D. Data · Data Literacy and ProbabilityMisleading Graphs, Challenging Ideas and Convincing Arguments
D1.6 D1.3 A1About 50 minutes
Learning goals- I can spot ways that histograms, broken-line graphs and bar graphs can be misleading.
- I can use data to challenge a preconceived notion.
- I can draw a conclusion and build a convincing argument with a claim, evidence and reasoning.
Words to know
misleading graph
A graph that gives the wrong idea about the data, on purpose or by mistake.
preconceived notion
An idea someone believes before looking at the evidence.
claim
The statement you are trying to show is true.
evidence
The data that supports a claim.
broken axis
A zigzag symbol on an axis that warns the scale does not start at 0.
Remember from Grade 5
You caught misleading bar graphs (a scale that doesn't start at 0) and claims based on small or biased samples, and you built arguments with data. Now you'll look for tricks in histograms and broken-line graphs too.
Same data, two stories
A winter carnival sold these cups of hot cocoa: 2023: 92, 2024: 95, 2025: 97, 2026: 100.
Both graphs below show exactly the same numbers. Look at them before reading on.
Graph 1 makes sales look like they exploded. The 2026 point sits 5 times as high above the bottom of the graph as the 2023 point. But sales only went from 92 to 100, an increase of 8 cups. Graph 2 shows the honest picture: a small, steady increase.
The trick in Graph 1 is that the scale starts at 90, not 0, and there is no broken-axis symbol to warn you. Its headline, "SOARING!", pushes you to believe it.
Tricks to watch for
- The scale doesn't start at 0 (with no warning), so small differences look huge.
- Squashed or stretched axes. A tall, narrow broken-line graph makes changes look dramatic; a short, wide one makes them look tiny.
- Uneven steps on a scale. For example, a broken-line graph whose years go 2010, 2020, 2024, 2025, 2026, equally spaced, so the change looks steady when it isn't.
- Unequal intervals in a histogram. A bar for "30 to 60 minutes" is 3 times as wide an interval as "0 to 10 minutes", so it catches more values just because it is wider.
- Cherry-picking: showing only the part of the data that supports a claim.
- Missing source, a tiny sample or a biased sample.
Worked exampleExample 1: Is the claim fair?
A poster uses Graph 1 and says: "Cocoa sales have more than doubled since 2023!" Is this claim fair?
- Read the numbers, not just the picture: 92 in 2023 and 100 in 2026.
- Doubling 92 would be 92 × 2 = 184. Sales are only 100.
- The increase is 100 − 92 = 8 cups, less than one tenth of 92.
- The graph only looks like doubling because its scale starts at 90.
No. Sales grew by only 8 cups, from 92 to 100. The graph's scale starts at 90, which makes the increase look much bigger.
Challenging preconceived notions
A preconceived notion is an idea people believe before checking the evidence, like "Kids don't read anymore" or "It always snows on the first day of winter break." Data lets us test these ideas.
To make a convincing argument, use:
- Claim: what you think is true.
- Evidence: numbers from your data (measures like the median, counts, fractions).
- Reasoning: explain how the evidence supports the claim.
- Limits: be honest about what your data can't tell you.
Worked exampleExample 2: Build a convincing argument
Someone says, "Grade 6 students don't read for fun anymore." Maya surveyed a random sample of 60 Grade 6 students at her school about the minutes they read for fun yesterday. Her histogram shows 8 students read 0 minutes, and the median was 25 minutes. Write an argument.
- Claim: Most Grade 6 students at our school do read for fun.
- Evidence: 52 of 60 students (60 − 8) read for fun yesterday. That's 52/60, more than 8/10. The median was 25 minutes.
- Reasoning: if most students didn't read, the median would be 0. A median of 25 minutes means at least half the students read for 25 minutes or more.
- Limits: this is one day at one school. A different day, or another school, might give different results.
"The data does not support the idea that Grade 6 students don't read for fun. In a random sample of 60 students, 52 read for fun yesterday, and the median was 25 minutes. This was only one day at our school, so we could collect more data to be sure."
Question respectfully
When a graph or claim seems wrong, be a respectful skeptic. Ask questions like "Where did this data come from?" or "What does the scale start at?" instead of "That's wrong!" And be ready to change your own mind when good evidence shows a preconceived notion of yours was wrong. That's what good thinkers do.
Just for funWith a vertical scale that starts at 99 and ends at 100, even a snail crawling from 99.1 m to 99.9 m would look like a rocket launch. Always check where the scale starts!
Hands-on activityTest a family notion
You need: paper and a pencil, and a family notion to test, such as "Our family is always late," "The dog sleeps all day" or "The bus is always late when it snows." Keep it kind!
- Write the notion as a question you can answer with data: "How many minutes late are we when we leave for school?"
- Collect data for at least 5 days (or use a secondary source, like weather records).
- Find the median and range, or make a graph.
- Write a claim, evidence, reasoning and limits paragraph.
- Share it. Did the data support the notion or challenge it?
Use this graph for the Your turn questions. The vertical scale starts at 44 and counts by 1s. Science centre 48 votes, Zoo 45 votes.Your turn
1Look at the bar graph. About how many times as tall as the Zoo bar does the Science centre bar look?
About 4 times as tall. The Zoo bar is 1 step high (44 to 45), and the Science centre bar is 4 steps high (44 to 48).
2How many votes did each trip really get? How many more votes did the science centre get?
Science centre 48, Zoo 45. The science centre got 3 more votes.
3What makes this graph misleading?
The scale starts at 44, not 0, with no broken-axis warning. A small difference of 3 votes looks huge.
4How would you redraw it fairly?
Start the vertical scale at 0 and count by 5s or 10s up to 50. Both bars would look almost the same height, which is the truth.
5A broken-line graph shows the years 2000, 2010, 2020, 2025, 2026, all equally spaced. Why could this be misleading?
The steps are uneven (10 years, 10 years, 5 years, 1 year) but they look equal, so the speed of change is shown wrongly.
6A claim says, "9 out of 10 students love our new snack!" But only 10 students were asked, all from the snack club. What two problems are there?
The sample is tiny (only 10 students), and it is biased: snack club members are likely to love snacks.
7Write a claim, evidence and reasoning about the cocoa data that would be fair.
Sample: Claim: Cocoa sales grew a little each year. Evidence: 92, 95, 97, then 100 cups. Reasoning: each year was higher than the last, but the total increase was only 8 cups, so the growth is small and steady.
More practice: warm-up
1What is a preconceived notion?
An idea someone believes before looking at the evidence.
2What should you check first on any graph's vertical axis?
Where the scale starts (is it 0?) and what steps it counts by.
3Cocoa data: by how many cups did sales increase from 2023 to 2026?
4What does a zigzag on an axis tell you?
That the scale doesn't start at 0 (it's a broken axis).
5Name the three main parts of a convincing argument.
Claim, evidence and reasoning (and it's good to add limits).
More practice: on your own
1A histogram of homework times uses the intervals 0–10, 10–20, 20–30 and 30–60 minutes, drawn the same width. The 30–60 bar is tallest. Why is this misleading?
The last interval is 30 minutes wide, three times as wide as the others, so it catches more values just because it is wider. All intervals should be the same width.
2A broken-line graph of a puppy's mass is drawn very tall and narrow. What impression might it give?
It might make the puppy's mass look like it is increasing very fast, because a narrow, tall graph makes the line look steep.
3A store's ad shows only the three warmest days of a cold week to claim "This week was warm!" What is this trick called?
Cherry-picking: choosing only the data that supports the claim.
4Notion: "It snows more in December than in March where we live." What data could test it?
Sample: the total snowfall for each December and each March over several years, from a secondary source such as Environment and Climate Change Canada's historical weather data. Compare the means or medians.
5Graph 2 above is fair. Which would you show if you were trying to convince someone sales were growing fast? Why is that a problem?
Graph 1 makes the growth look fast, but using it to convince people is misleading. A convincing argument should be honest; the real growth is only 8 cups.
Challenge
1Using the cocoa data, find the percent increase from 2023 to 2026, to the nearest whole percent. (Hint: increase ÷ starting value.)
About 9%. Increase: 8 cups. 8 ÷ 92 ≈ 0.087, which is about 9%. That's far from the "more than doubled" (over 100%) the poster claimed.
2A broken-line graph's vertical scale goes 0, 10, 20, 50, 100 with equal spacing. A value goes from 20 to 100. Explain how this hides the real change.
From 20 to 100 is an increase of 80, but because the steps get bigger (10, 10, 30, 50) while the spacing stays the same, the line climbs only 2 steps, the same as from 0 to 20. The big increase looks small.
3Write a convincing argument, using data, that challenges this notion: "Nobody in Room 6 likes skating." Use the sample infographic: 8 of 30 students chose skating as their favourite snow activity.
Sample: "The notion is not supported by the data. In Room 6's survey, 8 of 30 students (more than 1/4 of the class) chose skating as their favourite snow activity. Since 8 students picked it as their favourite, many students clearly like skating. Limits: the survey only asked for a favourite, so other students might like skating too, even if they liked something else more."
Quiz
1. A bar graph's scale starts at 50. Bar A shows 52 and Bar B shows 56. What is the real difference?
2. Which makes a histogram misleading?
3. What is a preconceived notion?
4. Which argument is most convincing?
5. "Cherry-picking" data means:
6. Cocoa sales went from 92 to 100 cups. Which statement is fair?
What this lesson teaches: analysing data presented in different ways, including misleading histograms, broken-line graphs and bar graphs; challenging preconceived notions; drawing conclusions; and making convincing arguments (D1.6, D1.3). Questioning respectfully and being willing to change one's mind are Strand A skills (A1).
How to help at home:
- When you see a graph in an ad or the news, ask together: "Where does the scale start? Where did the data come from? How many people were asked?"
- Model claim–evidence–reasoning: "I think... because the data shows..."
- Show that adults change their minds when evidence is good, too.
Common difficulties:
- Judging by bar height or line steepness without reading the scale.
- Accepting claims based on tiny or biased samples.
- Writing arguments with opinions but no numbers, or with numbers but no reasoning.