Interpreting and drawing frequency diagrams
🎯 In this topic you will
- Draw and interpret frequency diagrams for discrete and continuous data
🧠 Key Words
- class interval
- classes
- continuous data
- discrete data
- frequency diagram
- grouped data
Show Definitions
- class interval: A range of values used to group continuous data in a frequency table.
- classes: The groups into which data is divided when organizing into intervals.
- continuous data: Data that can take any value within a range (e.g., height, time).
- discrete data: Data that can only take specific, separate values (e.g., number of students).
- frequency diagram: A chart, often using bars, to show how often values occur in grouped data.
- grouped data: Data that has been organized into intervals or categories.
A frequency diagram shows how often particular values occur in a set of data. One example of a frequency diagram is a bar chart. In a bar chart, the bars are used to represent the frequency.
When you draw a bar chart for grouped data, you must use suitable classes and have equal class intervals.
When you draw a bar chart for discrete data, you should make sure:
- the bars are all the same width
- the gaps between the bars are equal
- you label each bar with the relevant data group
- you give the frequency diagram a title and label the axes
- you use a sensible scale on the vertical axis
When you draw a bar chart for continuous data, you should make sure:
- the class intervals are all the same width
- there are no gaps between the bars
- you use a sensible scale on the horizontal axis
- you give the frequency diagram a title and label the axes
- you use a sensible scale on the vertical axis
❓ EXERCISE
1. The frequency diagram shows the number of phone calls made by all the employees of a company on one day.
a) How many employees made 10–19 phone calls?
b) How many more employees made 30–39 phone calls than made 0–9 phone calls?
c) How many employees are there in the company? Explain how you worked out your answer.
👀 Show answer
a) 8 employees.
b) 8 more (30–39 has 10; 0–9 has 2 → 10 − 2 = 8).
c) 26 employees in total. Add the frequencies for each class: 2 + 8 + 6 + 10 = 26.
2. The frequency table shows the number of cups of coffee sold each day in a coffee shop during one month.
| Number of cups of coffee sold | Frequency |
|---|---|
| 0–19 | 2 |
| 20–39 | 3 |
| 40–59 | 6 |
| 60–79 | 12 |
| 80–99 | 5 |
a) Draw a frequency diagram to show the data.
b) Which month do you think your frequency diagram represents? Explain your answer.
c) Marcus says, “The frequency diagram shows that the most cups of coffee sold was 99.” Is he correct? Explain your answer.
👀 Show answer
a) Frequency diagram required (bars for each class interval).
b) The total frequency is 2 + 3 + 6 + 12 + 5 = 28 days, so it most likely represents February in a non-leap year.
c) Not correct. The diagram uses groups (e.g., 80–99). The tallest bar (60–79) means most days fell in that range, not that 99 cups was the most sold on a single day. We cannot tell the exact maximum from grouped data.
🧠 Think like a Mathematician
Task: Record Ryan’s data in a frequency table, draw a frequency diagram, and discuss which class intervals are most suitable.
Data (30 days):
23, 17, 19, 0, 16, 18, 7, 17, 15, 18, 12, 10, 18, 14, 14, 4, 12, 20, 9, 13, 20, 11, 19, 1, 20, 20, 24, 2
a) Frequency table (class width = 5)
| Class interval | Frequency |
| 0–4 | 4 |
| 5–9 | 2 |
| 10–14 | 9 |
| 15–19 | 8 |
| 20–24 | 7 |
b) Frequency diagram:
Draw bars with class intervals on the horizontal axis (0–4, 5–9, etc.) and frequency on the vertical axis.
c) Discussion of classes:
- The chosen intervals (width = 5) are equal, cover the full range (0–24), and make the data easy to interpret.
- Smaller class widths (e.g., width = 2) would show more detail but may scatter the data too much.
- Larger class widths (e.g., width = 10) would hide useful variation.
- Best choice: width = 5 gives a clear balance between detail and clarity.
👀 show answer
- a) Frequency table shown above.
- b) Frequency diagram would be a bar chart with class intervals of width 5. The tallest bar is 10–14 (frequency 9).
- c) Intervals of width 5 are most suitable. They group the data sensibly while still showing the main distribution.
❓ EXERCISE
4. Erin recorded the number of emails she sent each day for one month. Here are her results:
| 31 | 17 | 37 | 11 | 35 | 34 | 36 | 15 | 33 | 22 | 31 | 18 | 34 | 12 | 28 |
| 14 | 30 | 21 | 39 | 16 | 13 | 38 | 34 | 29 | 10 | 19 | 39 | 32 | 38 | 15 |
Marcus: “Use classes 0–4, 5–9, 10–14, …”
Arun: “Use classes 10–14, 15–19, 20–24, …”
Zara: “Use classes 10–19, 20–29, 30–39, …”
a) Who has chosen the most suitable classes — Marcus, Arun or Zara? Explain why.
b) Explain why the other two choices are not suitable.
c) Record the information in a frequency table.
d) Draw a frequency diagram to show the data.
👀 Show answer
a)Arun. The data range is 10–39 emails. Arun’s classes (width 5) start at the minimum (10) and give a clear, detailed picture without empty groups.
b) Marcus’s classes start at 0, so the first two groups (0–4, 5–9) are empty and waste space. Zara’s classes are width 10 (10–19, 20–29, 30–39), which are too wide and hide useful detail compared with width-5 groups.
c) Frequency table using Arun’s classes:
| Emails (per day) | Frequency |
|---|---|
| 10–14 | 5 |
| 15–19 | 6 |
| 20–24 | 2 |
| 25–29 | 2 |
| 30–34 | 8 |
| 35–39 | 7 |
| Total | 30 |
d) Draw a bar-style frequency diagram with the six class intervals on the horizontal axis and the frequencies above. (No gaps if you treat the classes as continuous.)
🧠 Think like a Mathematician
Frequency table (ages of choir members)
| Age class (years) | Frequency |
| 10 ≤ a < 20 | 12 |
| 20 ≤ a < 30 | 8 |
| 30 ≤ a < 40 | 15 |
| 40 ≤ a < 50 | 6 |
- a) Explain what the class 10 ≤ a < 20 means.
- b) Why shouldn’t we write the classes as 10–19, 20–29, etc.?
- c) In which class would someone aged exactly 30 years go?
- d) Draw a frequency diagram for the data.
👀 show answer
- a) It includes everyone aged from 10 up to but not including 20 (10 ≤ age < 20).
- b) Writing 10–19, 20–29 etc. causes boundary problems with whole vs. decimal ages (e.g., where does 19.9 go?) and risks overlaps or gaps. Using ≤ and < gives a clear, non-overlapping rule.
- c) Exactly 30 years goes in 30 ≤ a < 40.
- d) Draw a bar for each class with equal widths and heights equal to the frequencies: 12, 8, 15, 6. Label the horizontal axis with the class intervals (10–20, 20–30, 30–40, 40–50) and the vertical axis “Frequency”.
❓ EXERCISE — Grouped Frequency & Intervals
6. The frequency table shows the speeds of cars passing a speed camera on one day (km/h).
| Speed of car, $s$ (km/h) | Frequency |
|---|---|
| $50 < s \le 60$ | 2 |
| $60 < s \le 70$ | 3 |
| $70 < s \le 80$ | 6 |
| $80 < s \le 90$ | 12 |
| $90 \le s \le 100$ | 5 |
a) Draw a frequency diagram to show the data.
b) The speed limit is $80$ km/h. How many cars are travelling over the speed limit?
c) Sofia says, “The frequency diagram shows that the slowest car was travelling at $50$ km/h.” Is she correct? Explain.
👀 Show answer
a) Bar chart required with the five class intervals on the horizontal axis and frequencies on the vertical axis.
b) Cars over $80$ km/h are in the last two groups: $12 + 5 = \mathbf{17}$ cars.
c) Not correct. The first class is $50 < s \le 60$ so the slowest car is somewhere just above $50$ km/h. Grouped data do not give exact values.
7. Heights (cm) of some plants:
| 25 | 32 | 30 | 26 | 34 | 22 | 33 | 34 | 31 | 28 |
| 39 | 20 | 27 | 33 | 37 | 32 | 25 | 24 | 30 | 29 |
a) Record this information in a frequency table. Use the classes $20 \le h < 25$, $25 \le h < 30$, $30 \le h < 35$ and $35 \le h < 40$.
b) Draw a frequency diagram to show the data.
c) How many of the plants are at least $25$ cm high? Explain.
👀 Show answer
a) Frequency table:
| Height class (cm) | Frequency |
|---|---|
| $20 \le h < 25$ | 3 |
| $25 \le h < 30$ | 6 |
| $30 \le h < 35$ | 9 |
| $35 \le h < 40$ | 2 |
| Total | 20 |
b) Draw a bar chart with the four class intervals and the frequencies above.
c) “At least $25$ cm” means $h \ge 25$. That is all plants except the first class. So $6 + 9 + 2 = \mathbf{17}$ plants (or $20 - 3 = 17$).
8. The frequency diagrams show the population of a village by age group in 1960 and 2010.
a) Look at the graphs. Write two sentences to compare the age groups in the population of the village in 1960 and 2010.
b) Marcus says, “Approximately 25% of the population were over the age of 40 in 1960, compared with approximately 60% in 2010.” Is Marcus correct? Show your working to support your answer.
👀 Show answer
a) In 1960 most people were in the 20–40 group, with very few over 60, so the population was much younger. By 2010 the 40–60 and 60–80 groups are larger than before, showing an older population with fewer young people than in 1960.
b) From the bars (approximate counts):
- 1960 totals ≈ 30 (0–20) + 85 (20–40) + 35 (40–60) + 5 (60–80) = 155. Over 40 ≈ 35 + 5 = 40 → fraction ≈ 40/155 ≈ 26%.
- 2010 totals ≈ 25 + 40 + 55 + 35 + 5 = 160. Over 40 ≈ 55 + 35 + 5 = 95 → fraction ≈ 95/160 ≈ 59%.
Therefore Marcus is about right: roughly 25% over 40 in 1960 and about 60% in 2010.
⚠️ Be careful!
- Pick the right style:Discrete data → bars with gaps; Continuous (grouped) data → bars with no gaps.
- Equal class widths: grouped-data bars must be equal width. (If widths differ, use frequency density — advanced topic.)
- Clear class labels: avoid overlap like “10–20” & “20–30”. Use inequalities, e.g. $10<m\le20$, $20<m\le30$.
- Start at zero: the frequency (vertical) axis should start at $0$; truncating exaggerates differences.
- Same bar widths & equal gaps (discrete): uneven widths/gaps distort comparisons.
- Read height, not width: with equal-width classes, bar height shows frequency (don’t infer from area).
- Total check: the sum of all bar heights equals the total frequency; use this to spot mistakes.
- Axes & units: label axes (with units) and add a clear title; choose sensible, even scales.