The shape of the lesson
By the end of the lesson, learners can
- Three averages: In the frequency-table tab of the lab, change one frequency and watch which averages move.
- Estimates from grouped data: When data are grouped the raw values are lost, so every answer is an estimate.
- Measuring spread: The range uses only the two extreme values, so it is very sensitive to outliers.
- Comparing data sets: A comparison question wants two statements: one about average and one about spread (consistency), each written in the context of the question, and using matching measures for both sets.
- Mean: \(\bar{x} = \dfrac{\sum fx}{\sum f}\); for grouped data use midpoints.
- Missing or combined values: work with totals, \(\sum x = n\bar{x}\).
- Median and quartiles: find the position, then count along the cumulative frequencies.
- Choose: outliers or skew means median and IQR; otherwise mean and standard deviation.
- Compare: one average and one spread statement, in context.
One picture every idea returns to
The mean \(\bar{x} = \dfrac{\sum fx}{\sum f}\) uses every value, so one extreme value can drag it. The median and interquartile range use only positions, so they resist outliers and skew. Quote a centre AND a spread, and compare them in context.
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, Where it sits, how far it spreads.. A data set is summarised by one number for its centre and one for its spread. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Three averages | In the frequency-table tab of the lab, change one frequency and watch which averages move. Project tab 2 of the lab (Averages and charts lab, frequency tables) and let a student drive it while the class predicts. Model the worked example From a frequency table on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Estimates from grouped data | When data are grouped the raw values are lost, so every answer is an estimate. Model the worked example Estimated mean and median on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Measuring spread | The range uses only the two extreme values, so it is very sensitive to outliers. Model the worked example Choosing a measure on the board, one line at a time. Your turn: Find the interquartile range of 3, 7, 8, 11, 14, 15, 19, 22, 26, 30, 41. | Pairs |
| 23 to 30 min | Teach and check: Comparing data sets | A comparison question wants two statements: one about average and one about spread (consistency), each written in the context of the question, and using matching measures for both sets. Model the worked example Two runners on the board, one line at a time. | Round Table |
| 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, 21 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 Comparing data sets, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| Averaging the two means. \(\tfrac{1}{2}(64 + 59) = 61.5\) is wrong because the groups are different sizes; always go back to the totals \(\sum x\). | Return to Three averages: ask a pair to state the correct version and show it on the lab before moving on. |
| Using the upper class boundary (or the class width) instead of the midpoint in \(\sum fx\). | For \(30 \le t < 50\) the midpoint is 40, not 50 and not 20. |
| Quoting a quartile POSITION instead of a value: for 11 values \(Q_3\) is the 9th value (26), not 9. | Return to Measuring spread: ask a pair to state the correct version and show it on the lab before moving on. |
| Saying 'Ana's data are higher' without context. | In this question a lower time is better, so say 'faster', and refer to times, not just numbers. |
Measures of central tendency and variation: worksheet
6 questions, 21 marks. Written in the style of the papers; no past paper question is reproduced.
Find the mean, median and mode of 5, 8, 8, 9, 11, 12, 15.
The numbers of siblings \(x\) of 40 students are: \(x = 0\): 8, 1: 15, 2: 12, 3: 4, 4: 1. Find the mean, median and mode.
The masses \(m\) grams of 40 eggs are: \(50 \le m < 60\): 5, \(60 \le m < 65\): 12, \(65 \le m < 70\): 15, \(70 \le m < 80\): 8.
Estimate (a) the mean mass, (b) the median mass.
Five numbers have mean 14. Four of them are 10, 12, 15 and 19.
(a) Find the fifth number. (b) A sixth number \(x\) is added and the mean becomes 15. Find \(x\).
A class has 25 girls and 15 boys. The mean score of the girls is 58 and the mean score of the whole class is 55.3. Find the mean score of the boys.
The times, in seconds, of two sprinters in 9 races are:
Ali: 12.1, 12.3, 12.4, 12.4, 12.6, 12.7, 12.8, 13.0, 15.9
Bo: 12.0, 12.2, 12.5, 12.6, 12.7, 12.9, 13.0, 13.1, 13.2
(a) Find the mean, median and interquartile range of each sprinter's times. (b) Ali fell in one race. Which average should be used to choose the faster sprinter? Justify your choice and compare the two sprinters.
Answer key and marking notes
- Mean \(= \dfrac{68}{7} = 9.71\) (3 s.f.).
- Median 9 (4th value), mode 8.
- \(\sum fx = 0 + 15 + 24 + 12 + 4 = 55\), mean \(= 55 \div 40 = 1.375 = 1.38\) (3 s.f.).
- Median: the 20th and 21st values. Cumulative frequencies 8, 23, ... so both are 1. Median 1.
- Mode 1.
- (a) Midpoints 55, 62.5, 67.5, 75. \(\sum fm = 275 + 750 + 1012.5 + 600 = 2637.5\).
- Mean \(\approx 2637.5 \div 40 = 65.9\) g (3 s.f.).
- (b) Median: 20th egg. Cumulative frequencies 5, 17, 32, 40, so it is the 3rd of 15 in \(65 \le m < 70\).
- \(65 + \dfrac{3}{15} \times 5 = 66\) g.
- (a) Total \(= 5 \times 14 = 70\); \(70 - 56 = 14\).
- (b) New total \(= 6 \times 15 = 90\), so \(x = 90 - 70 = 20\).
- Class total \(= 40 \times 55.3 = 2212\); girls' total \(= 25 \times 58 = 1450\).
- Boys' total \(= 2212 - 1450 = 762\), so the boys' mean is \(762 \div 15 = 50.8\).
- (a) Ali: mean \(= 116.2 \div 9 = 12.9\) s; median 12.6 s; \(Q_1 = \tfrac{1}{2}(12.3 + 12.4) = 12.35\), \(Q_3 = \tfrac{1}{2}(12.8 + 13.0) = 12.9\), IQR 0.55 s.
- Bo: mean \(= 114.2 \div 9 = 12.7\) s (3 s.f.); median 12.7 s; \(Q_1 = 12.35\), \(Q_3 = 13.05\), IQR 0.7 s.
- (b) 15.9 is an outlier caused by the fall, so it distorts Ali's mean. Use the median (and IQR).
- Ali's median (12.6 s) is lower, so he is generally faster, and his smaller IQR shows he is more consistent. Choose Ali.
8 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.
The scores \(x\) of 25 players are: \(x = 1\): 3 players, \(x = 2\): 6, \(x = 3\): 9, \(x = 4\): 5, \(x = 5\): 2. Find the mean score.
- A2.88
- B14.4
- C3
- D5
The numbers \(x\) of pets owned by 40 families are: \(x = 0\): 6, \(x = 1\): 9, \(x = 2\): 5, \(x = 3\): 12, \(x = 4\): 8. Find the median.
- A2.5
- B2
- C3
- D8
The mean of 15 numbers is 22. Two more numbers are added and the mean of all 17 numbers becomes 23. Find the mean of the two numbers that were added.
- A30.5
- B61
- C24
- D22.5
A data set has 21 different values. The largest value is increased by 10 and nothing else changes. Which statement is true?
- AThe mean, range and standard deviation change, but the median and interquartile range do not
- BAll of the mean, median, range, interquartile range and standard deviation change
- COnly the range changes
- DThe mean and median change, but the range and interquartile range do not
The times \(t\) minutes taken by 40 students to travel to school are: \(0 \le t < 5\): 8, \(5 \le t < 10\): 14, \(10 \le t < 20\): 12, \(20 \le t < 40\): 6. Estimate the mean time.
- A12.1 minutes
- B16.5 minutes
- C7.75 minutes
- D121 minutes
Using the same data (\(0 \le t < 5\): 8, \(5 \le t < 10\): 14, \(10 \le t < 20\): 12, \(20 \le t < 40\): 6), estimate the median time by interpolation, taking the median as the 20th value.
- A9.29 minutes
- B9.46 minutes
- C7.5 minutes
- D10 minutes
In a class of 30 students, the 18 girls have a mean mark of 64 and the mean mark of the whole class is 61. Find the mean mark of the 12 boys.
- A56.5
- B58
- C59
- D22.6
The numbers \(x\) of goals scored by a team in 23 matches are: \(x = 0\): 3, \(x = 1\): 4, \(x = 2\): 6, \(x = 3\): 5, \(x = 4\): 3, \(x = 5\): 2. Find the interquartile range.
- A2
- B3
- C12
- D5
Quiz key
\(\sum fx = 3 + 12 + 27 + 20 + 10 = 72\) and \(\sum f = 25\), so \(\bar{x} = 72 \div 25 = 2.88\). The tempting 14.4 divides \(\sum fx\) by 5, the number of rows in the table. 3 is the mean of the five \(x\) values ignoring the frequencies, and 5 is the mean frequency.
Cumulative frequencies are 6, 15, 20, 32, 40. The 20th value is 2 and the 21st value is 3, so the median is \(\tfrac{1}{2}(2 + 3) = 2.5\). The tempting 2 takes only the 20th value. 3 is the mode (frequency 12), and 8 is the middle of the frequencies, which is not a data value at all.
Total before \(= 15 \times 22 = 330\); total after \(= 17 \times 23 = 391\). The two new numbers add to \(391 - 330 = 61\), so their mean is \(61 \div 2 = 30.5\). The tempting 61 is their total, not their mean. 22.5 averages the two means, and 24 assumes each new number is just one more than the new mean.
The largest value stays the largest, so the median (11th value) and the quartiles (6th and 16th values) are unchanged. The mean uses every value, so it rises by \(10 \div 21\); the range rises by 10; and the deviation of the largest value from the mean grows, so the standard deviation increases. The tempting "only the range" forgets that the mean and standard deviation use every value too.
Midpoints 2.5, 7.5, 15 and 30. \(\sum ft = 20 + 105 + 180 + 180 = 485\), so the mean is about \(485 \div 40 = 12.125\), or 12.1 minutes (3 s.f.). The tempting 16.5 uses the upper class boundaries 5, 10, 20, 40; 7.75 uses the lower boundaries, and 121 divides by the 4 classes instead of the 40 students.
Cumulative frequencies are 8, 22, 34, 40, so the 20th value is the 12th of the 14 values in \(5 \le t < 10\). Median \(\approx 5 + \dfrac{12}{14} \times 5 = 9.29\) minutes. The tempting 9.46 uses position 20.5, which is not the method asked for. 7.5 is just the midpoint of the median class, and 10 is its upper boundary.
Class total \(= 30 \times 61 = 1830\); girls' total \(= 18 \times 64 = 1152\). Boys' total \(= 678\), so their mean is \(678 \div 12 = 56.5\). The tempting 58 assumes 61 is the plain average of the two means (\(2 \times 61 - 64\)), which ignores the different group sizes. 59 swaps the group sizes (12 girls and 18 boys), and 22.6 divides the boys' total by 30.
Cumulative frequencies: 3, 7, 13, 18, 21, 23. The 6th value is 1, so \(Q_1 = 1\); the 18th value is 3, so \(Q_3 = 3\). IQR \(= 3 - 1 = 2\). The tempting 3 is the upper quartile on its own. 12 subtracts the positions (\(18 - 6\)), and 5 is the range.
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.