Misleading Graphs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Misleading Graphs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A misleading graph is a data visualization that distorts the true pattern through truncated axes, unequal intervals, cherry-picked data, or manipulated scales.

A graph can tell any story the creator wants by choosing which data to show, where to start the axis, and how to scale the bars — visual clarity requires honest design.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A misleading graph uses real data but distorts the design so you read the wrong story.

Common stuck point: The procedure for misleading graphs is the easy part; the trap is trusting a graph because the data is real. Asking "Does the picture exaggerate or hide a difference that the actual numbers don't support?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the picture exaggerate or hide a difference that the actual numbers don't support?

Worked Examples

Example 1

easy

A bar chart of company profits shows the y-axis starting at \$950M. Profit Year 1: \$960M, Year 2: \$970M. The bar for Year 2 appears to be twice as tall. Calculate the actual percentage increase and the visually implied increase.

Answer

Actual increase: ~1%. Visual impression: ~100%. Truncated y-axis creates massive deception.

First step

Actual increase:

\frac{970 - 960}{960} \times 100 = \frac{10}{960} \times 100 \approx 1.04\%

Full solution

2
Visual impression: Year 2 bar is twice as tall as Year 1 (y-axis starts at 950, not 0) → implies 100% increase
3
Deception: the truncated y-axis exaggerates the 1% actual increase to look like 100%
4
Fix: start y-axis at 0, or clearly mark the axis break with a symbol (//) to indicate non-zero start

Truncating the y-axis is one of the most common forms of misleading graphs. Area below the bars represents zero baseline; cutting it off makes small differences appear enormous. Always check the y-axis origin before interpreting bar charts.

Example 2

medium

A graph shows 'cases of disease over time' with no y-axis label or values. The line goes up sharply. List three questions you should ask before drawing conclusions from this graph.

Example 3

medium

A truncated bar chart shows two bars with true values

100

and

110

, but the y-axis starts at

95

. Compute the actual percent difference and the visual percent difference of bar heights above the baseline.

Example 4

medium

A pictogram doubles a square icon's side from

1

cm to

2

cm to represent a

2\times

increase. Compute the area ratio and explain what the honest scaling would be.

Example 5

hard

A chart claims 'support fell from

80\%

76\%

— a major collapse.' The y-axis runs

75\%

80\%

and bars sit at heights

5

and

1

. Compute the visual ratio, the true percentage-point drop, and the true relative drop.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A 3D pie chart shows three categories: A=50%, B=30%, C=20%. The chart is tilted so C appears largest. Explain why 3D effects distort pie charts.

Example 2

hard

A graph shows 'gun deaths rise after Stand Your Ground laws.' The y-axis is inverted (high values at bottom). Analyze how axis inversion misleads viewers.

Example 3

easy

A bar chart's y-axis starts at

90

instead of

0

. What visual effect does this create?

Example 4

easy

A pictogram doubles both the height AND width of an icon to show a value that doubled. What is wrong?

Example 5

easy

A 3D pie chart tilts so the front slice appears larger. Why is this misleading?

Example 6

easy

A line graph shows only the months where sales rose, omitting the months they fell. What technique is this?

Example 7

easy

Two bars look like one is twice the other, but the y-axis starts at

50

and the bars read

52

and

54

. What is the real ratio?

Example 8

easy

A graph uses unequal time intervals on the x-axis (

1990, 2000, 2005, 2006, 2007

) but equal spacing. What does this distort?

Example 9

easy

Is a bar chart with a y-axis starting at

0

guaranteed to be honest? Yes or no, and why?

Example 10

easy

A chart compares 'total crime' in two cities of very different population using raw counts. What is the flaw?

Example 11

medium

A bar chart's y-axis runs

80

100

. Bar A reads

82

, bar B reads

98

. The bars appear at heights

2

and

18

pixels above baseline. By what factor does the chart visually overstate the true ratio

B/A

Example 12

medium

A magazine shows incomes as stacked coins where coin stack height is proportional to income, but the coin diameter also grows with the stack. Income B is

3\times

income A. What visual area ratio results if both width and height scale by

3

Example 13

medium

A dual-axis chart plots revenue (left axis

0

–

100

) and complaints (right axis

0

–

10

) so the two lines cross. Why can this falsely suggest correlation?

Example 14

medium

A poll reports '

70\%

prefer Brand X!' but the sample was

10

people. Why is the graph of this result misleading?

Example 15

medium

A graph's bars are colored so the disliked option is bright red and the favored one is calm green, but axes are correct. Is this a data distortion?

Example 16

medium

A trend line is drawn through scattered points, but the line is extended far beyond the data range to predict the future. What is the danger?

Example 17

medium

A bar for

100

and a bar for

200

are drawn, but the

200

bar is also made twice as wide 'for emphasis.' What is the visual area ratio?

Example 18

medium

A chart shows 'crime up

100\%

!' going from

1

incident to

2

incidents in a small town. Why is the graph misleading despite the math being correct?

Example 19

medium

A pictogram of money bags: bag for $4M is drawn twice as tall as the bag for $1M. If the artist keeps the shape's proportions (width scales with height), what visual area ratio appears, and is it faithful to the

4\times

value?

Example 20

challenge

A bar chart truncates its y-axis to start at value

b

. Two bars have true values

v_1 > v_2 > b

. Show that the ratio of drawn pixel heights is

\frac{v_1-b}{v_2-b}

and explain why choosing

b

close to

v_2

makes the exaggeration unbounded.

Example 21

challenge

Design a single honest bar chart, then list the THREE independent edits that would each separately make it misleading without changing any underlying value. Name them.

Example 22

challenge

A pictogram represents quantity by a sphere's radius. Income doubles, so the artist doubles the radius. By what factor does the perceived volume change, and what should the radius factor have been for an honest doubling?

Example 23

easy

A bar chart's y-axis runs from

60

100

. The bars read

70

and

90

. Compute the true ratio of the larger to the smaller, and the visual ratio of the bar heights above the baseline.

Example 24

easy

A pictogram represents $10M with a coin of radius

1

cm and $40M with a coin of radius

2

cm. By what factor does the area of the larger coin exceed the smaller, and what factor would be honest for

4\times

value?

Example 25

easy

Honest bar chart: Brand A sells

200

units, Brand B sells

250

units, y-axis starts at

0

. By roughly what percent is B greater than A? And how does this look on the chart?

Example 26

easy

A pie chart shows

30\%

30\%

40\%

. The artist colors the

40\%

slice in a dim gray and the two

30\%

slices in bright yellow. Is the chart numerically misleading?

Example 27

medium

A pictogram represents population by stick figures of height

h

and width

w

. To honestly show double the population using a figure twice as tall, what must the width do?

Example 28

medium

A truncated-axis chart shows bars at heights

20

and

80

pixels above baseline

b

. The true values are

60

and

120

. Find

b

Example 29

medium

A dual-axis chart shows ice-cream sales (left axis,

0

100

) and shark attacks (right axis,

0

5

) tracking each other. Does the visual correlation prove causation?

Example 30

medium

A graph claims '

1000\%

rise in scam reports.' Reports went from

1

in 2020 to

11

in 2024. Is the claim numerically correct? Is the graph honest?

Example 31

medium

A bar chart's y-axis is logarithmic (

1, 10, 100, 1000

). Two bars sit at

10

and

100

. The bars look equally spaced from

1

. Is the chart misleading?

Example 32

medium

A school reports test scores per teacher but omits class sizes. Teacher A: mean

80

from

4

students. Teacher B: mean

78

from

40

students. Why is a bar chart of just the means misleading?

Example 33

medium

A trend line is fit to noisy points from

x=0

x=10

and then extended to

x=50

to predict the future. Why is this risky?

Example 34

medium

A bar chart of test scores shades the bars red for boys and pink for girls. Beyond color, the bars are otherwise correct. Is this a numerical distortion or a framing bias?

Example 35

hard

A bar chart shows true values

a

and

b

on a truncated y-axis with baseline

t

(with

t<a<b

). For the visual height ratio

\frac{b-t}{a-t}

to equal a given exaggeration factor

k>1

times the true ratio

b/a

, find

t

in terms of

a,b,k

Example 36

hard

A pictogram shows three-dimensional spheres whose volumes are meant to represent the values

1, 8, 27

. To be honest by volume, what is the ratio of the displayed radii?

Example 37

hard

A magazine's bar chart of average incomes uses three-dimensional rectangular blocks whose height, width, and depth all scale with the value. If incomes are in ratio

1:2

, what is the perceived volume ratio?

Example 38

hard

A line graph plots a stock price with the y-axis inverted (high prices at the bottom). The line slopes up to the right. Which way did the price actually go, and what is the deception?

Example 39

challenge

You see a bar chart with bars of heights

h_1

and

h_2

above baseline

b

. The legend hides the true values. List three pieces of information you must know to recover the true ratio of values.

Example 40

challenge

Design a chart that uses NO numerical distortion but still creates a strong false impression. Describe the technique.

← Back to Misleading Graphs Practice Mode

Related Concepts

Data Visualization Scale Distortion

Background Knowledge

These ideas may be useful before you work through the harder examples.

data visualization