Frequency diagrams and line graphs
🎯 In this topic you will
- Investigate questions by collecting and organising data.
- Represent data using frequency diagrams and line graphs.
- Identify patterns in data to answer statistical questions.
🧠 Key Words
- frequency diagram
- line graph
Show Definitions
- frequency diagram: A diagram that shows how often different values or categories occur in a data set.
- line graph: A graph that uses points connected by straight lines to show how a variable changes, often over time.
Collecting Measurement Data
S ometimes the data we collect is in the form of measurements rather than values that are counted or sorted into categories. These measurements can be compared to help solve problems and to answer statistical questions.
❓ EXERCISES
1. This is a frequency diagram showing the times it took runners to complete a race.
a. How many runners took between $15$ and $20$ minutes to complete the race?
b. How many runners took less than $15$ minutes to complete the race?
c. Arun says that the graph shows that the longest time taken to complete the race was $20$ minutes to complete the race. He might not be correct. Explain why he might not be correct.
👀 Show answer
a. $3$
b. $14$
c. The bar shows a group from $15$ to $20$ minutes, so the graph does not show anyone took exactly $20$ minutes. The longest time could be any value less than $20$ minutes (for example, $19$ minutes), because the data are grouped into intervals.
2. The heights of $20$ dogs are measured in centimetres. The results are recorded in a table.
| Height in centimetres | Frequency (number of dogs) |
|---|---|
| $30$ to less than $40$ | $6$ |
| $40$ to less than $50$ | $7$ |
| $50$ to less than $60$ | $4$ |
| $60$ to less than $70$ | $2$ |
| $70$ to less than $80$ | $1$ |
Copy and complete this frequency diagram using the data in the table.
👀 Show answer
Complete the frequency diagram by drawing bars with these heights:
• $30$ to less than $40$: frequency $6$
• $40$ to less than $50$: frequency $7$
• $50$ to less than $60$: frequency $4$
• $60$ to less than $70$: frequency $2$
• $70$ to less than $80$: frequency $1$
3.
a. Describe one way that bar charts and frequency diagrams are different.
b. Describe one way that bar charts and frequency diagrams are similar.
c. Discuss the differences and similarities between bar charts and frequency diagrams with a partner. Write down any further differences and similarities you discuss.
👀 Show answer
a. In a frequency diagram (for grouped continuous data), the bars touch because the intervals join; in a bar chart (for categories), the bars are usually separated with gaps.
b. Both use bars to show frequencies, and the taller the bar, the greater the frequency.
c. Possible further points: frequency diagrams often use numerical intervals on the horizontal axis (e.g., $30$ to less than $40$), while bar charts often use named categories; both use a vertical frequency axis and can be used to compare how common different values/categories are.
4. The temperature in Zoe's playground was measured five times during one day. These are the results. This line graph shows the results.
| Time | Temperature ($^\circ\text{C}$) |
|---|---|
| $9$ a.m. | $8$ |
| $11$ a.m. | $12$ |
| $1$ p.m. | $14$ |
| $3$ p.m. | $15$ |
| $5$ p.m. |
$11$ |
a. What was the temperature at $5$ p.m.?
b. At what time was the temperature $8^\circ\text{C}$?
c. Use the line graph to estimate the temperature at $10$ a.m.
d. Use the line graph to estimate the two times when the temperate was $13^\circ\text{C}$.
👀 Show answer
a. $11^\circ\text{C}$
b. $9$ a.m.
c. About $10^\circ\text{C}$
d. About $12$ p.m. and about $4$ p.m.
5. Mike put two cups of water in different places. He measured the temperature of the water every $10$ minutes. These are his results.
| Time | Cup $1$ | Cup $2$ |
|---|---|---|
| $0$ minutes | $10^\circ\text{C}$ | $10^\circ\text{C}$ |
| $10$ minutes | $10^\circ\text{C}$ | $9^\circ\text{C}$ |
| $20$ minutes | $11^\circ\text{C}$ | $7^\circ\text{C}$ |
| $30$ minutes | $12^\circ\text{C}$ | $5^\circ\text{C}$ |
| $40$ minutes | $14^\circ\text{C}$ | $4^\circ\text{C}$ |
| $50$ minutes | $17^\circ\text{C}$ | $3^\circ\text{C}$ |
| $60$ minutes | $18^\circ\text{C}$ | $3^\circ\text{C}$ |
a. Draw a line graph showing the temperature of cup $1$.
b. Draw a line graph showing the temperature of cup $2$.
c. Describe the pattern on the graph for cup $1$.
d. Describe the pattern on the graph for cup $2$.
e. Explain why there might be a different pattern for cup $1$ and cup $2$.
f. Share your explanation with your class. What different explanations do other learners have for the different patterns?
👀 Show answer
a. Plot the points for cup $1$: $(0,10)$, $(10,10)$, $(20,11)$, $(30,12)$, $(40,14)$, $(50,17)$, $(60,18)$ (temperature in $^\circ\text{C}$), then join with straight lines.
b. Plot the points for cup $2$: $(0,10)$, $(10,9)$, $(20,7)$, $(30,5)$, $(40,4)$, $(50,3)$, $(60,3)$ (temperature in $^\circ\text{C}$), then join with straight lines.
c. The temperature of cup $1$ increases overall, rising slowly at first and then more quickly later on.
d. The temperature of cup $2$ decreases overall, dropping quickly at first and then leveling off around $3^\circ\text{C}$.
e. The cups were in different places, so one could be warming (for example, near a heater or in sunlight) while the other could be cooling (for example, in a cooler place, shade, near a window, or near something cold). Different surroundings lead to different heat transfer.
f. Examples other learners might suggest: one cup was closer to a heat source; one was in sunlight; one was in a draft or near an open window; different containers/lids/insulation; different starting conditions or measurement error.
6. Work on this investigation with a partner or in a small group. Discuss how you will collect data to answer these questions:
How does the temperature change outside your classroom during the day?
How does the temperature change inside your classroom during the day?
Draw a table to collect the data.
Draw two line graphs to display your data.
Describe the pattern in your graph showing the temperature outside your classroom.
Describe the pattern in your graph showing the temperature inside your classroom.
Write about what is similar and different about how the temperature changes outside and inside your classroom during the day.
👀 Show answer
One suitable plan:
• Choose regular times (for example, every $30$ minutes or every $1$ hour) across the school day.
• Measure outside temperature in the same outdoor spot each time (shade, away from direct sun if possible), and measure inside temperature in the same place in the classroom.
• Record results in a table with columns: time, outside temperature ($^\circ\text{C}$), inside temperature ($^\circ\text{C}$).
• Draw two separate line graphs (or one graph with two lines): time on the horizontal axis and temperature on the vertical axis.
• Describe patterns (increasing, decreasing, peaks, steady periods) for outside and inside.
• Compare similarities and differences (for example, outside may change more quickly; inside may be steadier due to walls, heating/air conditioning, and occupancy).
🧠 Think like a Mathematician
Work independently. Choose one of the investigations below and collect data to answer the question. You may also choose a suitable question of your own. Check your question with your teacher.
- Do nine-year-olds have longer feet than ten-year-olds?
- If you put a stick in the ground outside, how does the length of its shadow change over one day?
- How does the temperature of an object change on white and black paper left in sunshine?
Decide what data you need to collect and how you will collect it. Then collect your data.
Represent the data you have collected using a frequency diagram or a line graph.
Answer the following questions:
👀 show answer
1. Example: Ten-year-olds tend to have slightly longer feet than nine-year-olds.
2. The data show an overall increasing trend as age increases.
3. Older children are generally taller and have larger body measurements, which explains the increase.
4. A frequency diagram or line graph clearly shows how values change and makes comparisons easy.
5. The sample size may be small, or measurements could vary due to measurement error; collecting more data would improve reliability.
When you think of possible explanations for patterns, you are conjecturing. When you judge whether your graph or chart choice was best, you are critiquing.