At a glance. One lesson of about 60 minutes. Students draw and read box plots, find the range and IQR from them, and compare two groups on the same scale with a sentence that uses the numbers.
Calculator: no calculator. The data is short and the summaries are chosen to be exact. The learning is in reading the picture and writing the comparison.
What students will learn
Know
- A box plot shows the five-number summary: smallest, Q1, median, Q3, largest.
- The box holds the middle half; each whisker holds a quarter.
- Compare box plots on a shared scale by median, IQR and range.
Understand
- The same median can hide very different spread.
- A long whisker can come from a single extreme value.
- A comparison needs numbers and a context: what a larger or smaller value means here.
Do
- Draw a box plot from a list or from a five-number summary.
- Read the median, IQR and range from a box plot.
- Compare two data sets in a sentence that uses medians and spread.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: reading and calculating from box plots (questions 2 to 7).
- Criterion B. Investigating patterns: the village figure. Ask what changes, and what stays the same, between the three comparisons.
- Criterion C. Communicating: writing a comparison with the median, the IQR and a context in one sentence (the faded example and question 9).
- Criterion D. Applying mathematics in real-life contexts: water, hospitals and batteries, where a consistent service is the fair one.
The one thing to do with the figure
Do the prediction before the first button: both medians are 18, so will the box plots look alike? Then press the first button and let them see the wide second box.
Then use the third button: the box is identical and only the whisker differs. Ask: would you call these two villages the same? Which measure shows the difference (the range), and which does not (the IQR)? That contrast is the reason to report both.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Build the box plot with the first figure; students predict the number inside the box. |
| 10 min | Key card and the second figure with the village predictions. |
| 10 min | Worked examples; students write the comparison sentence themselves before reading it. |
| 5 min | Question 1, the check. Half, or a quarter? |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Hook | 5 of the 9: 14, 16, 18, 20 and 22 lie between Q1 = 12.5 and Q3 = 23.5, about half. |
|---|---|
| Warm-up | 7.5; 10; it makes small differences look large. |
| Worked example 1 | 28, 33.5, 38, 43.5, 50; IQR 10; range 22. |
| Worked example 2 | Same median 13; IQR 5 and 2; Q is more consistent. |
| Faded example | M is faster by 2 ms (median 36 against 38); IQRs 14 and 9, so N is more consistent. |
| 1. What the box holds | B, 20. A is one quarter; C adds a whisker; D forgets the whiskers. |
| 2. Range from the five numbers | 24. 30 − 6. |
| 3. IQR from a box plot | 10. 22 − 12. |
| 4. Difference of medians | 4. 62 − 58. |
| 5. Comparing two hospitals | A. B reads one extreme; C and D misread the medians. |
| 6. An extreme value and the range | 62. 70 − 8; the IQR stays 7.5. |
| 7. Spread of the top quarter | 24. 60 − 36. |
| 8. How many values between the marks | 5. 20 ÷ 4. |
| 9. Judging a claim from two box plots | B. A judges by one extreme day; C reads a small spread as busy; D reads box width as an amount. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Mixed up what the box and whiskers hold. | Count the dots in the first figure: how many are inside the box? |
| 10 or 36 in question 2 | Gave the IQR, or added the ends. | Ask where on the picture the range starts and ends. |
| 24, 34 or 12 in question 3 | Gave the range, added the ends of the box, or gave Q1. | Colour the box and ask for its width. |
| 7 or 120 in question 4 | Used the IQRs, or added the medians. | Underline the word median in the question. |
| A, C or D in question 9 | A: one extreme value. C: spread read as level. D: box width read as quantity. | Ask: what does each café sell on a typical day? Which number on the plot says that? |
| B in question 5 | Read the smallest value as the typical time. | Ask what each hospital's median says, and why the smallest value is not typical. |
| 13 or 49 in question 6 | Gave the old range, or the increase. | Ask which value is now the largest. |
| 12, 4 or 40 in question 7 | Gave the IQR, the lower whisker, or the whole range. | Mark Q3 and the largest on a line and measure between them. |
| 10 or 4 in question 8 | Gave the whole box, or divided by 5. | Count the marks: five marks make four sections. |
Questions worth asking
- What can you not see in a box plot? The number of values, the individual values, and the exact shape inside the box.
- Can two box plots with the same five numbers come from different data? Yes: only the five numbers are shown. 10 values and 100 values can give the same plot.
- When would the width of the box matter more than the median? When predictability is the point: delivery times, buses, waiting lists.
- Why must the plots share a scale? A longer box on a different scale may be a smaller IQR.
Differentiation
A way in for anyone stuck
- Give a printed number line and the five numbers; students only draw the box and whiskers.
- Use a sentence frame: “The medians are … so …; the IQRs are … so …”.
- Allow the figure's step list to be read aloud before the sliders are used.
Stretch
- Question 6, then: replace 21 by 70 and find the new Q3 and IQR. (Q3 stays 18 and the IQR stays 7.5.) Why? (The upper half's median uses its two middle values only.)
- Ask students to sketch two box plots for the same group, one before and one after a training programme, with a shift in the median but no change in spread, and describe it in a sentence.
Students learning in English as an additional language
A box plot is also called a box-and-whisker plot: the whiskers are the thin lines like a cat's whiskers. Point to each part of the figure as you say it.
| Word | Plain meaning | Example |
|---|---|---|
| box plot | a picture of the five-number summary on a number line | box from Q1 to Q3, line at the median |
| whisker | the line from the box out to the smallest or largest value | the line from Q3 to the maximum |
| five-number summary | smallest, Q1, median, Q3, largest | 2, 5, 8, 15, 20 for 2, 5, 7, 8, 12, 15, 20 |
| consistent | close together, with little spread | a bus that is always 2 or 3 minutes late is consistent |
| compare | say what is the same and what is different | say what two groups share and where they differ |
| scale | the numbers marked along an axis | 0, 10, 20, … minutes |
Sentence starters for explaining aloud or in writing:
- The medians are … and …, so …
- The IQR of … is … and of … is …, so … is more …
- Both box plots are on the same scale, which means …
Two things not to say
“The bigger box is better.” Bigger means more spread, which can be good or bad depending on the context.
“The median line is in the middle of the box.” It is not always: it moves with the data, and a lopsided box shows lopsided data.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ReadingA box plot has summary 3, 8, 12, 17, 25. State the median, the IQR and the range.
Median 12, IQR 9 (17 − 8), range 22 (25 − 3). - HalvesWhat fraction of the data lies between Q1 and the median?
A quarter.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ComparingTeam X: 40, 46, 50, 54, 60. Team Y: 38, 45, 50, 57, 64. Compare their scores.
The medians are both 50. X's IQR is 8 and Y's is 12, and the ranges are 20 and 26. X is more consistent. - DrawingFind the five-number summary of 3, 5, 6, 8, 10, 11, 13 and say where the median line sits in the box.
3, 5, 8, 11, 13. The median is 8, the middle of the box (the box is 5 to 11).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningGroup A's summary is 10, 20, 30, 40, 50 and group B's is 0, 20, 30, 40, 80. Which statements are true? (i) same median (ii) same IQR (iii) same range.
(i) and (ii) are true: both have median 30 and IQR 20. (iii) is false: 40 and 80. - From lesson 3Which of the mean, median and mode would most change if B's 80 became 800?
The mean. The median, Q1 and Q3 do not change.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider and the buttons work with the keyboard (tab, then the arrow keys or Enter) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Draw both box plots on squared paper on a shared axis, five numbers at a time.