Boxplots and the five-number summary
Comparing the centre, spread and shape of numerical data with the minimum, quartiles, median and maximum
About three lessons of 45 to 60 minutes
Two classes both averaged 75%. Were they really the same?
A teacher tells you two classes scored a median of 75% on the same test. Sounds identical, until you see the actual marks: Class A ranges from 40% to 98%, with students scattered right across that range. Class B mostly sits tightly between 68% and 82%, with barely anyone outside it. Same middle, completely different story.
The median alone hides this. A boxplot, built from five key numbers, shows the middle AND the spread at a glance, so you can see instantly whether a group is consistent or all over the place, and compare two or more groups fairly.
- Two classes' test scoressame median, but one class is far more spread out than the other
- House prices in two suburbsboxplots reveal typical price and how wide the range really is
- Runners' race timesspot which runner is more consistent, not just who is fastest on average
- Monthly rainfall over a decadecompare a wet year's spread with a dry year's, not just the yearly total
What students will be able to do
Students will find the five-number summary (minimum, lower quartile, median, upper quartile, maximum) of a numerical data set, calculate the interquartile range, and use these measures to compare the centre, spread and shape of two or more distributions.
- I can sort a data set and identify its minimum and maximum values.
- I can find the median of a sorted data set.
- I can find the lower quartile (Q1) and upper quartile (Q3) as the median of the lower and upper halves of the data.
- I can calculate the interquartile range (IQR) as Q3 minus Q1, and explain what it measures.
- I can compare two data sets using their five-number summaries, and say which is more spread out.
Standards this unit teaches
- AC9M10ST02Australian Curriculum v9 (ACARA)Boxplots and distributions
Compare distributions of continuous numerical data with displays such as boxplots, discussing centre, spread, shape and outliers.
- AC9M9ST03Australian Curriculum v9 (ACARA)Compare data distributions (Year 9 bridge)
Represent several numerical data sets with comparative displays and compare their centre, spread, shape and the effect of outliers. This unit's five-number summary work also reaches toward this Year 9 descriptor.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Five-number summary
- the minimum, lower quartile, median, upper quartile and maximum of a sorted data set
- Median
- the middle value of sorted data
- Lower quartile (Q1)
- the median of the lower half of the sorted data, below which about a quarter of values fall
- Upper quartile (Q3)
- the median of the upper half of the sorted data, below which about three quarters of values fall
- Interquartile range (IQR)
- Q3 minus Q1, the spread of the middle 50 per cent of the data
- Boxplot
- a diagram built from the five-number summary that shows a data set's centre and spread at a glance
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. The five-number summary
ConcreteSort a data set from smallest to largest, and five values tell you almost everything about its shape: the minimum, the lower quartile, the median, the upper quartile, and the maximum.
The median is the middle of the whole sorted list. The lower quartile (Q1) is the median of everything below the overall median; the upper quartile (Q3) is the median of everything above it. Together with the minimum and maximum, these five numbers are the five-number summary.
Find the five-number summary of Class A's sorted scores: 12, 15, 18, 22, 25, 29, 33, 38.
- Minimum = 12, maximum = 38 (the first and last sorted values).
- There are 8 values (even), so the median is the mean of the 4th and 5th: (22 + 25) / 2 = 23.5.
- Lower half is 12, 15, 18, 22; its median (Q1) is (15 + 18) / 2 = 16.5.
- Upper half is 25, 29, 33, 38; its median (Q3) is (29 + 33) / 2 = 31.
Answer: Five-number summary: min 12, Q1 16.5, median 23.5, Q3 31, max 38.
- Why do you sort the data before finding any of the five numbers?
- How is finding Q1 similar to finding the overall median?
2. The interquartile range: measuring the middle spread
PictorialThe interquartile range (IQR) is Q3 minus Q1. It measures how spread out the MIDDLE 50 per cent of the data is, ignoring the most extreme quarter at each end, which makes it a more robust spread measure than the full range when there are outliers.
For Class A, IQR = 31 - 16.5 = 14.5. A small IQR means the middle half of the data is tightly bunched; a large IQR means it is spread widely. Comparing IQRs is often more informative than comparing full ranges, because one extreme value cannot distort it.
Class B's sorted scores are 20, 21, 22, 23, 24, 26, 27, 28. Find the five-number summary and the IQR.
- Minimum = 20, maximum = 28.
- 8 values, so median = (23 + 24) / 2 = 23.5.
- Lower half 20, 21, 22, 23: Q1 = (21 + 22) / 2 = 21.5.
- Upper half 24, 26, 27, 28: Q3 = (26 + 27) / 2 = 26.5.
- IQR = Q3 - Q1 = 26.5 - 21.5 = 5.
Answer: Five-number summary: min 20, Q1 21.5, median 23.5, Q3 26.5, max 28. IQR = 5.
- Class A's IQR is 14.5 and Class B's is 5. Which class's middle 50% of scores is more tightly clustered?
- Why might the IQR be a fairer spread measure than the full range if one student scored very low by mistake?
3. Comparing distributions: centre, spread and shape
AbstractBack to the hook: Class A and Class B both have a median of 23.5 out of 40, exactly the same centre. But Class A's IQR (14.5) is nearly three times Class B's (5), so Class A's scores are far more spread out even though the 'typical' student did the same in both classes.
When comparing two boxplots, always talk about all three things: centre (compare the medians), spread (compare the IQRs or ranges), and shape (is the box centred in the whiskers, or shifted toward one end, suggesting the data is not symmetric?). A single number, like the median alone, never tells the whole story.
- Two data sets have the same median but different IQRs. What does that tell you about how they differ?
- If a boxplot's box sits much closer to one whisker than the other, what does that suggest about the shape of the data?
Common misconceptions and how to address them
MisconceptionThe quartiles split the data into four groups of exactly equal size no matter what.
Why it happens: Students expect a perfectly even split, without accounting for how the median is handled when the count is odd.
How to address it: Quartiles are found by taking medians of halves. With an odd total count, whether the overall median itself is included in both halves is a convention some methods differ on; the version taught here (median of the strictly lower and strictly upper values) is the one to use consistently.
MisconceptionA bigger range always means a bigger IQR.
Why it happens: Students assume all spread measures move together.
How to address it: A single extreme outlier can make the range huge while the IQR, which ignores the outer quarters, stays small. Show a data set with one huge outlier: the range balloons but the IQR barely changes.
MisconceptionYou cannot compare two data sets unless they have the same number of values.
Why it happens: Students confuse comparing counts with comparing summaries.
How to address it: The five-number summary and IQR work for data sets of any size, including different sizes, because they describe the shape of the distribution, not how many values there are.
Guided practice (with answers)
1. Find the median of the sorted data 4, 7, 9, 12, 15, 20 (6 values).
Answer: 10.5, because with an even count the median is the mean of the two middle values, 9 and 12: (9+12)/2 = 10.5.
2. Data set (sorted): 2, 5, 8, 11, 14, 17, 20, 23. Find Q1.
Answer: 6.5, because the lower half is 2, 5, 8, 11 and its median is (5+8)/2 = 6.5.
3. Using the same data set (2, 5, 8, 11, 14, 17, 20, 23), find Q3.
Answer: 18.5, because the upper half is 14, 17, 20, 23 and its median is (17+20)/2 = 18.5.
4. Find the IQR for that data set.
Answer: 12, because IQR = Q3 - Q1 = 18.5 - 6.5 = 12.
5. Data set A has IQR 3 and data set B has IQR 18, with the same median. Which set is more consistent?
Answer: Data set A, because a smaller IQR means the middle 50% of its values are packed closer together.
Independent practice worksheets
Practise finding the five-number summary and comparing distributions with computed, never-wrong answer keys.
Differentiation
- Start with small, already-sorted data sets (6 to 8 values) so students focus on the quartile logic, not sorting.
- Colour-code the lower half, median and upper half of the sorted list before finding Q1 and Q3.
- Use a simple table with columns for min, Q1, median, Q3, max so the five numbers are never confused with each other.
- Physically fold a sorted list of number cards in half, then in half again, to make the quartile positions tangible.
- Introduce data sets with an odd number of values, and discuss how the quartile convention handles the middle value.
- Ask students to construct a data set with a specified five-number summary.
- Compare three or more real data sets (e.g. three classes' results) using all five numbers and the IQR, and justify which is most consistent.
- Investigate how adding one extreme value changes the median and IQR very differently from how it changes the mean and range.
Assessment: exit ticket
A three-question exit ticket sampling the five-number summary, the IQR, and comparing two distributions.
1. Data set (sorted): 3, 6, 9, 12, 15, 18. Find the median.
Answer: 10.5, because with 6 values the median is the mean of the two middle values, 9 and 12: (9+12)/2 = 10.5.
2. Using the same data set, find Q1 and Q3.
Answer: Q1 = 6 (median of 3, 6, 9), Q3 = 15 (median of 12, 15, 18).
3. Class X has median 60 and IQR 8. Class Y has median 60 and IQR 22. Which class's scores are more spread out, and how do you know?
Answer: Class Y, because a larger IQR means the middle 50% of its scores are spread across a wider range, even though both classes share the same median.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 the five-number summary (section 1), Lesson 2 the IQR (section 2), Lesson 3 comparing distributions plus the exit ticket (section 3 and assessment).
- This unit assumes comfort with finding a median (Year 7 statistics unit). Revisit that first if sorting and locating the middle value is still shaky.
- The convention used here for quartiles is 'median of the strictly lower/upper half', which matches how this site's own boxplot worksheets are generated (lib/content_ausecondarymath.ts's fiveNumberSummary). Some textbooks include the overall median in both halves for an odd count; either convention is acceptable if applied consistently.
- Language to repeat: the five-number summary is min, Q1, median, Q3, max, in that order, and IQR is Q3 minus Q1, the spread of the middle half.
- Curriculum note: AC9M10ST02 (Australian Curriculum v9) is the Year 10 descriptor for boxplots; AC9M9ST03 introduces comparing distributions at Year 9. This unit's independent practice links to both year levels' worksheets.
- Present and print both work: use the Print button for a clean handout, or project the number-line diagram and build the five-number summary with the class live.