At a glance. One lesson of about 60 minutes. Students find the range, the quartiles, the IQR and the five-number summary of an ordered list by halving it at the median.
Calculator: no calculator. Nothing here needs one; halving a short list is the skill being built.
What students will learn
Know
- Range = largest − smallest.
- Q1 is the median of the lower half and Q3 the median of the upper half; the median goes in neither half for an odd count.
- IQR = Q3 − Q1 and the five-number summary is smallest, Q1, median, Q3, largest.
Understand
- The range depends on two extreme values and the IQR on the middle half.
- Quartiles are medians of halves, so the same method repeats.
- Two data sets with the same average can differ a lot in spread.
Do
- Order the data and find the five-number summary.
- Calculate the range and the IQR.
- Compare two data sets by both measures of spread.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding quartiles, range and IQR (questions 2 to 7).
- Criterion B. Investigating patterns: the figure. Ask what is the same and what is different between data sets 1 and 2, which have the same range.
- Criterion C. Communicating: writing the five-number summary in order, and using “spread” and “middle half” in a sentence (question 9).
- Criterion D. Applying mathematics in real-life contexts: reliability of routes and deliveries, where spread matters as much as the typical value.
The one thing to do with the figure
Ask for the prediction first: route A or route B? Most will say A, and some will say “it depends how late I can be”. Welcome that: it is the idea of spread.
Then use the figure to ask: sets 1 and 2 have the same range, 15. Do they have the same IQR? (9 and 8.) And in set 3, if the 30 were 23 instead, which would change, the range or the IQR? (The range falls from 18 to 11; the IQR stays 7, because Q3 is still 21.)
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook with the two routes; compute each route's range on the board; predictions. |
| 10 min | Key card and the figure: split the data into halves, read Q1 and Q3. |
| 10 min | Worked examples, odd then even, with students halving the lists themselves. |
| 5 min | Question 1, the check. Hands up for each letter. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Hook | Route A: its range is 19 − 17 = 2 minutes against 28 − 8 = 20 for route B, so A is the one to plan around. The number besides the mean is a measure of spread. |
|---|---|
| Warm-up | The mean; 8; 10.5. |
| Worked example 1 | Summary 6, 9, 14, 18, 23; range 17; IQR 9. |
| Worked example 2 | Median 11, Q1 6.5, Q3 15, IQR 8.5. |
| Faded example | Median 30, Q1 25, Q3 38, range 23, IQR 13. |
| 1. Lower quartile of an odd list | A, 5. C leaves the median in the lower half; D gives Q2; B gives the top of the half. |
| 2. Range | 23 mm. 31 − 8. |
| 3. IQR from the quartiles | 8. 20 − 12. |
| 4. Upper quartile of an odd list | 16.5. (15 + 18) ÷ 2. |
| 5. IQR of an even list | 7.5. 17 − 9.5. |
| 6. Comparing range and IQR | B. Ranges 30 and 40; IQRs 10 and 6. |
| 7. Finding a quartile from the IQR | 10. 19 − 9. |
| 8. A missing value from a quartile | 8. (5 + k) ÷ 2 = 6.5. |
| 9. Reading IQR in context | A. B reads spread as speed; C reads the IQR as the range; D has no basis. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Left the median in the lower half, or gave Q2 or the end of the half. | Cross the median out of the list and ask for the middle of what is left. |
| 39 or 31 or 8 in question 2 | Added the extremes, or gave one of them. | Ask how far apart the largest and smallest are on a number line. |
| 32 or 16 in question 3 | Added the quartiles or took their mean. | Ask what the bracket in the figure measures. |
| 15 or 18 in question 4 | 15: median left in the upper half. 18: took one of the two middle values. | Cross the median out first, then find the middle of four. |
| 13 or 26.5 in question 5 | Gave the range or added the quartiles. | Ask which numbers the IQR bracket starts and ends at. |
| 28 or 5 in question 7 | Added the IQR to Q3, or subtracted it from the median. | Draw Q1, the median and Q3 on a line and mark the 9. |
| 13 or 6.5 in question 8 | Solved half the equation, or gave the quartile. | Write Q1 = (5 + k) ÷ 2 first, then solve. |
Questions worth asking
- Why does the median go in neither half when the count is odd? It is in the middle: it cannot belong to the lower half or the upper half more than the other.
- Which is more affected by one extreme value, the range or the IQR? The range: it uses the largest and smallest directly.
- Can two data sets have the same range but different IQRs? Yes. Sets 1 and 2 in the figure.
- Can the IQR be zero? Yes, when Q1 and Q3 land on the same value, such as 2, 5, 5, 5, 5, 5, 5, 9 (Q1 = Q3 = 5). Half the values being equal is not enough: 1, 2, 3, 4, 5, 5, 5, 5 has an IQR of 2.5.
Differentiation
A way in for anyone stuck
- Provide the list written on a strip that students cut in half, then cut each half in half.
- Give lists already in order for the first questions; ordering is a separate skill.
- Colour the lower and upper halves differently in the working, as in the figure.
Stretch
- Find a data set of 8 values with range 20 and IQR 2. (For example 0, 10, 10, 10, 12, 12, 12, 20.)
- Question 8, then: what values of k make the lower quartile of 3, 5, k, 9, 12, 14, 17, 18, 20 as large as possible, if k must stay in order? (k = 9 gives Q1 = 7.)
Students learning in English as an additional language
Quartile comes from the Latin for four. Say it with “quarter” and point to the four groups in the figure.
| Word | Plain meaning | Example |
|---|---|---|
| range | largest value minus smallest value | in 4, 9, 15 the range is 11 |
| quartile | one of three values that cut ordered data into four parts of about equal size | Q1, Q2 (the median), Q3 |
| lower half | the values below the median | 2 and 4 in 2, 4, 7, 10, 13 |
| interquartile range (IQR) | Q3 minus Q1, the spread of the middle half | 35 − 24 = 11 |
| spread | how far apart the values are | marks from 20 to 95 are more spread than marks from 60 to 70 |
| five-number summary | smallest, Q1, median, Q3, largest | 2, 5, 8, 15, 20 for 2, 5, 7, 8, 12, 15, 20 |
Sentence starters for explaining aloud or in writing:
- The median is …, so the lower half is … and Q1 is …
- The range is … but the IQR is …, so the middle half is …
- … is more spread out because its IQR is …
Two things not to say
“Quartiles are at 25%, 50% and 75% of the way along the range.” They are positions in the ordered list, not on the number line.
“The IQR is the range of the middle.” Say “Q3 minus Q1” and name the middle half.
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.
- RangeFind the range of 14, 9, 22, 18, 11.
13. 22 − 9. - IQRQ1 is 33 and Q3 is 51. Find the IQR.
18. 51 − 33.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- QuartilesFind Q1 and Q3 of 5, 8, 9, 12, 14, 15, 20.
Q1 = 8 and Q3 = 15. The median 12 goes in neither half. - Even listFind the median and the IQR of 10, 12, 15, 19, 20, 26.
Median 17; Q1 = 12, Q3 = 20, so the IQR is 8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA set of 8 values has Q1 = 6 and Q3 = 14. The largest value is replaced by one twice as big. What happens to Q3?
Nothing: Q3 is halfway between the 6th and 7th values, and only the 8th value changes. - From lesson 2The mean of 5 values is 12. Four of them are 8, 10, 14 and 18. Find the fifth.
10. The total is 60 and the four add to 50.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard (tab, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives all four data sets in words.
No device? Print the four data sets as strips and let students fold them in half.