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

Outliers in Two-Variable Data

D1.5 D1.6About 45 minutes
Learning goals
  • I can spot an outlier on a scatter plot.
  • I can describe a relationship with and without its outliers.
  • I can decide what to do with an outlier by looking for its cause.

Words to know

outlier
A point that is far away from the pattern of the other points.
measurement error
A mistake made while measuring or recording, like reading the wrong number or switching digits.
cluster
A group of points that are close together.

Remember from Grade 7

In Grade 7, an outlier was a value much higher or lower than the rest of a one-variable data set, and you saw how it pulls the mean up or down. On a scatter plot, an outlier is a point that doesn't fit the pattern of the other points, even if neither of its numbers is unusual on its own.

The point that doesn't fit

A science teacher asked 11 students how many minutes they studied for a quiz and recorded their scores. Ten points rise in a neat line. One point sits far below them: a student who studied 50 minutes but scored only 42%.

That point is an outlier. 50 minutes is not unusual, and other students scored in the 50s. What is unusual is the pair: lots of studying with a very low score.

Study Time and Quiz Score (11 Students)01020304050604050607080901000Minutes studyingQuiz score (%)Source: Grade 8 science class survey
Ten points rise from (10, 52) to (55, 88). The highlighted point at (50, 42) is the outlier: far below the others at 50 minutes. The score axis starts at 40, after a zigzag.

How outliers change the description

  • Without the outlier: the 10 points rise steadily and stay close to a line. Strong positive.
  • With the outlier: the trend is still positive, but one point is far from the line. The relationship looks weaker.

So good data analysts describe the relationship both ways: "There is a strong positive relationship between study time and quiz score, with one outlier."

Outliers can also create a pattern that isn't really there, or hide a pattern that is.

A: one point far awaymakes a fake patternB: strong negativewith one outlier
A: ten points in a blob with no pattern, plus one point far up and to the right. That single point could fool you into seeing a positive relationship. B: a strong negative pattern with one highlighted point far below the others.

What should you do with an outlier?

Don't just erase it. Look for a reason.

  1. Is it a mistake? A wrong reading, a typo (switched digits, a missing decimal point), or the wrong units. If you can fix it, fix it. If you can't, you may remove it, but say so in your report.
  2. Did something unusual happen? The student was sick, the ball hit a rug, there was a tournament at the rink. The point is real but measured something different. You might leave it out of the pattern and explain why.
  3. Is it real and just unusual? Keep it. It's part of the story. Describe the relationship with and without it.
Worked example

Example 1: Describe with and without

Describe the study-time data with and without the outlier at (50, 42).

  1. Without it: 10 points from (10, 52) to (55, 88), always rising and close to a line. Strong positive.
  2. With it: still rising overall, but one point is about 40 points below where the others would predict at 50 minutes. The relationship looks weaker.
  3. Find out more: the teacher learned that this student was home sick the day before and studied from the wrong chapter.

Strong positive without the outlier; positive but weaker with it. Because the student studied the wrong chapter, the point doesn't measure the same thing, so it makes sense to leave it out of the trend and say why.

Worked example

Example 2: A typing mistake

In the Okafor table, one person's data was typed as (165, 561). What probably happened?

  1. An arm span of 561 cm is over 5 metres. That is impossible for a person.
  2. The digits 1, 6, 5 look like 165 reversed. The person's height is also 165 cm.
  3. Check the original notes: the arm span was really 163 cm, so the typed value was a mistake.

A recording error. Fix it to the correct value from the notes (163 cm). Never keep an impossible value, and never guess: check the source.

Worked example

Example 3: A real outlier

On one more day at the rink, it was −2°C but the canteen sold 85 cups of hot chocolate. That day there was a hockey tournament with hundreds of visitors. What should the canteen manager do with this point?

  1. On the other −2°C day, 38 cups were sold. 85 is far above the pattern.
  2. It's not a mistake: the tournament brought extra customers.
  3. The tournament day measured something different (a huge crowd), so it doesn't fit the temperature pattern.

Keep it in the records but describe it separately: "Strong negative relationship between temperature and sales, with one outlier on a tournament day." It also gives the manager useful information: order extra supplies for tournament days.

Be honest with data

Removing points just because they don't match what you hoped to find is not honest. Remove a point only when you have a good reason, and always tell your reader what you removed and why. This is part of thinking critically and making convincing arguments.

Just for fun

Every class photo has one: 29 people smiling at the camera and one person looking at a squirrel. That's an outlier. Before you crop it out, find out what the squirrel was doing.

Hands-on activity

Outlier on purpose

You need: a ball that bounces (a tennis ball works), a metre stick or measuring tape, tape, and a partner.

  1. Tape the measuring tape to a wall. Drop the ball (don't throw it) from 25, 50, 75, 100, 125 and 150 cm on a hard floor. Record the bounce height to the nearest cm.
  2. Now drop it from 100 cm onto a rug, towel or cushion. Record that bounce too.
  3. Plot all 7 points. Which one is the outlier? Highlight it.
  4. Describe the relationship with and without the outlier. Write one sentence explaining what you would do with it and why.

What to look for: the hard-floor points rise in a strong positive pattern, and the soft-surface point sits far below it. It's an outlier with a known cause.

Your turn

1

What is an outlier on a scatter plot?

2

Data: (2, 4), (3, 6), (4, 8), (5, 30), (6, 12), (7, 14). Which point is the outlier?

3

Describe the relationship in the data above with and without the outlier.

4

A student's height is recorded as 16.5 cm. What probably happened?

5

In the study-time graph, at what score would you expect the 50-minute student to be, based on the other students?

6

Give two good reasons to remove an outlier.

More practice: warm-up

1

True or false: you should always delete outliers.

2

Data: (1, 10), (2, 20), (3, 30), (4, 5), (5, 50). Which point is the outlier?

3

If you remove an outlier, what must you do?

4

An arm span of 561 cm is recorded. Possible or impossible?

5

Does an outlier usually make a relationship look stronger or weaker?

More practice: on your own

1

Data on age (years) and height (cm) of children: (4, 103), (6, 116), (8, 128), (10, 138), (12, 51), (14, 163). Find the outlier and suggest what happened.

2

Describe the age and height data with and without the outlier.

3

In panel A of the outlier picture, why is the single point at the top right dangerous?

4

A runner's time for 5 km is recorded as 12 minutes, while everyone else in the club took 22 to 35 minutes. Should you remove it?

5

Write a complete description of the study-time data that mentions the outlier.

Challenge

1

The study-time graph's 10 regular points have a mean score of (52 + 58 + 60 + 66 + 70 + 71 + 78 + 80 + 85 + 88) ÷ 10. Find it, then find the mean score with the outlier included.

2

Panel B shows a strong negative relationship with one outlier far below the line. Make up a real-life story that could explain such a point for the variables "temperature" and "hot chocolate sold".

3

Avi says, "If I remove enough points, my data will show the strong relationship I predicted." Explain why this is a problem.

Check your understanding

Quiz

1.

Data: (1, 3), (2, 5), (3, 7), (4, 25), (5, 11). Which point is the outlier?

2.

What should you do first when you find an outlier?

3.

An outlier in a strong positive pattern usually makes the relationship look...

4.

A recorded height of 1 650 cm is most likely...

5.

Which description is the most complete?

6.

In the study-time graph, the outlier is at...

For parents and teachers

Big idea: The Ontario curriculum (D1.5) asks students to describe relationships between two variables with and without outliers, and D1.6 asks them to analyse data and draw honest conclusions. The key habit is: find the cause before deciding what to do with an outlier, and always report any data that was removed.

Common difficulties:

  • Erasing outliers automatically, or keeping obvious typos.
  • Looking only at one number. On a scatter plot, a point can be an outlier even when each of its numbers is normal on its own.
  • Letting one point decide the pattern (panel A).

How to help: The ball-bounce activity makes outliers memorable. Ask, "Is this point a mistake, an unusual event, or a real part of the data?" The lesson also builds A1: honesty, critical thinking and clear explanations.