Representing data
🎯 In this topic you will
- Choose how to represent data
🧠 Key Words
- continuous data
- discrete data
- justify
Show Definitions
- continuous data: Data that can take any value within a given range (e.g., height, time).
- discrete data: Data that can only take specific, separate values (e.g., number of students).
- justify: To explain or give reasons to support an answer, method, or conclusion.
When you represent data using a diagram, graph or chart, you must decide which type is best to use. This table will help you decide.
| Type of diagram/graph/chart | When to use it | What it looks like |
|---|---|---|
| Venn or Carroll diagram | When you want to sort data or objects into groups that have some common features. |
|
| Bar chart | When you want to compare discrete data. |
|
| Dual bar chart | When you want to compare two sets of discrete data. |  |
| Compound (stacked) bar chart | When you want to combine two or more quantities into one bar to see parts and the total. |  |
| Frequency diagram (histogram) | When you want to compare continuous data. |  |
| Line graph | When you want to see how data changes over time. |  |
| Scatter graph | When you want to compare two sets of data points and look for correlation. |  |
| Pie chart | When you want to compare the proportions of each sector with the whole amount. |  |
| Infographic | When you want to show information quickly in a way that is easy to understand. |
|
| Stem-and-leaf diagram | When you want to compare data that is grouped, but you still want to see the actual values. |
|
❓ EXERCISES
1. Look at the following sets of data. Which type of diagram, graph or chart do you think is best to use to display the data? Justify your choice.
a. The number of ice creams sold in a shop each day for one week.
b. The height and the shoe size of 20 students.
c. The total number of cakes, sandwiches and drinks sold in a café on two different days.
d. The proportion of students that travel to college by car, bus, bicycle or on foot.
👀 Show answer
a. A bar chart or line graph, since it shows change over days of the week.
b. A scatter graph, to show the relationship between height and shoe size.
c. A comparative bar chart, to compare sales of items on two days.
d. A pie chart, to show proportions of students using different transport methods.
2. Ten students are asked which sports they play out of a choice of football, hockey and cricket.
The ten students are: Aaron, Brad, Chloe, Dian, Eralia, Fayard, Guang, Harper, Irine and Jengo.
The students that play football are: Aaron, Chloe, Eralia, Irine and Jengo.
The students that play hockey are: Brad, Chloe, Eralia, Fayard and Harper.
The students that play cricket are: Chloe, Guang, Harper and Jengo.
a. Draw a diagram, graph or chart to represent the data.
b. Justify your choice.
c. Make one comment about what information your diagram, graph or chart shows.
👀 Show answer
a. A Venn diagram is best, with three overlapping circles (football, hockey, cricket) showing shared players.
b. A Venn diagram is appropriate because some students play more than one sport, and overlaps need to be shown clearly.
c. The diagram shows, for example, that Chloe plays all three sports, and that Harper plays both hockey and cricket.
3. The table shows how the amount of air in a scuba tank changes during a dive.
| $Amount\ of\ air\ (litres)$ | $15$ | $14$ | $12$ | $8$ | $7$ | $4$ |
|---|---|---|---|---|---|---|
| $Time\ (minutes)$ | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ |
🧠 Tip
A scuba tank is a metal cylinder used to store air for a diver to use under water.
a. Draw a diagram, graph or chart to represent the data.
b. Justify your choice.
c. Make one comment about what information your diagram, graph or chart shows.
👀 Show answer
a. A line graph with $Time$ (minutes) on the $x$-axis and $Amount\ of\ air$ (litres) on the $y$-axis.
b. A line graph is most appropriate as both variables are continuous and show how air decreases over time.
c. The graph shows that the air in the tank decreases steadily, with a faster drop between $20$ and $30$ minutes, indicating increased usage during that period.
🧠 Think like a Mathematician
Data: The table shows the number of dentist appointments for two dentists on one day.
| Type of appointment | Check-up | Filling | Extraction |
| Dentist A | 14 | 8 | 2 |
| Dentist B | 7 | 10 | 3 |
Questions:
- a) Using a dual bar chart:
- i) What parts of the data are easier to compare?
- ii) What parts of the data are more difficult to compare?
- b) Using a compound bar chart:
- i) What parts of the data are easier to compare?
- ii) What parts of the data are more difficult to compare?
- c) Complete:
- i) In general, to compare total amounts it is best to use a …
- ii) In general, to compare individual amounts it is best to use a …
👀 show answers
- a) Dual bar chart:
- i) Easier to compare individual appointment types between dentists (e.g., check-ups A vs B).
- ii) Harder to see overall totals for each dentist.
- b) Compound bar chart:
- i) Easier to compare total appointments between dentists.
- ii) Harder to compare each appointment type separately.
- c)
- i) To compare totals → use a compound bar chart.
- ii) To compare individual amounts → use a dual bar chart.
❓ EXERCISES
5. The table shows the length and mass of 10 hedgehogs.
a. Explain why a scatter graph is the best way to represent this data.
b. Copy the grid and draw a scatter graph to show this data.
c. Draw a line of best fit on your scatter graph.
d. Use your line of best fit to estimate the length of a hedgehog with a mass of $725\ g$.
👀 Show answer
a. A scatter graph is best because it shows how two continuous variables (length and mass) are related.
b. Points plotted on the scatter graph show each hedgehog’s length and mass.
c. The line of best fit shows the overall trend (mass increases with length).
d. From the line of best fit, when $mass \approx 725\ g$, the estimated length is about $28\ cm$.
6. Javed records the distances he cycled each day in May. This frequency table shows his results.
| $Distance\ in\ km$ | $Frequency$ |
|---|---|
| $0-5$ | $4$ |
| $5-10$ | $7$ |
| $10-15$ | $14$ |
| $15-20$ | $6$ |
🧠 Tip
If Javed cycled $5\ km$ he would record this in the $5-10\ km$ group. If he cycled $10\ km$ he would record this in the $10-15\ km$ group, etc.
a. Draw a diagram, graph or chart to represent the data.
b. Justify your choice.
c. Make one comment about what information your diagram, graph or chart shows.
👀 Show answer
a. A bar chart or histogram is suitable to represent the grouped distance data.
b. A bar chart/histogram is appropriate as it shows frequency of distances in grouped intervals.
c. The diagram shows that Javed cycled most frequently between $10$ and $15\ km$, and fewer times at shorter and longer distances.
7. Eight people were asked to run $100\ m$, and their time was recorded in seconds. They were also given a spelling test of ten words. The table shows their results.
| $Time\ to\ run\ 100m\ (seconds)$ | $16$ | $18$ | $20$ | $22$ | $19$ | $23$ | $24$ | $17$ |
|---|---|---|---|---|---|---|---|---|
| $Spelling\ test\ result\ (out\ of\ 10)$ | $7$ | $10$ | $6$ | $4$ | $3$ | $9$ | $7$ | $2$ |
a. Draw a diagram, graph or chart to represent the data.
b. Justify your choice.
c. Make one comment about what information your diagram, graph or chart shows.
👀 Show answer
a. A scatter graph is the best diagram, plotting $Time$ on the $x$-axis and $Spelling\ score$ on the $y$-axis.
b. A scatter graph is appropriate because it shows how two continuous variables are related and whether there is a correlation between them.
c. The scatter graph shows a negative correlation: those who took longer to run $100\ m$ generally scored lower on the spelling test.
🔗 Learning Bridge
You’ve just reviewed how to choose a display for different kinds of data (discrete vs continuous, paired vs over time). Next, you’ll apply that logic to new scenarios and practise justifying your choice.
- Identify the data type: discrete categories → bar/dual/stacked; continuous groups → histogram; time series → line graph.
- Check pairing: two measurements per item → use a scatter graph to explore correlation.
- Think “total vs parts”: compare totals → bar/dual; compare composition + total → stacked (compound) bars; compare proportions of a whole → pie.
- Keep exact values? Use a stem-and-leaf to show grouped shape while retaining individual data points.
Quick matches
- Value each Monday: line (time series) to show trend.
- Two dressings’ ingredients (per 100 ml): stacked bars to compare parts & totals.
- Ages on a bus: stem-and-leaf to keep exact ages but see the distribution.
Tip: In your justification, link a feature of the data (e.g. “paired measurements”) to a strength of the graph (“reveals correlation”).
❓ EXERCISES
1. Look at the following sets of data. Which type of diagram, graph or chart do you think is best to use to display the data? Justify your choice.
a. The percentage of the members of two running clubs that are men, women, girls and boys.
b. The ages and heights of the horses at a riding school.
c. The scores, out of 50, of 30 students in a spelling test.
d. The mass of a baby chimpanzee each week.
👀 Show answer
a. A comparative bar chart or two pie charts to show proportions clearly.
b. A scatter graph to show the relationship between age and height.
c. A bar chart or histogram to show distribution of test scores.
d. A line graph to show how the chimpanzee’s mass changes over time.
2. A group of 30 students study science at advanced level.
Four students study physics, biology and chemistry.
Five students study only chemistry and biology, three study only chemistry and physics, and two study only physics and biology.
Six students study only physics, seven study only biology and three study only chemistry.
a. Draw a diagram, graph or chart to represent this data.
b. Justify your choice of diagram, graph or chart.
c. Make one comment about what your diagram, graph or chart shows you.
👀 Show answer
a. A Venn diagram with three circles (physics, biology, chemistry).
b. A Venn diagram is best because it shows overlaps between students taking different combinations of subjects.
c. The diagram shows, for example, that four students take all three sciences and that physics-only has more students than chemistry-only.
3. The table shows the monthly average mass of a baby girl from newborn to one year old.
| $Month$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $Mass\ (kg)$ | $3.2$ | $4.2$ | $5.1$ | $5.8$ | $6.4$ | $6.9$ | $7.3$ | $7.6$ | $7.9$ | $8.2$ | $8.5$ | $8.7$ | $8.9$ |
a. Draw a diagram, graph or chart to represent this data.
b. Justify your choice of diagram, graph or chart.
c. Make one comment about what your diagram, graph or chart shows you.
👀 Show answer
a. A line graph with month on the $x$-axis and mass on the $y$-axis.
b. A line graph is best because both variables are continuous and it shows how mass changes over time.
c. The graph shows that the baby’s mass increases steadily, with rapid growth in the first few months and slower growth later in the year.
🧠 Think like a Mathematician
Data: The ingredients of two different cans of beans.
| Ingredient | Beans | Water | Tomato paste | Sugar | Salt |
| Can A | 48 g | 32 g | 17 g | 2 g | 1 g |
| Can B | 67 g | 30 g | 18 g | 8 g | 2 g |
Questions:
- a) Using a compound bar chart:
- i) What parts of the data are easier to compare?
- ii) What parts of the data are more difficult to compare?
- b) Using a pie chart:
- i) What parts of the data are easier to compare?
- ii) What parts of the data are more difficult to compare?
- c) Complete:
- i) When comparing individual and total amounts, it is better to use a …
- ii) When comparing proportions, it is better to use a …
👀 show answers
- a) Compound bar chart:
- i) Easier to compare individual ingredient amounts and totals for each can.
- ii) More difficult to see relative proportions of each ingredient within a can.
- b) Pie chart:
- i) Easier to compare proportions of ingredients within each can.
- ii) More difficult to compare absolute quantities between cans.
- c)
- i) Use a compound bar chart.
- ii) Use a pie chart.
❓ EXERCISES
5. These are the numbers of pages of a book that Daylen reads each day for four weeks.
| $25$ | $5$ | $18$ | $34$ | $16$ | $35$ | $12$ | $12$ | $20$ | $14$ | $8$ | $27$ | $39$ | $9$ |
| $30$ | $11$ | $22$ | $19$ | $7$ | $27$ | $10$ | $32$ | $27$ | $33$ | $11$ | $24$ | $17$ | $22$ |
a. Draw a diagram, graph or chart to represent this data.
b. Justify your choice of diagram, graph or chart.
c. Make one comment about what your diagram, graph or chart shows you.
d. Work out
i. the mode ii. the median iii. the range of the data.
👀 Show answer
a. A line graph (or bar chart) with $Day$ on the $x$-axis and $Pages$ on the $y$-axis.
b. A line graph shows how a daily value changes over time (28 consecutive days), revealing trends and peaks.
c. Daylen’s reading fluctuates, with several peaks around $30$–$35$ pages and some low days near $5$–$10$ pages.
d.
i. Mode: $27$ (occurs $3$ times).
ii. Median: Sort the $28$ values; median $=\dfrac{19+20}{2}=19.5$ pages.
iii. Range: $39-5=34$ pages.
6. Zara recorded the number of minutes she spent doing homework each evening for one month. The frequency table shows her results.
| $Time,\ t\ (minutes)$ | $Frequency$ |
|---|---|
| $0 \le d < 20$ | $1$ |
| $20 \le d < 40$ | $6$ |
| $40 \le d < 60$ | $2$ |
| $60 \le d < 80$ | $8$ |
| $80 \le d < 100$ | $14$ |
a. Draw a diagram, graph or chart to represent this data.
b. Justify your choice of diagram, graph or chart.
c. Make one comment about what your diagram, graph or chart shows you.
👀 Show answer
a. A histogram (or bar chart) using the class intervals $[0,20), [20,40), [40,60), [60,80), [80,100)$.
b. A histogram is suitable because $t$ is continuous and the data are grouped into equal-width intervals.
c. Most evenings Zara spent between $80$ and $100$ minutes on homework (highest frequency $14$); very few evenings were below $20$ minutes.
7. A scientist measured the length and mass of $12$ sea turtles. The table shows her results.
| $Length\ (cm)$ | $87$ | $99$ | $92$ | $84$ | $108$ | $105$ | $109$ | $94$ | $85$ | $95$ | $100$ | $90$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $Mass\ (kg)$ | $125$ | $150$ | $135$ | $112$ | $175$ | $163$ | $188$ | $132$ | $115$ | $144$ | $158$ | $128$ |
a. Draw a diagram, graph or chart to represent this data.
b. Justify your choice of diagram, graph or chart.
c. Make one comment about what your diagram, graph or chart shows you.
👀 Show answer
a. A scatter graph with $Length\ (cm)$ on the $x$-axis and $Mass\ (kg)$ on the $y$-axis.
b. A scatter graph is appropriate because it shows the relationship between two continuous variables (length and mass).
c. The points show a positive correlation: longer turtles tend to have greater mass.
⚠️ Be careful!
- Match display to data: discrete → bar/dual/stacked; continuous (grouped) → histogram; over time → line; paired values → scatter; proportions → pie.
- Don’t mix up bar charts and histograms: bars have gaps for discrete data; histograms have no gaps and use class intervals.
- Totals vs parts: compare totals with bars/duals; compare parts + totals with stacked (compound) bars; compare proportions with a pie chart.
- Avoid pies for fine comparisons: small differences in angles are hard to judge—use bars when precision matters.
- Use a scatter graph only for paired data: one point per item with (x,y); a line graph is for values tracked over time.
- Equal class widths: if class widths differ, don’t use raw frequency—use frequency density (advanced) or re-bin.
- Label and key: give a clear title, axis labels (with units), and a legend when plotting multiple series.
- Be consistent with categories/colors: keep the same order and color mapping across charts you want to compare.
- Don’t mix counts and percentages: convert to the same unit before comparing or combining on a single chart.
- Time belongs on the x-axis: use equal time steps; don’t join points when it’s not a time series.
- Mind sample sizes: a bigger percentage of a smaller group can be fewer items than a smaller percentage of a larger group.
- Justify your choice: always link a feature of the data (discrete/continuous/paired/time) to why the chosen display reveals the comparison best.