The shape of the lesson
By the end of the lesson, learners can
- Histograms with unequal widths: Open the histogram tab of the lab and widen one class.
- Stem-and-leaf and box plots: A stem-and-leaf diagram keeps every raw value while showing the shape.
- Cumulative frequency graphs: Plot each cumulative frequency at the UPPER class boundary and join the points (with a curve or straight lines).
- Choosing a display: Paper 5 asks you to choose a representation and to give an advantage or disadvantage.
- Boundaries: for rounded data, extend each class by half a unit each way before finding widths.
- Histograms: height = frequency ÷ width; frequency = width × height.
- Stem-and-leaf: order the data, give a key for both sides, read left leaves backwards.
- Cumulative frequency: plot at upper boundaries; read at \(\tfrac{n}{4}\), \(\tfrac{n}{2}\), \(\tfrac{3n}{4}\) or \(\tfrac{k}{100}n\).
- Compare: one comment on average, one on spread, both in context.
One picture every idea returns to
In a histogram the AREA of a bar is the frequency, so the height is the frequency density: \(\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}\). On a cumulative frequency graph, read across from a position (such as \(\tfrac{n}{2}\) for the median) and down to the value.
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, Pictures of data.. Four displays come up again and again on Paper 5. 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: Histograms with unequal widths | Open the histogram tab of the lab and widen one class. Project tab 2 of the lab (Cumulative frequency lab, histograms) and let a student drive it while the class predicts. Model the worked example Frequency densities on the board, one line at a time. Your turn: A class \(10 \le x < 25\) has frequency 36. Its bar is drawn 4.8 cm high. The bar for \(25 \le x < 30\) is 5.6 cm high. Find the frequency of \(25 \le x < 30\). | Whole class, then pairs |
| 11 to 17 min | Teach and check: Stem-and-leaf and box plots | A stem-and-leaf diagram keeps every raw value while showing the shape. Project tab 4 of the lab (Averages and charts lab, stem-and-leaf) and let a student drive it while the class predicts. Model the worked example Quartiles from a back-to-back diagram on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Cumulative frequency graphs | Plot each cumulative frequency at the UPPER class boundary and join the points (with a curve or straight lines). Project tab 1 of the lab (Cumulative frequency lab, cumulative frequency) and let a student drive it while the class predicts. Model the worked example Median, quartiles and a percentile on the board, one line at a time. Your turn: Using the same table, estimate the number of students with heights between 155 cm and 175 cm. | Pairs |
| 23 to 30 min | Teach and check: Choosing a display | Paper 5 asks you to choose a representation and to give an advantage or disadvantage. | Mix-Pair-Share |
| 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 Choosing a display, 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.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
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 |
|---|---|
| Plotting frequency (not frequency density) on the vertical axis when the class widths differ. | It makes wide classes look far too important and scores no marks for the heights. |
| Forgetting to reverse the left-hand leaves. | In the row \(8\ 6\ 5\ 2\ 0 \mid 3\), the smallest Class A value is 30 (the leaf next to the stem), not 38. |
| Plotting cumulative frequencies at the class midpoints. | A cumulative frequency counts everything UP TO the end of a class, so it belongs at the upper boundary. |
Stem-and-leaf, box plots, histograms and cumulative frequency: worksheet
6 questions, 21 marks. Written in the style of the papers; no past paper question is reproduced.
Lengths of leaves are measured to the nearest centimetre and grouped as 5-9, 10-14, 15-24. State the class boundaries and class width of the class 15-24.
The times \(t\) seconds taken by 60 people are: \(0 \le t < 5\): 10, \(5 \le t < 10\): 16, \(10 \le t < 20\): 24, \(20 \le t < 40\): 10.
On a histogram the bar for \(5 \le t < 10\) is 8 cm high. Find the heights of the other three bars.
The reaction times, in hundredths of a second, of 11 boys and 11 girls are:
Boys: 21, 24, 25, 28, 30, 31, 33, 35, 36, 40, 46
Girls: 19, 22, 23, 23, 26, 27, 27, 29, 32, 34, 38
(a) Draw a back-to-back stem-and-leaf diagram with a key. (b) Find the median and interquartile range for each group and compare the two groups.
The speeds \(v\) km/h of 120 cars give cumulative frequencies \(v \le 40\): 8, \(v \le 50\): 30, \(v \le 60\): 78, \(v \le 70\): 108, \(v \le 80\): 120. No car travelled slower than 30 km/h.
Estimate (a) the median speed, (b) the number of cars travelling faster than 65 km/h, (c) the 80th percentile.
A histogram of the masses \(m\) grams of some apples has bars \(0 \le m < 20\) height 0.6, \(20 \le m < 30\) height 2.8, \(30 \le m < 35\) height 4 and \(35 \le m < 50\) height 1.2 (heights are frequency densities).
(a) Find the total number of apples. (b) Estimate the number of apples with masses between 25 g and 40 g.
The lengths, in cm, of 60 fish are summarised: minimum 12, lower quartile 18, median 21, upper quartile 27, maximum 40.
(a) Describe the skewness of the distribution, giving a reason. (b) A second sample of 60 fish is to be compared with this one. Give one advantage and one disadvantage of using box-and-whisker plots rather than histograms for the comparison.
Answer key and marking notes
- Boundaries 14.5 and 24.5.
- Width \(= 24.5 - 14.5 = 10\).
- Frequency densities: \(10 \div 5 = 2\), \(16 \div 5 = 3.2\), \(24 \div 10 = 2.4\), \(10 \div 20 = 0.5\).
- 3.2 is drawn as 8 cm, so 1 unit of density is 2.5 cm.
- Heights: \(2 \times 2.5 = 5\) cm, \(2.4 \times 2.5 = 6\) cm, \(0.5 \times 2.5 = 1.25\) cm.
- (a) Stems 1 to 4. Boys (left): \(\mid 2 \mid\) leaves 8 5 4 1; \(\mid 3 \mid\) leaves 6 5 3 1 0; \(\mid 4 \mid\) leaves 6 0. Girls (right): 1 | 9; 2 | 2 3 3 6 7 7 9; 3 | 2 4 8. Key: 1 | 2 | 6 means 21 for boys and 26 for girls.
- (b) Boys: \(Q_1 = 25\) (3rd), median 31 (6th), \(Q_3 = 36\) (9th), IQR 11.
- Girls: \(Q_1 = 23\), median 27, \(Q_3 = 32\), IQR 9.
- The girls reacted faster on average (median 27 against 31) and their times were slightly less spread out (IQR 9 against 11).
- (a) 60th car: \(50 + \dfrac{60 - 30}{48} \times 10 = 56.25\), about 56.3 km/h.
- (b) At 65 km/h: \(78 + 0.5 \times 30 = 93\), so \(120 - 93 = 27\) cars.
- (c) 80% of 120 is 96: \(60 + \dfrac{96 - 78}{30} \times 10 = 66\) km/h.
- (a) Frequencies: \(20 \times 0.6 = 12\), \(10 \times 2.8 = 28\), \(5 \times 4 = 20\), \(15 \times 1.2 = 18\). Total 78.
- (b) \(5 \times 2.8 + 20 + 5 \times 1.2 = 14 + 20 + 6 = 40\) apples.
- (a) \(Q_3 - Q_2 = 6\) is greater than \(Q_2 - Q_1 = 3\), and the upper whisker (13) is longer than the lower (6), so the data are positively skewed.
- (b) Advantage: two box plots on one scale show the medians, quartiles and spreads side by side, which makes comparison easy.
- Disadvantage: the detailed shape of each distribution (for example, how many fish are in each length band) is lost.
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 heights of 115 plants, measured to the nearest centimetre, are grouped as 150-159 (40 plants), 160-164 (30 plants) and 165-179 (45 plants). Find the frequency density of the class 160-164.
- A6
- B7.5
- C30
- D0.167
A back-to-back stem-and-leaf diagram has the key 3 | 4 | 1, meaning 43 for set P and 41 for set Q. One row reads \(7\ 5\ 2 \mid 5 \mid 0\ 8\). What is the smallest value of set P in this row?
- A52
- B57
- C50
- D25
Find the interquartile range of these 15 ordered values: 11, 14, 15, 18, 20, 22, 23, 25, 27, 29, 30, 34, 38, 41, 56.
- A16
- B13
- C45
- D8
The times taken by 500 runners to finish a race are recorded to the nearest second. The organisers want to estimate the time beaten by the fastest 10% of runners. Which display is most suitable?
- AA cumulative frequency graph
- BA back-to-back stem-and-leaf diagram
- CA box-and-whisker plot
- DA bar chart of the raw times
The histogram shows the distribution of a continuous variable \(x\). The frequency density of each bar is marked on it. Estimate the number of values with \(12 \le x < 26\).
- A55
- B74
- C39
- D12.2
The cumulative frequencies of the times \(t\) minutes taken by 160 people are \(t \le 10\): 14, \(t \le 20\): 46, \(t \le 30\): 98, \(t \le 40\): 140, \(t \le 60\): 160. Joining the points with straight lines, estimate the number of people who took more than 35 minutes.
- A41
- B119
- C21
- D62
The masses of some parcels, to the nearest kilogram, are grouped. The class 10-29 has frequency 30 and the class 30-34 has frequency 18. On a histogram the bar for 10-29 is 3 cm high. How high is the bar for 30-34?
- A7.2 cm
- B9 cm
- C1.8 cm
- D3.6 cm
A box-and-whisker plot of some waiting times, in minutes, has minimum 5, lower quartile 12, median 15, upper quartile 24 and maximum 30. Which statement is correct?
- AThe data are positively skewed, because \(Q_3 - Q_2\) is greater than \(Q_2 - Q_1\)
- BThe data are negatively skewed, because the upper whisker is shorter than the lower whisker
- CThe data are negatively skewed, because the median is closer to \(Q_1\) than to \(Q_3\)
- DThe data are symmetrical, because the two whiskers are almost the same length
Quiz key
The class 160-164 runs from 159.5 to 164.5, so its width is 5 and the frequency density is \(30 \div 5 = 6\). The tempting 7.5 uses a width of \(164 - 160 = 4\), forgetting the class boundaries. 30 is the frequency itself, and 0.167 is width divided by frequency, the wrong way round.
The P leaves are 7, 5 and 2, written outwards from the stem, so the P values in this row are 52, 55 and 57, and the smallest is 52 (the leaf next to the stem). The tempting 57 reads the left leaves from left to right as if they were in increasing order. 50 is the smallest Q value in the row, and 25 puts the leaf in front of the stem.
With 15 values the median is the 8th value (25). \(Q_1\) is the median of the lower 7 values, the 4th value, 18; \(Q_3\) is the 12th value, 34. IQR \(= 34 - 18 = 16\). The tempting 13 includes the median in each half (8 values each), giving \(Q_1 = 19\) and \(Q_3 = 32\). 45 is the range, and 8 subtracts the positions 12 and 4 instead of the values.
The fastest 10% finish below the 10th percentile, which is read from a cumulative frequency graph by going across from \(0.1 \times 500 = 50\) and down to the time. A stem-and-leaf diagram of 500 values would be far too unwieldy. The tempting box-and-whisker plot shows only the quartiles, median and extremes, so the 10th percentile cannot be read from it. A bar chart is for discrete or categorical data, not hundreds of different times.
From 12 to 15: \(3 \times 4.4 = 13.2\). From 15 to 20: the whole bar, \(5 \times 5.2 = 26\). From 20 to 26: \(6 \times 2.6 = 15.6\). Total \(13.2 + 26 + 15.6 = 54.8\), about 55. The tempting 74 uses the whole of the three bars from 10 to 30. 39 stops at 20 and leaves out the part from 20 to 26, and 12.2 adds the heights \(4.4 + 5.2 + 2.6\) as if they were frequencies.
35 is halfway from 30 to 40, so the cumulative frequency there is \(98 + 0.5 \times (140 - 98) = 119\). More than 35 minutes: \(160 - 119 = 41\). The tempting 119 is the number who took 35 minutes or less. 21 is \(140 - 119\), only those between 35 and 40 minutes, and 62 is \(160 - 98\), those taking more than 30 minutes.
Boundaries give widths of \(29.5 - 9.5 = 20\) and \(34.5 - 29.5 = 5\). Densities: \(30 \div 20 = 1.5\) and \(18 \div 5 = 3.6\). 1.5 is drawn as 3 cm, so 1 unit is 2 cm and the second bar is \(3.6 \times 2 = 7.2\) cm. The tempting 9 cm uses a width of 4 for 30-34 (density 4.5). 1.8 cm scales by frequency only, \(\tfrac{18}{30} \times 3\), ignoring the widths, and 3.6 cm is the density without the scale.
\(Q_3 - Q_2 = 9\) and \(Q_2 - Q_1 = 3\): the upper half of the box is three times as long, so the middle half of the data is stretched towards high values. That is positive skew. The whiskers (7 below, 6 above) are close in length, which tempts some to say negative skew or symmetry, but the box carries the main evidence here. A median close to \(Q_1\) indicates positive skew, not negative.
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.