The shape of the lesson
By the end of the lesson, learners can
- Mean, median, mode and range: In the lab, drag one dot far to the right and watch the four measures.
- Which average fits?: Each average has a job.
- Quartiles and the interquartile range: Extended only.
- Order the data before finding the median or quartiles.
- Mean: total \(\div\) number of values; to find a missing value, work with totals (mean \(\times\) n).
- Median at position \(\dfrac{n+1}{2}\); (Extended) quartiles at \(\dfrac{n+1}{4}\) and \(\dfrac{3(n+1)}{4}\).
- Spread: range = max \(-\) min; (Extended) IQR = UQ \(-\) LQ.
- Choosing: median when there is an outlier; mode for categories.
One picture every idea returns to
Mean \(= \dfrac{\text{sum of values}}{\text{number of values}}\); median = middle value once the data are IN ORDER, at position \(\dfrac{n+1}{2}\); mode = most common; range = largest \(-\) smallest. (Extended) Interquartile range = upper quartile \(-\) lower quartile.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, The middle, the most and the spread.. Three averages describe where the data sit, and the range and interquartile range describe how spread out they are. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Mean, median, mode and range | In the lab, drag one dot far to the right and watch the four measures. Project tab 1 of the lab (Averages and charts lab, averages and range) and let a student drive it while the class predicts. Model the worked example All four measures on the board, one line at a time. Your turn: A sixth number is added to the five numbers above, and the mean rises to 9. What is the sixth number? | Whole class, then pairs |
| 13 to 21 min | Teach and check: Which average fits? | Each average has a job. Model the worked example An outlier on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Quartiles and the interquartile range | Extended only. Model the worked example Quartiles of 11 values (Extended) on the board, one line at a time. Your turn: (Extended) Find the interquartile range of 14, 9, 22, 17, 11, 25, 19. | Numbered Heads Together |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 19 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Quartiles and the interquartile range, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Order the data BEFORE finding the median. | The middle of the unsorted list 12, 7, 9, 15, 7, 10, 18, 8 is 15 and 7, which gives a wrong 'median' of 11. And \(\dfrac{n+1}{2}\) is the POSITION of the median, not its value. |
| 'The mean, because it uses all the values' is not enough when there is an outlier. | Look at the data: if one value is far from the rest, the median is the better choice, and you must say why. |
| The IQR is one number, UQ \(-\) LQ. | Writing 'the IQR is 11 to 22' loses the mark, and so does finding quartiles from an unordered list. |
Averages, range and interquartile range: worksheet
6 questions, 19 marks. Written in the style of the papers; no past paper question is reproduced.
Find the mean, median, mode and range of 6, 2, 9, 4, 9, 3, 8.
The mean of 8 numbers is 12.5. One of the numbers, 20, is removed. Find the mean of the remaining 7 numbers.
(Extended) The marks of 15 students are
41 35 52 47 38 60 44 49 55 33 46 58 40 51 43
Find the median and the interquartile range.
The numbers of minutes eight buses were late are 2, 0, 3, 1, 2, 4, 1, 45.
Explain why the median is a better average to use than the mean, and find it.
Four whole numbers have a mode of 5, a median of 6 and a mean of 7. Find the four numbers.
(Extended) The heights, in cm, of plants grown with fertiliser X and fertiliser Y are
X: 31 24 28 35 22 30 26
Y: 27 33 18 29 40 21 36
(a) Find the median and IQR for each fertiliser. (b) A gardener wants tall plants of a similar height. Which fertiliser should she choose? Give two reasons.
Answer key and marking notes
- In order: 2, 3, 4, 6, 8, 9, 9 (\(n = 7\)).
- Mean \(= \dfrac{41}{7} = 5.86\) (3 s.f.)
- Median = 4th value = 6. Mode = 9. Range \(= 9 - 2 = 7\).
- Total \(= 8 \times 12.5 = 100\). New total \(= 100 - 20 = 80\).
- New mean \(= \dfrac{80}{7} = 11.4\) (3 s.f.)
- In order: 33, 35, 38, 40, 41, 43, 44, 46, 47, 49, 51, 52, 55, 58, 60.
- Median = 8th value = 46.
- LQ = 4th value = 40, UQ = 12th value = 52, so IQR \(= 52 - 40 = 12\).
- 45 is an extreme value that drags the mean up (mean \(= \dfrac{58}{8} = 7.25\) minutes, more than seven of the eight values).
- In order: 0, 1, 1, 2, 2, 3, 4, 45. Median \(= \dfrac{2+2}{2} = 2\) minutes.
- Total \(= 4 \times 7 = 28\).
- The mode is 5, so 5 appears at least twice. The median 6 is more than 5, so the two smallest numbers are 5 and 5.
- Median \(= \dfrac{5 + c}{2} = 6\), so the third number \(c = 7\).
- Fourth number \(= 28 - 5 - 5 - 7 = 11\). The numbers are 5, 5, 7, 11.
- X in order: 22, 24, 26, 28, 30, 31, 35. Median 28, LQ 24, UQ 31, IQR 7.
- Y in order: 18, 21, 27, 29, 33, 36, 40. Median 29, LQ 21, UQ 36, IQR 15.
- (b) The medians are almost equal (28 cm and 29 cm), so Y is only slightly taller on average.
- X has a much smaller IQR (7 cm against 15 cm), so its plants are far more consistent in height.
- Fertiliser X is the better choice for plants of a similar height.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Find the median of 14, 9, 21, 6, 17, 11.
- A12.5
- B13.5
- C13
- D3.5
The mean of seven numbers is 9. Six of the numbers are 4, 12, 7, 10, 15 and 8. Find the seventh number.
- A7
- B-2
- C9.33
- D63
A shop records the sizes of the last 9 pairs of shoes it sold: 4, 5, 5, 6, 6, 8, 8, 8, 11. The manager wants to know which size to order the most of. Which average should she use, and what is its value?
- AThe mode, size 8
- BThe median, size 6
- CThe mean, size 6.78
- DHalfway between the smallest and largest, size 7.5
The midnight temperatures, in °C, on six nights were \(-3\), 5, \(-7\), 2, 0, 4. Find the range.
- A12 °C
- B2 °C
- C-2 °C
- D8 °C
(Extended) The marks of 15 students are
27 41 35 19 52 30 44 23 38 48 33 26 56 40 31
Find the interquartile range.
- A17
- B37
- C7
- D9
(Extended) The points scored by two basketball players in seven games are
Priya: 24, 12, 30, 18, 35, 21, 26
Quinn: 22, 28, 17, 25, 19, 27, 23
Which statement is correct?
- APriya has the higher median, but Quinn's scores are more consistent.
- BPriya has the higher median and her scores are more consistent.
- CQuinn has the higher median, but Priya's scores are more consistent.
- DQuinn's scores are less consistent, because his IQR is smaller.
Quiz key
In order: 6, 9, 11, 14, 17, 21. There are 6 values, so the median is at position \(\dfrac{6+1}{2} = 3.5\), halfway between the 3rd and 4th values: \(\dfrac{11 + 14}{2} = 12.5\). The tempting 13.5 takes the middle two of the UNORDERED list, 21 and 6. 13 is the mean (\(78 \div 6\)), and 3.5 is the position of the median, not its value.
Total of all seven \(= 9 \times 7 = 63\). Total of the six known numbers \(= 4 + 12 + 7 + 10 + 15 + 8 = 56\). Seventh number \(= 63 - 56 = 7\). The tempting \(-2\) uses \(9 \times 6 = 54\), multiplying by the number of KNOWN values instead of all seven. 9.33 is the mean of the six numbers, and 63 is the total, not the missing number.
The most popular size is the one that occurs most often: size 8 appears three times, so the mode, 8, is the useful average. The median (the 5th value, 6) and the mean (\(61 \div 9 = 6.78\)) describe the centre, but no customer can buy size 6.78, and size 6 sold only twice. 7.5 is halfway between the smallest and largest sizes, \(\dfrac{4 + 11}{2}\), which ignores every other value and is not one of the averages.
The largest value is 5 and the smallest is \(-7\), so the range is \(5 - (-7) = 5 + 7 = 12\) °C. The tempting 2 and \(-2\) come from \(7 - 5\) and \(5 - 7\), treating \(-7\) as if it were 7. 8 uses \(-3\), the first negative in the list, as the smallest value.
In order: 19, 23, 26, 27, 30, 31, 33, 35, 38, 40, 41, 44, 48, 52, 56. With \(n = 15\), the LQ is at position \(\dfrac{16}{4} = 4\), so LQ = 27, and the UQ is at position \(\dfrac{3 \times 16}{4} = 12\), so UQ = 44. IQR \(= 44 - 27 = 17\). The tempting 37 is the range (\(56 - 19\)). 7 comes from taking the 4th and 12th values of the UNORDERED list (19 and 26), and 9 is UQ \(-\) median (\(44 - 35\)), only a quarter of the data.
Priya in order: 12, 18, 21, 24, 26, 30, 35, so median 24, LQ 18, UQ 30 and IQR 12. Quinn in order: 17, 19, 22, 23, 25, 27, 28, so median 23, LQ 19, UQ 27 and IQR 8. Priya's median is higher (\(24 > 23\)); Quinn's IQR is smaller (\(8 < 12\)), so his scores are more consistent. The tempting second option forgets that a LARGER IQR means more spread, so less consistent. The fourth option has the same mix-up the other way round.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.