. In safety and fairness questions the extreme values often matter more than the middle half.
Exit ticket
9. Daily sales at two cafés (smallest, Q1, median, Q3, largest). Café J: 20, 35, 40, 45, 90. Café K: 30, 38, 44, 50, 56. The manager says: “J is the busier café, because it had the busiest day.” Which reply is best?