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

Misleading Graphs and Misleading Claims

D1.6 D1.3 A1About 50 minutes
Learning goals
  • I can find features that make a bar graph, line graph or scatter plot misleading.
  • I can spot cherry-picked data, small or biased samples, and cause-and-effect claims that the data doesn't support.
  • I can redraw or rewrite a misleading graph or claim so it is fair.

Words to know

misleading graph
A graph that gives a wrong impression of the data, on purpose or by accident.
truncated axis
An axis that starts above 0, which can make small differences look huge.
cherry-picking
Choosing only the data that supports what you want to show and leaving out the rest.
biased sample
A sample that does not fairly represent the whole population.

Remember from Grades 6 and 7

You caught misleading bar graphs and broken-line graphs (scales that don't start at 0, uneven scales, missing labels) and misleading circle graphs (sectors not drawn to size, 3-D tilting, percentages that don't add to 100%). In Grade 8 you'll add scatter plots and misleading claims about relationships.

Same data, different story

Three Grade 8 classes ran a food drive for the local food bank. Class 8A posted Graph A on the school website: "8A crushed the competition!" Look at both graphs below. They use exactly the same numbers.

Food Drive: Kilograms Collected by ClassGraph A46485052548A8B8CGraph B01020304050608A8B8CSame data: 8A 52 kg, 8B 48 kg, 8C 50 kg
Graph A's scale starts at 46 kg with no zigzag, so 8A's bar (52 kg) looks three times as tall as 8B's (48 kg). Graph B starts at 0 and shows the real picture: the three classes collected similar amounts.
Worked example

Example 1: How much bigger does it look?

In Graph A, how many times as tall does 8A's bar look compared with 8B's? How many times as much did 8A really collect?

  1. Graph A starts at 46. 8A's bar shows 52 − 46 = 6 kg of height. 8B's bar shows 48 − 46 = 2 kg of height.
  2. 6 ÷ 2 = 3. So 8A's bar looks 3 times as tall.
  3. Really: 52 ÷ 48 ≈ 1.08. 8A collected only about 8% more than 8B, which is 4 kg more.

It looks 3 times as much, but 8A really collected only 4 kg (about 8%) more. The truncated axis makes a small difference look huge. A fair headline: "All three classes collected about 50 kg. Great teamwork!"

Ways graphs and claims can mislead

  1. Truncated axis without a zigzag (like Graph A).
  2. Uneven intervals, like 0, 10, 20, 50, 100.
  3. Stretched or squashed scales. Stretching the vertical axis makes a slope look steep and dramatic; squashing it makes a big change look flat.
  4. Cherry-picking: showing only some of the data, or only a short time window, that supports a point.
  5. Small or biased samples: "4 out of 5 people agree!" when only 5 friends were asked.
  6. Cause claims from relationships: "Drinking our juice raises your marks!" from a scatter plot.
  7. Missing information: no title, no labels, no units, no source, no sample size.
  8. Pictures sized wrong: in a pictograph, doubling the height and width of a picture makes it look 4 times as big.
All 12 studentsno relationshipOnly 4 students shownlooks strong positive
Left: all 12 students' data shows no pattern. Right: only 4 of those same points, picked because they rise, look like a strong positive relationship.
20231 carton(1 unit of sales)2025: misleading2 times as tall and wide= 4 times the area!2025: fair2 cartons, same size= 2 times the sales
Sales doubled from 2023 to 2025. Doubling the carton's height and width makes it cover 4 times the area (count the 4 dashed parts). The fair picture uses two cartons of the same size.
Worked example

Example 2: A juice ad

An ad says: "Students who drink SuperJuice every morning get higher marks! Our scatter plot proves it." The scatter plot shows 5 students, has no axis labels or source, and the marks axis starts at 80%. List what is misleading.

  1. Small sample: only 5 students. Were they chosen fairly? (Possibly cherry-picked.)
  2. Missing information: no labels, units or source, so we can't tell what was measured or by whom.
  3. Truncated axis: starting at 80% stretches small differences in marks.
  4. Cause claim: even if there were a relationship, it wouldn't prove the juice causes higher marks. A hidden variable (like eating a good breakfast, or family routines) could explain it.
  5. Who made it? The juice company benefits from the claim.

Five problems: tiny sample, missing labels and source, truncated axis, cause claimed from a relationship, and a source with a reason to be biased. A fair version would test many randomly chosen students, show the full scale, and say "related" rather than "causes".

Worked example

Example 3: Rewrite a misleading headline

A town's data shows a strong positive relationship between the number of outdoor rinks open each week and the number of hospital visits for wrist injuries. A headline says: "Rinks Cause Broken Wrists! Close Them Now!" Write a fairer headline and explain.

  1. The data shows a relationship, not proof of cause.
  2. A hidden variable could be icy winter weather: more rinks open in the coldest weeks, and icy sidewalks also cause falls.
  3. Skating may cause some injuries, but the data can't say how many. Closing rinks is a big decision that needs better evidence.

Sample: "Wrist Injuries Rise in Weeks When More Rinks Are Open: Is It Skating or Icy Sidewalks?" It reports the relationship honestly and points out a possible hidden variable.

Question respectfully

When you find a misleading graph, point to the feature, not the person: "The scale starts at 46, so the differences look bigger than they are. Could we redraw it from 0?" Also be fair to yourself: check your own graphs for these problems before you share them.

Just for fun

Pizza Palace's ad: "Our pizzas are WAY bigger!" Their graph's axis starts at 29 cm. Their pizza: 31 cm. The other shop's: 30 cm. Always check where the scale starts before you get excited about pizza.

Hands-on activity

Make it mislead, then fix it

You need: graph paper, a ruler, pencil, and the soccer water data (or your own data).

  1. Draw a misleading graph on purpose: a bar graph of the water at three practices (12 L, 14 L and 15 L) with the axis starting at 11 L and no zigzag. Write a dramatic headline.
  2. Draw the fair version: axis from 0 L with equal intervals. Write an honest headline.
  3. Show both to a family member without telling them which is which. Ask what each graph says. Then reveal the trick.

What to look for: with the axis starting at 11, the 15 L bar looks 4 times as tall as the 12 L bar (4 units versus 1 unit), even though 15 L is only 1.25 times 12 L.

Your turn

1

In Graph A, how tall does 8C's bar look compared with 8B's? How much did 8C really collect compared with 8B?

2

What is cherry-picking?

3

A poster says "9 out of 10 kids prefer Brand Q snowpants!" What question should you ask first?

4

Why is a stretched vertical axis misleading on a scatter plot?

5

A graph has no title, no axis labels and no source. Why can't you trust it?

6

A pictograph shows a small milk carton for 2023 and a carton twice as tall and twice as wide for 2025. What impression does that give, and what's the truth?

More practice: warm-up

1

What does a zigzag on an axis warn you about?

2

True or false: a scatter plot can prove that one variable causes another.

3

Name one feature that should always be on a graph.

4

Is "0, 5, 10, 20, 40" an even scale?

5

Why might a company's own survey be biased?

More practice: on your own

1

A bar graph's axis starts at 90. Bars show 92 and 98. How many times as tall does the second bar look? What is the true ratio?

2

A news site shows house prices for only the 3 months when prices rose, and says "Prices are skyrocketing!" Over the whole year, prices went down. What kind of misleading is this?

3

Look at the cherry-picking picture. Describe the relationship in each panel.

4

A scatter plot of 8 Grade 8 students shows a positive relationship between hours of video games and reaction time scores. A headline says "Video games make teens faster!" Give two reasons to be careful.

5

Fix this graph plan: "Bar graph of litres of water, scale 10, 12, 14, 18, 24, starting at 10 with no zigzag."

Challenge

1

Mila wants to make 8A's 52 kg look only a little bigger than 8B's 48 kg, but Jonah wants it to look much bigger. Describe the scale each would choose.

2

A company surveyed 1 000 people at its own stores and found 70% prefer its brand. Explain why the large sample doesn't make the result fair for all Canadians.

3

A graph of a town's average January temperature from 2000 to 2025 rises slowly by about 1°C. One version shows the vertical axis from −10°C to −8°C; another from −30°C to +30°C. Which one exaggerates and which one hides the change? Which is fairest?

Check your understanding

Quiz

1.

Graph A starts at 46 kg. 8A's bar (52 kg) looks how many times as tall as 8B's (48 kg)?

2.

Showing only the data that supports your point is called...

3.

"Students who eat breakfast get better marks, so breakfast causes good marks." What's the problem?

4.

Which sample is most likely biased?

5.

A picture's height and width are both doubled. Its area becomes...

6.

Which is the fairest way to draw a bar graph that compares amounts?

For parents and teachers

Big idea: The Ontario curriculum (D1.6) asks students to analyse data in various forms, including scatter plots and misleading graphs, to challenge preconceived notions, and to make convincing arguments and informed decisions. This lesson trains a habit your child will need for life: checking graphs and claims in ads, news and social media.

Common difficulties:

  • Trusting a graph because it looks official. Look for labels, sources and sample sizes.
  • Believing big samples are always fair. How people were chosen matters as much as how many.
  • Accepting "causes" claims from relationships.

How to help: Keep an eye out for graphs in flyers, sports broadcasts and news. Ask, "Where does the scale start? Who made this? How many people were asked? Does it prove a cause?" The lesson builds A1 skills such as critical thinking and questioning respectfully.