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Teacher Guide: Misleading Graphs

Learn to spot graphs that trick the eye: truncated axes, stretched scales, uneven intervals, oversized pictures and pie charts that do not add up to 100%.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Statistical Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify truncated axes, stretched or squashed scales and uneven intervals on a graph
  • Compute how many times taller a bar looks on a truncated axis and compare with the real ratio
  • Calculate the real percent change from values read on a graph
  • Explain why pictures scaled in two or three directions exaggerate differences
  • Check that a pie chart adds up to 100% and spot cherry-picked time ranges
Prerequisites
  • • Reading bar graphs and line graphs (lessons Reading Bar Graphs and Reading Line Graphs)
  • • Reading pie charts (lesson Reading Pie Charts)
  • • Computing percentages and ratios
  • • Area of rectangles
Discussion Starters
  • 1. Where have you seen a graph that made something look bigger or smaller than it really was?
  • 2. Why might a company want its axis to start at 95 instead of 0?
  • 3. If a graph has true numbers on it, can it still lie? How?
  • 4. When would it be fair for a line graph not to start at 0?
  • 5. Why do our eyes compare the area of pictures instead of their height?
Common Misconceptions

If the numbers on a graph are correct, the graph cannot be misleading

Remediation: Show the two visitor graphs from this lesson side by side. The numbers are identical, yet one shows a rocket and the other a flat line. Ask which one matches a 5% change.

A bar that looks twice as tall means twice as much

Remediation: Cover the bottom of a fair bar graph with paper so only the top shows. Students see the bars suddenly look very different, although the values did not change.

Percent change is the difference divided by the bigger number

Remediation: Use the price of a snack: from 2 to 3 is a 50% rise, from 3 to 2 is a 33% fall. The base is always the starting value.

Differentiation Ideas

For Struggling Students:

  • • Start with bar graphs only and ask: where does the axis start?
  • • Give the drawn heights and the real values in a table and compare them
  • • Use whole-number percent changes (10%, 20%, 25%, 50%)

For On-Level Students:

  • • Compute visual ratios on truncated axes and real percent changes
  • • Find the mistake in pie charts and picture graphs
  • • Redraw a misleading graph fairly

For Advanced Students:

  • • Find where a truncated axis starts from the drawn bar heights
  • • Explore 3D graphs: a scale factor k multiplies the volume by k3k^3
  • • Analyse a real news graph and write a short report on what it hides
Standards Alignment
  • 7.RP.A.3 (CCSS.MATH.CONTENT.7.RP.A.3)

    Use proportional relationships to solve multistep ratio and percent problems, including percent increase and decrease

  • 6.SP.B.5 (CCSS.MATH.CONTENT.6.SP.B.5)

    Summarize numerical data sets in relation to their context

  • 7.G.A.1 (CCSS.MATH.CONTENT.7.G.A.1)

    Solve problems involving scale drawings of geometric figures, including computing actual areas from a scale drawing

Lesson Resources
  • visualSame Data, Two Axes

    Two line graphs of the same visitor numbers: one axis starts at 980, the other at 0

  • activityGraph Detective

    Students bring a graph from the news or an advert, find the trick and compute the real change

  • activityMake It Look Big

    Groups get the same small data set; half draw a fair graph, half draw the most dramatic graph they can, then compare

  • worksheetFix the Graph

    Redraw five misleading graphs fairly and write one honest sentence about each

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A misleading graph shows correct numbers in a way that gives the wrong impression. The data can be true while the picture lies.
The most common tricks:
  • Truncated axis: the vertical axis does not start at 0, so small differences look huge
  • Stretched or squashed scale: the same change looks steep or flat depending on how far apart the axis numbers are
  • Uneven intervals: the steps on an axis are not equal (for example 2000, 2010, 2020, 2022), so slopes cannot be compared
  • Picture and 3D graphs: an icon that is twice as tall *and* twice as wide has 4 times the area
  • Broken pie charts: the slices must add up to exactly 100%
  • Cherry-picked time range: only the part of the data that supports a claim is shown
  • Missing labels: no title, no units or no scale, so you cannot check anything
The cure is to ignore the picture for a moment and compute with the numbers:
percent change=new−oldold×100%\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%

Worked Examples

In a school vote, Juice A got 52 votes and Juice B got 48 votes. A poster shows a bar graph whose vertical axis starts at 45. How many times taller does bar A look, and how much bigger is A really?
1

Find the drawn heights above the start of the axis

A: 52−45=752 - 45 = 7, B: 48−45=348 - 45 = 3 → Drawn heights 7 and 3
2

Compare the drawn heights

73≈2.33\frac{7}{3} \approx 2.33 → Bar A looks about 2.3 times as tall
3

Compare the real values

5248≈1.08\frac{52}{48} \approx 1.08 → A is really about 1.08 times B
4

Write the real difference as a percent

52−4848×100%≈8.3%\frac{52 - 48}{48} \times 100\% \approx 8.3\% → About 8.3% more votes

Common Mistakes

Judging the size of a difference by the bar heights alone

Why it's wrong: If the axis starts at 45 instead of 0, the bars show only the part above 45. A bar for 52 then looks more than twice as tall as a bar for 48.

Correct: Always look at where the axis starts. Compare the real values: 5248≈1.08\frac{52}{48} \approx 1.08, only about 8% more.

Dividing by the new value when computing percent change

Why it's wrong: Percent change always compares with the starting point. From 1000 to 1050, dividing by 1050 gives about 4.8%, which is not the change.

Correct: Divide the change by the old value: 1050−10001000×100%=5%\frac{1050 - 1000}{1000} \times 100\% = 5\%.

Thinking a steep line always means a big change

Why it's wrong: How steep a line looks depends on the scale. A stretched axis makes 5% look like a rocket, and a squashed axis makes 25% look flat (see the rent graph above, whose axis goes up to 10000).

Correct: Read the numbers at the start and the end of the line and compute the change yourself.

Comparing picture or 3D graphs by height only

Why it's wrong: Our eyes judge the area (or the volume) of a picture. An icon twice as tall and twice as wide has 4 times the area; a 3D box with every edge doubled has 8 times the volume.

Correct: Compare the numbers the pictures stand for, not the size of the pictures.

Why It Matters

Graphs are everywhere: news reports, adverts, election polls, company reports and social media posts. Many are made to persuade, not just to inform.
  • Adverts: a battery that lasts 21 hours instead of 20 can be drawn as a bar twice as tall
  • News and politics: a poll lead of 2 points can be made to look like a landslide
  • Money: an investment fund can show only its two best years and hide five bad ones
  • Science and health: a stretched axis can make a tiny effect look dramatic, or a squashed one can hide a real problem
If you know the tricks, you can check the numbers in a few seconds and decide for yourself. That is one of the most useful math skills for everyday life.

Real World Applications

Adverts and Product Claims

Adverts often use bar graphs with a truncated axis to make a small advantage look huge.

Example:

A battery advert shows 21 hours for its battery and 20 hours for a rival. With the axis starting at 19, the bars are drawn 2 and 1 units tall, so one bar looks twice as long.

1Try It Yourself

Battery A lasts 21 hours and battery B lasts 20 hours. The advert's bar graph starts its axis at 19.

By what percent does battery A really last longer than battery B?

Step 1: Write the mathematical expression

Compute the percent: change divided by the value of B, times 100

Investment Funds and Cherry-Picked Time Ranges

A fund can make itself look good by showing only the years when its price went up.

Example:

A fund price was 80 in 2019 and fell to 50 in 2022, then rose to 60 in 2024. The advert shows only 2022 to 2024: "up 20%!"

2Try It Yourself

The fund price was 80 at the start of 2019 and 60 at the end of 2024. The advert shows only the last two years, when the price rose from 50 to 60.

What is the real percent change over the whole time, from 80 to 60?

Step 1: Write the mathematical expression

Compute: (new - old) / old × 100

Infographics With Pictures

News infographics often show amounts as pictures (bags, coins, people) and scale the pictures in both directions.

Example:

A school collected 2 tons of paper for recycling last year and 3 tons this year. The infographic makes the second paper bag 1.5 times as tall and 1.5 times as wide.

3Try It Yourself

The amount grew from 2 tons to 3 tons, so the real ratio is 1.5. The picture of this year's bag is 1.5 times as tall and 1.5 times as wide.

How many times bigger is the area of the new bag picture?

Step 1: Write the mathematical expression

Multiply the two scale factors:

Key Takeaways

  • 1A misleading graph can show true numbers and still give a false impression
  • 2Always check where the vertical axis starts: an axis that does not start at 0 makes small differences look big
  • 3Stretched or squashed scales change how steep a line looks, so read the numbers instead of the slope
  • 4Check that the steps on each axis are equal, and that the time range is not cherry-picked
  • 5Pictures that grow in two directions exaggerate: double height and width means 4 times the area
  • 6Pie chart slices must add up to 100%; compute the real change with new−oldold×100%\frac{\text{new} - \text{old}}{\text{old}} \times 100\%

Frequently Asked Questions

Is a graph whose axis does not start at 0 always misleading?

Not always. For line graphs of things like body temperature or a share price, starting near the data can be useful to see small changes, as long as the axis is clearly labelled. For bar graphs it is almost always misleading, because the length of the bar is the message.

How do I find the real percent change from a graph?

Read the starting value and the final value from the axis, then compute new−oldold×100%\frac{\text{new} - \text{old}}{\text{old}} \times 100\%. Ignore how steep or tall the picture looks.

Why is a 3D graph worse than a flat one?

3D effects add depth, so our eyes compare volumes. A box with every edge twice as long has 23=82^3 = 8 times the volume. 3D pie charts also tilt the circle, so front slices look bigger than back slices.

Can a misleading graph contain only true numbers?

Yes, and that is what makes it tricky. The numbers can all be correct while the scale, the pictures or the chosen time range create a false story. That is why you should always check the numbers yourself.

Glossary

Misleading graph
A graph that gives a wrong impression of the data, even if the numbers on it are true
Truncated axis
An axis that does not start at 0, so only the top part of each value is drawn
Scale
The numbers on an axis and the distance between them; it decides how big a change looks
Interval
The step between two neighbouring numbers on an axis; a fair axis uses equal intervals
Percent change
The change divided by the starting value, times 100%
Cherry-picking
Showing only the part of the data that supports a claim and hiding the rest

Formula Card

Percent change

new−oldold×100%\frac{\text{new} - \text{old}}{\text{old}} \times 100\%

The real size of a change, whatever the graph looks like

How tall bars look on a truncated axis

a−sb−s\frac{a - s}{b - s}

Bars for values a and b on an axis starting at s

Picture scaled by k in both directions

area×k2\text{area} \times k^2

Twice as tall and twice as wide: 4 times the area; in 3D the volume is multiplied by $k^3$

More in This Topic

Reading Bar Graphs

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Reading Line Graphs

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Reading Pie Charts

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Dot Plots

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