IGCSE / O Level Mathematics · Topic 9 Statistics · Chapter 36 · pp. 510–531
Collecting, displaying and interpreting data
Statistics starts with a pile of numbers. This chapter shows how to organise them (tally charts, stem and leaf, grouped tables), draw them (bar charts, pie charts, histograms), and look for patterns between two sets of data (scatter diagrams and correlation).
Learning objectives
- Collect data in a tally chart and a frequency table, and show it as a pictogram or bar chart.
- Draw and read stem and leaf diagrams, including back-to-back diagrams, and find the median from them.
- Group data into a grouped frequency table with no overlaps.
- Draw pie charts and work out sector angles and frequencies.
- Plot scatter diagrams, describe correlation and use a line of best fit.
- Draw histograms, using frequency density when the class widths are unequal.
Book exercises: 36.1 to 36.5 and Student assessment 1 (pp. 510–531).
Tally charts and frequency tables
🔑 Key ideas
- A tally chart records each value as you go: strokes in groups of five.
- Adding the tallies gives the frequency. The chart is then a frequency table.
- A pictogram uses a symbol (with a key) for a number of items. A bar chart uses bars, with gaps between them.
- The mode is the most common value: the tallest bar.
| Buttons in packet | Tally | Frequency |
|---|---|---|
| 35 | |||| | 4 |
| 36 | |||| || | 7 |
| 37 | ||| | 3 |
| 38 | |||| | | 6 |
💡 Real tallies draw the fifth stroke across the first four. Here the groups are written as spaced strokes so they print clearly. 4 + 7 + 3 + 6 = 20 packets.
Stem and leaf diagrams
🔑 Key ideas
- For discrete data. Each value is split into a stem (tens) and a leaf (units).
- Leaves go in order, smallest first. Always give a key, e.g. 2 | 5 means 25.
- Unlike a bar chart, you can read the original data back.
- Median of n ordered values: the middle one, or the mean of the two middle ones.
Back-to-back diagrams
- Put one data set on each side of the same stem. The left leaves run outwards, in order, from the stem.
- Key: 5 | 3 | 6 means 35 on the left and 36 on the right.
- It is the easy way to compare two sets. Here the golf coach is older: its median is 50, against 27.5.
📘 Example 1 · Make a stem and leaf diagram for 34, 27, 35, 22, 41, 27, 38 and find the median
- The stems are 2, 3 and 4 (the tens digits).
- Write the leaves beside each stem: 2 | 7 2 7, 3 | 4 5 8, 4 | 1.
- Put each row in order: 2 | 2 7 7, 3 | 4 5 8, 4 | 1. Key: 2 | 2 means 22.
- There are 7 values, so the median is the 4th: 22, 27, 27, 34.
- Check: the 4th value has three below it and three above it. ✔
Grouped frequency tables
🔑 Key ideas
- With a big range, put values into groups (classes). No value may fit in two groups.
- Whole-number (discrete) data: 71−75, 76−80, 81−85. You cannot score 75.5, so there is no gap problem.
- Continuous data (time, length): use 0−, 10−, 20−. A value of 10 belongs to the group 10−.
| Golf score | 71−75 | 76−80 | 81−85 | 86−90 | 91−95 | 96−100 | 101−105 | Total |
|---|---|---|---|---|---|---|---|---|
| Frequency | 5 | 6 | 8 | 3 | 1 | 3 | 4 | 30 |
Surveys
✏️ Try it
- Make a frequency table for 2, 3, 3, 5, 3, 5, 2, 3, 5, 3.
- A stem and leaf diagram (key 2 | 1 means 21) reads 2 | 1 4 4 9 and 3 | 0 5. Find the range and the median.
- Why are the groups 70−75 and 75−80 a bad choice?
Answers
1. 2 occurs 2 times, 3 occurs 5 times, 5 occurs 3 times (total 10) · 2. The values are 21, 24, 24, 29, 30, 35: range 35 − 21 = 14, median (24 + 29) ÷ 2 = 26.5 · 3. A score of 75 is in both groups, so it would be counted twice.
Pie charts
🔑 Key ideas
- A pie chart is a circle split into sectors. The whole circle is 360° and stands for the whole total.
- Each sector is in proportion to its frequency. The angle does not show the actual number.
- Draw: angle = × 360°.
- Read: frequency = × total.
- Check: the angles add to 360° (or the frequencies to the total).
📘 Example 2 · 240 children voted for a holiday destination. Spain has a 120° sector. How many votes?
- The whole circle, 360°, is 240 votes.
- Spain's share of the circle: 120 ÷ 360 = .
- Spain's votes: × 240 = 80 votes.
- Check: the book's other sectors (75°, 45°, 90°, 30°) give 50, 30, 60, 20. 80 + 50 + 30 + 60 + 20 = 240. ✔
📘 Example 3 · 5 million votes were cast. The Social Democrats got 45%. How many votes?
- Percentage of the total: 45 ÷ 100 × 5 million.
- 0.45 × 5 = 2.25 million votes.
- Check all four parties: 45% + 36% + 15% + 4% = 100%, and 2.25 + 1.8 + 0.75 + 0.2 = 5 million. ✔
📘 Example 4 · Draw a pie chart for 72 students' favourite sport
| Sport | Frequency | Working | Angle |
|---|---|---|---|
| Football | 35 | 35 × 5 | 175° |
| Tennis | 14 | 14 × 5 | 70° |
| Volleyball | 10 | 10 × 5 | 50° |
| Hockey | 6 | 6 × 5 | 30° |
| Basketball | 5 | 5 × 5 | 25° |
| Other | 2 | 2 × 5 | 10° |
- 72 students are 360°, so 1 student is 360 ÷ 72 = 5°.
- Multiply each frequency by 5 (the table).
- Check: 175 + 70 + 50 + 30 + 25 + 10 = 360°. ✔
- Draw a circle, start at a radius, and measure each angle in turn with a protractor, using the same starting line each time.
🧪 Try it: chart maker (bar chart + pie chart)
Type one category per line as name, frequency (up to 8 lines). The tool draws both charts and shows each angle = f ÷ total × 360°.
✏️ Try it
- Of 60 students, 15 walk to school. What angle shows them on a pie chart?
- On a pie chart of 300 pupils, the bus sector is 120°. How many pupils take the bus?
- A sector is 72°. What percentage of the whole is that?
Answers
1. 15 ÷ 60 × 360 = 90° · 2. 120 ÷ 360 × 300 = 100 pupils · 3. 72 ÷ 360 = 0.2 = 20%
Scatter diagrams and correlation
🔑 Key ideas
- A scatter diagram plots pairs of values, one point for each pair, to see if two sets of data are related.
- Correlation is the relationship: positive (both rise), negative (one rises, the other falls) or none.
- Strong means the points hug the line of best fit. Weak means they are spread out.
- The line of best fit is drawn by eye, straight, with about as many points on each side. It is used to estimate values.
| Example pair | Expect |
|---|---|
| Age of a motorcycle and resale price | Negative |
| Temperature and cold drinks sold | Positive |
| Hair colour and distance to school | None |
The book's height and mass example
| Height | 32 | 34 | 45 | 46 | 52 | 59 | 63 | 64 | 71 | 73 |
|---|---|---|---|---|---|---|---|---|---|---|
| Mass | 5.8 | 3.8 | 9.0 | 4.2 | 10.1 | 6.2 | 9.9 | 16.0 | 15.8 | 9.9 |
| Height | 86 | 87 | 95 | 96 | 96 | 101 | 108 | 109 | 117 | 121 |
|---|---|---|---|---|---|---|---|---|---|---|
| Mass | 11.1 | 16.4 | 20.9 | 16.2 | 14.0 | 19.5 | 15.9 | 12.0 | 19.4 | 14.3 |
📘 Example 5 · Estimate the mass of a child who is 80 cm tall
- Plot the 20 points and see they rise: positive correlation.
- Draw a straight line of best fit by eye, through the middle of the points.
- Go up from 80 on the height axis to the line.
- Go across to the mass axis and read it: about 13 kg.
- Say it is an estimate: another child of 80 cm could be a bit different.
📘 Example 6 · Why not extend the line to 200 cm?
- The data only covers children under five, about 32 to 121 cm.
- We have no evidence the trend carries on outside that range.
- Masses do not keep rising in a straight line: a 200 cm line would predict nonsense. Do not extrapolate far beyond the data.
💡 Correlation does not prove one thing causes the other. A weak link, or a third hidden factor, can explain a pattern. That is why a conclusion such as “overweight people win more gold” needs care.
✏️ Try it
- What correlation would you expect between a child's height and age?
- A line of best fit passes through (60, 10) and (100, 30). Estimate y when x = 80.
- The data covers x from 60 to 100. Is an estimate at x = 300 reliable?
Answers
1. Positive (and fairly strong) · 2. Halfway between the two points: (10 + 30) ÷ 2 = 20 · 3. No. It is far outside the range of the data, so the trend may not hold.
Histograms
🔑 Key ideas
- A histogram shows continuous (or grouped) data. The bars are joined, with no gaps.
- The bars can have different widths. The area of a bar is the frequency.
- Equal widths: use frequency on the vertical axis. Unequal widths: use frequency density.
- Frequency density = frequency ÷ class width, so frequency = frequency density × class width.
| Height (m) | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 ≤ h < 1.0 | 6 | 1 | 6 ÷ 1 = 6 |
| 1.0 ≤ h < 1.5 | 3 | 0.5 | 3 ÷ 0.5 = 6 |
| 1.5 ≤ h < 2.0 | 4 | 0.5 | 4 ÷ 0.5 = 8 |
| 2.0 ≤ h < 2.25 | 3 | 0.25 | 3 ÷ 0.25 = 12 |
| 2.25 ≤ h < 2.50 | 5 | 0.25 | 5 ÷ 0.25 = 20 |
| 2.50 ≤ h < 2.75 | 4 | 0.25 | 4 ÷ 0.25 = 16 |
💡 Why not plot frequency? The 0 to 1 bar would be 6 tall and very wide, wrongly suggesting most sunflowers are under 1 m. In fact only about a quarter (6 of 25) are.
📘 Example 7 · Equal widths: lizards (32 of them), 10 cm classes
- Every class is 10 cm wide, so the bars all have the same width.
- So plot frequency straight up the vertical axis.
- Frequencies from 0 to 100 cm: 0, 0, 1, 2, 5, 8, 7, 6, 2, 1. Check: they add to 32. ✔
- Draw nine joined bars of equal width, the tallest (8) over 50 to 60 cm.
📘 Example 8 · Time to travel to school (40 students): complete the frequency density
| Time (min) | 0− | 10− | 15− | 20− | 25− | 30− | 40−60 |
|---|---|---|---|---|---|---|---|
| Frequency | 6 | 3 | 13 | 7 | 3 | 4 | 4 |
| Class width | 10 | 5 | 5 | 5 | 5 | 10 | 20 |
| Freq. density | 0.6 | 0.6 | 2.6 | 1.4 | 0.6 | 0.4 | 0.2 |
- The class widths come from the next start: 10 − 0 = 10, 15 − 10 = 5, and so on. The last class is 60 − 40 = 20.
- Divide each frequency by its width: 6 ÷ 10 = 0.6, 3 ÷ 5 = 0.6, 13 ÷ 5 = 2.6, 7 ÷ 5 = 1.4, 3 ÷ 5 = 0.6, 4 ÷ 10 = 0.4, 4 ÷ 20 = 0.2.
- Draw joined bars with these heights, over the real widths.
- Check: areas 6 + 3 + 13 + 7 + 3 + 4 + 4 = 40. ✔
🧪 Try it: histogram maker (frequency density)
Type one class per line as lower, upper, frequency (for example 0, 1, 6 for 0 ≤ h < 1). Up to 10 classes.
✏️ Try it
- The class 10 ≤ x < 15 has frequency 20. Find the frequency density.
- A bar is 20 wide with frequency density 0.5. What is the frequency?
- Classes 0 ≤ t < 10 (frequency 5), 10 ≤ t < 30 (frequency 30), 30 ≤ t < 60 (frequency 15): find each frequency density.
Answers
1. 20 ÷ 5 = 4 · 2. 0.5 × 20 = 10 · 3. 5 ÷ 10 = 0.5, 30 ÷ 20 = 1.5, 15 ÷ 30 = 0.5
Chapter summary
- Tally, then add to get the frequency table. Bar chart bars have gaps; the tallest is the mode.
- Stem and leaf: leaves in order, with a key. Median is the middle value (or the mean of the middle two). Back-to-back diagrams compare two sets.
- Grouped tables: no value may belong to two groups.
- Pie chart angle = f ÷ total × 360°. Frequency from an angle = angle ÷ 360 × total. Angles add to 360°.
- Scatter diagram: positive, negative or no correlation; strong or weak. Use a line of best fit to estimate, but not far outside the data.
- Histogram: joined bars, area = frequency. Frequency density = frequency ÷ class width.
📄 Practice questions
The book's Exercises 36.1 to 36.5 and Student assessment 1 are in the printable assignment for this chapter.
Mini-quiz
Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 36 (Collecting, displaying and interpreting data). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.