Frequency diagrams, line graphs and scatter graphs
🎯 In this topic you will
- Interpret and represent data using frequency diagrams, line graphs, and scatter graphs.
- Plan and carry out investigations using data that includes measures.
- Predict the outcome of an investigation, look for patterns, and check predictions.
🧠 Key Words
- frequency diagram
- line graph
- scatter graph
Show Definitions
- frequency diagram: A graph that shows how often different values occur in a data set.
- line graph: A graph where data points are connected by lines to show how a quantity changes over time or across values.
- scatter graph: A graph that displays pairs of numerical data as points to show possible relationships between two variables.
📊 Looking for Links in Data
We can use some graphs and charts to show if there is a link between two sets of data. For example, there might be a link between how tall someone is and how long their arms are.
📈 Investigating Data Connections
A class collected these sets of data about themselves. Which of these sets of data might have a link?
❓ EXERCISES
$1$. Halima measured the speed of vehicles passing her school for $30$ minutes. This frequency diagram represents the data she collected.
a. How many vehicles were travelling between $60$ and $80$ km per hour?
b. How many vehicles were travelling less than $60$ km per hour?
c. How many vehicles passed the school in total?
👀 Show answer
b. Vehicles travelling less than $60$ km/h are in the groups $0$–$20$, $20$–$40$, and $40$–$60$. Adding the frequencies (about $3 + 0 + 5$) gives approximately $8$ vehicles.
c. Adding all frequency bars ($3 + 0 + 5 + 11 + 2$) gives a total of about $21$ vehicles.
$2$. Imagine you represented the speed of vehicles passing your school for $30$ minutes.
a. Describe what equipment you would need and how you would collect the data.
b. Predict what would be similar about your frequency diagram and the frequency diagram in question $1$. Explain your prediction.
c. Predict what would be different about your frequency diagram and the frequency diagram in question $1$. Explain your prediction.
d. Share your predictions with a partner or in a small group. What do you agree and disagree about?
👀 Show answer
b. The diagram might show similar speed ranges and similar patterns, such as most vehicles travelling at medium speeds.
c. The frequencies could be different because traffic conditions, time of day, and location affect how fast vehicles travel.
d. Students may agree or disagree depending on their predictions about traffic patterns and vehicle speeds.
$3$. A class measured how high each of them could jump vertically. These are the results in centimetres:
$25,\ 31,\ 33,\ 18,\ 28,\ 36,\ 29,\ 28,\ 30,\ 27,\ 25,\ 29,\ 32,\ 19,\ 28,\ 24,\ 24,\ 24,\ 24,\ 26,\ 31,\ 28,\ 29,\ 23,\ 28,\ 31,\ 20,\ 25,\ 29,\ 26,\ 29$
a. Decide on five equal groups for the measurements.
b. Draw and complete a tally chart of the results.
c. Draw a frequency diagram of the heights jumped.
d. Write two sentences to describe the data in your frequency diagram.
👀 Show answer
b. A tally chart would count how many values fall in each group.
c. A frequency diagram can then be drawn using these groups on the horizontal axis and frequencies on the vertical axis.
d. For example: Most students jumped between about $26$ cm and $31$ cm. Very few students had jumps below $20$ cm or above $34$ cm.
❓ EXERCISES
$4$. Cheng left two thermometers in different places in the classroom. He recorded the temperature on the thermometers every half an hour. These line graphs show his results.
a. What was the temperature for thermometer $1$ at $1$ o’clock?
b. What was the time when thermometer $2$ first showed $23^\circ$?
c. Use the line graphs to estimate the temperature on both thermometers at $11{:}15$am.
d. Describe the patterns in the two graphs. How are they different?
e. Suggest an explanation for the difference in the two graphs.
👀 Show answer
b. Thermometer $2$ first shows about $23^\circ\text{C}$ at approximately $10{:}30$am.
c. At $11{:}15$am, thermometer $1$ is roughly $18.5^\circ\text{C}$ and thermometer $2$ is roughly $30^\circ\text{C}$ (estimated between the plotted points).
d. Thermometer $1$ increases slightly and then stays almost constant. Thermometer $2$ rises sharply until around $12$ noon and then falls afterwards.
e. The thermometers were placed in different locations. One may have been closer to sunlight, a heater, or an open window, causing different temperature patterns.
❓ EXERCISES
$5$. Dee measured her pulse rate every $10$ minutes on a $1$ hour run and for $20$ minutes afterwards. These are her results:
| Time | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ | $60$ | $70$ | $80$ |
| Pulse rate | $66$ | $102$ | $102$ | $118$ | $106$ | $130$ | $130$ | $88$ | $68$ |
Draw a line graph to represent the data in the table. Join the points on your graph with straight lines.
a. At what time was Dee’s pulse rate $118$ beats per minute?
b. What happened to Dee’s pulse rate between $40$ and $50$ minutes?
c. Describe the pattern of the line in your graph.
d. Use your line graph to estimate Dee’s pulse rate at:
i $15$ minutes
ii $35$ minutes
iii $75$ minutes
👀 Show answer
b. Between $40$ and $50$ minutes, Dee’s pulse rate increased from $106$ to $130$ beats per minute.
c. The line rises overall at first, stays level in some parts, dips at $40$ minutes, rises again to a peak, then falls sharply after the run and finally drops close to the starting value.
d. Estimated pulse rates:
i. At $15$ minutes, about $102$ beats per minute.
ii. At $35$ minutes, about $112$ beats per minute.
iii. At $75$ minutes, about $78$ beats per minute.
❓ EXERCISES
$6$. Izzy has measured the hand spans and foot length of the children in her class and plotted them onto a scatter graph. The red line is her line of best fit.
a. What is the longest hand span in Izzy’s class?
b. What is the shortest foot length in Izzy’s class?
c. One child has a foot length of $26$ cm, what is the measurement of their hand span?
d. A new child joins the class. Their hand span is $17$ cm. Use the line of best fit to estimate the length of the new child’s foot.
👀 Show answer
b. The shortest foot length is about $21$ cm.
c. A foot length of $26$ cm matches a hand span of about $18$ cm.
d. Using the line of best fit, a hand span of $17$ cm gives an estimated foot length of about $24$ cm.
$7$. $11$ plants were grown. Each plant was measured and its number of leaves was counted. This table shows the data that was collected.
| Height (cm) | $6$ | $11$ | $15$ | $8$ | $12$ | $17$ | $15$ | $18$ | $9$ | $11$ | $13$ |
| Number of leaves | $2$ | $4$ | $7$ | $3$ | $6$ | $8$ | $8$ | $9$ | $4$ | $6$ | $6$ |
a. Draw a scatter graph of the data in the table. Put the number of leaves along the horizontal axis and the height on the vertical axis.
b. Does it look like there is a link between the height of the plants and the number of leaves? Describe the link.
The taller the plant the ...
c. Draw a line of best fit on the graph.
d. Use your line of best fit to estimate how many leaves a plant might have if it was $14$ cm tall.
e. With your partner assess each of your lines of best fit. Are the lines:
• In the right direction
• Not too steep
• Steep enough
• Not too high
• Not too low.
👀 Show answer
b. Yes, there is a positive link. The taller the plant, the more leaves it tends to have.
c. The line of best fit should slope upwards through the middle of the points.
d. A plant that is $14$ cm tall might have about $6$ or $7$ leaves.
e. A good line of best fit should be in the correct direction, have a sensible slope, and pass through the middle of the points without being too high or too low.
❓ EXERCISES
$8$. Which graph would you use to represent the data in each of these investigations?
a. Investigation: How quickly does hot water cool to room temperature?
Would you use a frequency diagram, line graph or a scatter graph?
b. Investigation: What is the most common height for children in Stage $6$?
Would you use a frequency diagram, line graph or scatter graph?
c. Investigation: Is there a link between a person’s height and how well they do in a science test?
Would you use a frequency diagram, line graph or scatter graph?
👀 Show answer
b. A frequency diagram would be used because it shows how often different heights occur.
c. A scatter graph would be used because it shows the relationship between two variables (height and test results).
🧠 Think like a Mathematician
Choose one of these statistical questions to investigate.
- How does the temperature of water change in sunlight and in shade?
- Is there a link between head circumference and height?
- What is the most common distance that a person in your class can jump?
- Is there a link between how long a person’s arm is and how far they can throw?
You could investigate your own problem where the data will be measures.
Ask your teacher to check your question before you start investigating.
Write a sentence explaining what you think will be the result of your investigation and why.
Collect your data in a table.
Choose a way to represent your data. You could choose a frequency diagram, line graph or a scatter graph.
Explain why you chose that way of representing your data.
Describe any patterns you can see in your data.
Does your data suggest that your prediction was correct?
Use the information in your table, graph and diagrams to answer your statistical question.
Show Example Answer
- Prediction: If water is placed in sunlight it will heat up faster than water placed in the shade because it receives more heat energy from the Sun.
- Representation: A line graph is suitable because temperature changes continuously over time.
- Pattern: The graph may show temperature increasing steadily in sunlight while remaining lower or increasing more slowly in shade.
- Conclusion: The collected data can be used to confirm whether sunlight causes faster warming compared with shade.