D. Data · Two-Variable Data, Scatter Plots and Probability

Choosing and Justifying the Best Graph

D1.3 A1About 45 minutes

Learning goals

I can match a type of data with the graphs that suit it.
I can choose a scatter plot when I want to show a relationship between two numerical variables.
I can justify my choice of graph using the type of data and the question.

Words to know

justify
Give reasons that explain why your choice makes sense.

numerical data
Data that is numbers: counted (discrete) or measured (continuous).

categorical data
Data that is names or categories, like favourite sport. Also called qualitative data.

histogram
A bar graph for one continuous variable grouped into equal intervals, with no gaps between the bars.

broken-line graph
A graph whose points are joined to show how one variable changes over time.

Remember from Grade 7

In Grade 7 you chose between bar graphs, histograms, stem-and-leaf plots, broken-line graphs and circle graphs, and justified your choice. Grade 8 adds the scatter plot to your toolkit. The skill is the same: match the graph to the data and the question.

Ask two questions

Before you draw, ask:

How many variables? One or two?
What kind of data is each? Categories (qualitative), counted numbers (discrete) or measured numbers (continuous)? Is one variable time?

Then ask what the reader needs to see: a comparison of categories, parts of a whole, the shape of a distribution, a change over time, or a relationship between two variables.

Graph	Data it suits	Best for showing
Bar graph	One variable, categories or small discrete values	Comparing counts or amounts of categories
Circle graph	One variable, categories that make up a whole	Parts of a whole (percentages that add to 100%)
Histogram	One variable, continuous (grouped in equal intervals)	The shape of a distribution
Stem-and-leaf plot	One variable, numerical, small data sets	Every value plus the shape
Broken-line graph	Two variables, one of them is time	Change over time
Scatter plot	Two numerical variables measured on the same individuals	A relationship between the two variables

A guide for choosing a graph. Often more than one graph can work; your job is to pick the best one and say why.

Each graph has a job: comparing categories, parts of a whole, the shape of one variable, change over time, or a relationship between two variables. Pick the one whose job matches your question.

A circle graph is a good choice here: one categorical variable, and the four parts make the whole class. 18 of 40 is 45% (162°), 10 of 40 is 25% (90°), 8 of 40 is 20% (72°) and 4 of 40 is 10% (36°).

Worked example

Example 1: Height and arm span

Which graph best shows the Okafor family's height and arm-span data? Justify.

Two variables, both continuous (measured in cm), on the same 10 people.
The question is about a relationship: do taller people have longer arm spans?
A bar graph would need 20 bars and would hide the pairing. A broken-line graph is for time, which isn't a variable here.

A scatter plot, because it shows each person's two measurements as one point, so the reader can see the relationship between height and arm span.

Worked example

Example 2: Snowfall over a winter

A weather station records the total snowfall in each month from November to March. Which graph is best?

Two variables: month and snowfall. One of them is time.
The interesting question is how snowfall changes through the winter.
A scatter plot could show it, but joining the points in time order makes the rise and fall easier to follow. A bar graph could also work because there are only 5 months.

A broken-line graph (or a bar graph), because the data shows change over time. Explain your choice: the line makes the trend from month to month easy to see.

Worked example

Example 3: Wait times at the clinic

A walk-in clinic records the waiting time (in minutes) of 120 patients. Which graph shows how waiting times are spread out?

One variable: waiting time, which is continuous.
120 values is a lot for a stem-and-leaf plot or a line plot.
Group the times in equal intervals (0–9 min, 10–19 min, and so on).

A histogram, because it shows the shape of the distribution of one continuous variable with many values.

Justify like a mathematician

A strong justification names the number of variables, the type of data, and what the reader will see. Try this frame: "I chose a ___ because the data has ___ variable(s) that are ___, and this graph shows ___." Explaining your reasoning clearly is part of communicating in math (Strand A).

Just for fun

Why is the circle graph always invited to parties?

It brings the pie, and it promises every slice adds up to 100%.

Hands-on activity

Graph critic

You need: a newspaper, flyer, magazine, or a news or weather website (with a grown-up's OK).

Find three graphs.
For each one, write: how many variables, what type of data, and what the graph is trying to show.
Decide whether the creator chose the best graph. If not, name a better one and justify it in one sentence.
Look for at least one graph with two variables. Is time one of them?

What to look for: good reasons that mention the data type and the purpose, not just "it looks nice".

Your turn

Which graph would you choose for the number of students who chose each lunch option (pizza, wraps, soup, salad)? Justify.

Which graph would you choose for each student's hours of sleep and reaction time? Justify.

Which graph would you choose for a puppy's mass each week for 12 weeks? Justify.

Which graph would you choose for the masses of 200 apples from an orchard? Justify.

Why is a circle graph a poor choice for the heights of 25 students?

Check the circle graph: does 162° + 90° + 72° + 36° make a full circle?

More practice: warm-up

Which graph shows parts of a whole?

Which graph shows a relationship between two numerical variables?

Which graph shows change over time?

Which graph shows the shape of one continuous variable with many values?

How many variables does a scatter plot show?

More practice: on your own

Nadia wants to show whether taller sunflowers have wider flower heads. She draws a double bar graph with one pair of bars per plant. Suggest a better graph and justify it.

In a survey of 40 students, 18 chose skating. Show how to get the 162° angle for the skating sector.

Give one reason a histogram is better than a bar graph for the masses of 200 apples.

A town tracks the number of people at the outdoor rink and the daily high temperature for 30 winter days. Name two graphs that could show the data and say which is better for the question "Does temperature affect rink attendance?"

Choose a graph for the percentage of a family's monthly budget spent on housing, food, transportation, savings and other. Justify.

Challenge

Data: each Grade 8 student's favourite sport and their hours of exercise per week. Explain why a scatter plot won't work and suggest a graph that will.

Another class's circle graph has sectors of 40%, 35% and 30%. What is wrong, and what might have caused it?

A scientist measures the temperature of a cup of hot chocolate every minute for 20 minutes. One student says scatter plot, another says broken-line graph. Who is right? Justify.

Check your understanding

Quiz

Which graph is best for showing whether longer pendulums take more time to swing?

Which graph is best for the population of a town from 2000 to 2025?

Which data suits a circle graph?

In a circle graph about 40 students, a category has 10 students. What is its angle?

Which justification is strongest?

A histogram is used for...

For parents and teachers

Big idea: The Ontario curriculum (D1.3) asks students to select from a variety of graphs, including scatter plots, the type that is best suited to the data, and to justify the choice. The key is to talk about the number of variables, the type of data, and what the reader should see.

Common difficulties:

Choosing a graph because it looks nice rather than because it fits the data.
Using a circle graph for data that isn't parts of a whole (heights, or survey questions where people chose more than one answer).
Using a scatter plot when one variable is a category.

How to help: When you see a graph in the news or a flyer, ask, "Why do you think they chose that graph? Would another one be better?" The lesson also builds A1 skills: communicating and justifying mathematical choices.