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

Making Scatter Plots

D1.3About 50 minutes

Learning goals

I can put the independent variable on the horizontal axis and the dependent variable on the vertical axis.
I can choose a sensible scale for each axis, including a broken axis when the data doesn't start near 0.
I can plot ordered pairs accurately and include a title, labels and a source.
I can read the values of a point on a scatter plot.

Words to know

scatter plot
A graph that shows two-variable data. Each ordered pair is plotted as one point. The points are not joined.

x-axis
The horizontal axis. In a scatter plot it usually shows the independent variable.

y-axis
The vertical axis. In a scatter plot it usually shows the dependent variable.

scale
The numbers along an axis, going up by equal steps (intervals).

broken axis
A small zigzag near the start of an axis that shows the scale skips from 0 to a larger number.

Remember

You already know how to plot points on a coordinate grid: go across to the first number, then up to the second number. A scatter plot uses exactly the same skill, with real data and real units.

One point per ordered pair

A scatter plot shows two-variable data. Each person, object or trial becomes one point. Person D in the Okafor table is (162, 165): go across to 162 cm on the height axis, then up to 165 cm on the arm-span axis, and make a dot.

The points are not joined with lines. They are separate people or trials, and nothing happens "between" Person D and Person E. (A broken-line graph joins points because it shows one thing changing over time.)

When all the points are plotted, you can see the pattern of the data as a whole.

Steps to make a scatter plot

Choose the axes. Independent variable on the x-axis (horizontal). Dependent variable on the y-axis (vertical).
Find the least and greatest value of each variable.
Choose a scale for each axis with equal intervals that covers all the values and uses most of the grid. If the values are far from 0, start at a convenient number just below the least value and draw a broken axis (a zigzag) to show the jump from 0.
Label each axis with the variable name and unit.
Plot each point carefully: across, then up.
Add a title that names both variables, and a source that says where the data came from.

The Okafor data: 10 points, one per person. Heights go from 150 to 180 cm and arm spans from 148 to 181 cm, so both axes start at 145 with a zigzag to show the scale skips from 0. Each grid step is 5 cm.

Worked example

Example 1: Choosing the scales for the Okafor data

Height goes from 150 to 180 cm. Arm span goes from 148 to 181 cm. Choose scales for both axes.

If the axes started at 0, all 10 points would be squeezed into the top-right corner. That wastes most of the graph.
Both variables are between about 145 and 185, so start both axes at 145, end at 185, and go up by 5. That gives 8 equal intervals.
Draw a zigzag on each axis near 0 to show the break.
Plot Person A: across to 150, up to 148. Person J: across to 180, up to 181. And so on.

Both axes: 145 to 185 cm, intervals of 5, with broken axes. The finished graph is shown above.

Worked example

Example 2: Plotting Maya's pendulum data

Make a scatter plot of Maya's means: (20, 9.1), (40, 12.6), (60, 15.6), (80, 17.8), (100, 20.2). Length in cm, time in s.

x-axis: string length (cm), the independent variable. Values go from 20 to 100. A scale of 0 to 100 by 20s fits well.
y-axis: time for 10 swings (s). Values go from 9.1 to 20.2. A scale of 0 to 25 by 5s fits well.
Plot each point. For (60, 15.6): across to 60, then up to just above the 15 line (15.6 is only a little more than 15, well below halfway to 20).
Title: "Pendulum Length and Time for 10 Swings". Add the source.

See the graph below. The points rise from left to right: longer strings take more time.

Maya's pendulum data. The x-axis goes from 0 to 100 cm by 20s, and the y-axis from 0 to 25 s by 5s. The 5 points are (20, 9.1), (40, 12.6), (60, 15.6), (80, 17.8) and (100, 20.2).

Worked example

Example 3: Reading a scatter plot

A soccer team manager recorded the daily high temperature and the litres of water the team drank at 10 practices. Use the graph below. (a) How much water did the team drink on the hottest day? (b) On how many practices did the team drink more than 15 L?

(a) The point farthest to the right is the hottest day. It sits above 31°C and at 21 L.
(b) Look for points above the 15 L grid line. They are at 26°C (17 L), 28°C (18 L), 30°C (19 L) and 31°C (21 L). The point at 25°C is on the 15 L line, so it is not more than 15.
Count them: 4 practices.

(a) 21 L at 31°C. (b) 4 practices.

10 practices. The temperature axis starts at 10°C after a zigzag; the water axis starts at 0. Points: (14, 9), (17, 10), (19, 12), (21, 12), (23, 14), (25, 15), (26, 17), (28, 18), (30, 19), (31, 21).

Common mistakes

Switching the order of a pair. (162, 165) is not the same as (165, 162). The first number always goes across.
Uneven scales, like 0, 10, 20, 50, 100. Every step must be the same size.
Joining the dots. Scatter plots show separate individuals.
No zigzag when an axis doesn't start at 0. Without it, readers may think the axis starts at 0.
Missing units in the axis labels.

Just for fun

Why won't the points on a scatter plot hold hands?

They're separate people. Joining them would be weird for everyone.

Hands-on activity

Plot your own data

You need: your pendulum table from last lesson (or the Okafor table), graph paper or squared paper, a ruler and a pencil. A free spreadsheet program also works.

Decide which variable goes on each axis.
Find the least and greatest values. Choose scales with equal intervals. Decide whether you need a broken axis.
Label the axes with names and units, and add a title and a source.
Plot every point. Check each one by reading it back: "across to __, up to __".
If you have a spreadsheet, enter the two columns, select them and insert a scatter chart. Compare it with your hand-drawn graph. Did the computer choose the same scales?

What to look for: every point matches a row of the table, and the graph fills most of the grid.

Your turn

On a scatter plot of "hours of practice" and "test score", which variable goes on the x-axis?

Use the height and arm-span graph. Which person has an arm span shorter than 150 cm?

Values of a variable go from 3.2 to 8.9. Suggest a scale.

Values of a variable go from 512 to 589. Would you start the axis at 0? Explain.

Why are the points on a scatter plot not joined?

Use the soccer graph. What was the temperature on the practice when the team drank 9 L?

More practice: warm-up

In the ordered pair (40, 12.6), which number do you find on the x-axis?

What does the zigzag on an axis mean?

Name three things every scatter plot needs besides the points.

Is 0, 5, 10, 15, 20 an equal-interval scale?

Use the pendulum graph. How long did 10 swings take with a 40 cm string?

More practice: on your own

The arm-span graph uses steps of 5 cm. How many grid steps is it from 145 cm to 185 cm?

A student plots the soccer point (21, 12) at 12 across and 21 up. Explain the mistake and where the point should be.

Choose scales for this data: (2, 140), (4, 155), (5, 162), (7, 171), (9, 186). The first value is age (years); the second is height (cm).

Use the arm-span graph. Which two people have the same height as each other, or is everyone different?

Use the soccer graph. On which two practices did the team drink the same amount of water? How can you tell?

Challenge

Points (165, 163) and (163, 165) are both plotted on the arm-span graph. Describe where each one sits compared with the other.

Mia draws a scatter plot where the x-axis goes 0, 10, 20, 30, 60, 90. What is wrong, and how does it change the look of her graph?

In the soccer graph, about what fraction of the practices had a high of 25°C or more? Write it as a percent too.

Check your understanding

Quiz

On a scatter plot, the dependent variable usually goes on...

Where is the point (60, 15.6)?

Data values go from 850 to 940. Which y-axis scale is best?

Why do we use a zigzag on an axis?

Which graph should not have its points joined?

In the soccer graph, how much water did the team drink when the high was 23°C?

For parents and teachers

Big idea: The Ontario curriculum (D1.3) asks Grade 8 students to select from a variety of graphs, including scatter plots, and to display data with appropriate sources, titles, labels and scales. This lesson teaches the scatter plot itself; the next lesson is about choosing between graph types.

Common difficulties:

Switching coordinates. Say "across, then up" out loud.
Starting every axis at 0, which squeezes the points into a corner, or the opposite: starting above 0 without a zigzag.
Unequal intervals when the data has a gap.
Joining the dots because broken-line graphs do.

How to help: Graph paper and a sharp pencil help a lot. If you have a spreadsheet program, let your child compare a computer-made scatter plot with their hand-drawn one and talk about the scales the computer chose.