Scale Distortion Examples in Math

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

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

Concept Recap

Scale distortion occurs when a graph's axis does not start at zero or uses inconsistent intervals, making small differences appear large or large differences appear small.

Zoom in on tiny differences to make them look huge, or zoom out to hide them.

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: Scale distortion is changing where an axis starts or how its intervals run to fake the size of a difference.

Common stuck point: The procedure for scale distortion is the easy part; the trap is calling every nonzero axis a distortion. Asking "Is the axis baseline or interval spacing making a difference look bigger or smaller than it really is?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the axis baseline or interval spacing making a difference look bigger or smaller than it really is?

Worked Examples

Example 1

easy

Two graphs show the same data (unemployment: 4% to 5%). Graph A: y-axis from 0–10%. Graph B: y-axis from 3.5%–5.5%. Describe what each graph communicates visually and which is more honest.

Answer

Graph A (wider scale) shows true magnitude honestly; Graph B's compressed scale makes the change look much larger.

First step

Graph A (0–10%): the 1% increase looks small — the bar rises modestly from 40% to 50% of the axis height

Full solution

2
Graph B (3.5–5.5%): the same 1% increase looks enormous — the bar rises from near the bottom to near the top of the compressed axis
3
True change: $\frac{5-4}{4} \times 100 = 25\%$ relative increase — significant but not catastrophic
4
Graph A is more honest for showing absolute magnitude; Graph B exaggerates relative change

Scale selection dramatically affects visual impression. There is no universally 'right' scale, but starting at 0 for bar charts prevents area distortion. For line charts, the appropriate scale depends on the question being asked (absolute change vs. relative change).

Example 2

medium

A graph uses a logarithmic scale for a dataset ranging from 1 to 1,000,000. Explain when a log scale is appropriate vs. misleading, and how to label it correctly.

Example 3

medium

A chart's y-axis starts at

80

and ends at

100

. Bars represent test scores

84

and

92

. Compare the visual pixel ratio of the bars to their true ratio.

Practice Problems

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

Example 1

easy

A pictograph shows a 2021 salary twice as large as a 2020 salary by doubling both the height AND width of a dollar bill icon. If the actual increase was 2×, by what factor does the icon area increase?

Example 2

hard

A company's revenue chart has an x-axis with uneven time intervals (2010, 2012, 2016, 2017, 2018). The line shows constant growth. Explain how uneven spacing creates scale distortion.

Example 3

easy

A y-axis starts at

20

instead of

0

. Two bars read

22

and

26

. What does the truncation do to the difference?

Example 4

easy

To HIDE a real difference between

1000

and

1200

, should you stretch or compress the y-axis?

Example 5

easy

A chart's y-axis goes

0, 1, 2, 4, 8

at equal pixel spacing. What kind of scale is this?

Example 6

easy

A dual-axis chart uses left axis

0

–

10

and right axis

0

–

1000

to overlay two lines. What distortion risk arises?

Example 7

easy

On a linear y-axis, the gap from

0

50

100

pixels. How many pixels is the gap from

50

100

Example 8

easy

A graph shows GDP in trillions on one axis and unemployment percent on another, scaled so they overlap perfectly. Is the overlap meaningful?

Example 9

easy

True or false: a y-axis starting at

0

guarantees there is no scale distortion anywhere in the chart.

Example 10

easy

A bar chart's y-axis is truncated to start at

90

. True values are

92

and

96

. What is the true percent difference?

Example 11

medium

A linear chart maps value

v

to pixels by

p = 10v

. A distorted version truncates the baseline to

b=80

p' = 10(v-80)

. For

v_1=90, v_2=85

, find the true height ratio and the distorted height ratio.

Example 12

medium

On a log

_{10}

axis, two points sit

2

tick units apart (each tick

=

one power of

10

). What is the true ratio of their values?

Example 13

medium

A news chart shows the y-axis inverted (high values at the bottom). Sales 'rise' on screen but the numbers are falling. What is the distortion called and why is it effective?

Example 14

medium

Two charts of the SAME data are shown. Chart A's y-axis is

0

–

1000

; chart B's is

400

–

500

. Which makes a change from

450

460

look bigger, and roughly how many times bigger?

Example 15

medium

A chart claims a stock 'crashed' using a y-axis from

99.50

100.00

where price went

99.95 \to 99.70

. What is the true percent change?

Example 16

medium

A scatterplot uses a log scale on BOTH axes and the points fall on a straight line. What kind of relationship does a straight line on a log-log plot indicate?

Example 17

medium

A bar chart compares 2020 and 2021 with the y-axis starting at

0

, but the 2021 bar is drawn

3\times

wider 'for design.' Values are

50

and

55

. Is the chart honest? Why or why not?

Example 18

medium

A temperature chart uses a y-axis

0

–

100

but the data (

68

–

72

) clusters tightly. A critic says 'this hides real variation.' Is using the full

0

–

100

range a distortion here?

Example 19

medium

A chart's y-axis is truncated to start at

40

with the top at

50

. A value goes from

44

46

. What fraction of the chart height does this change occupy, and what is the true percent change?

Example 20

challenge

A bar of true value

v

is drawn on a linear axis with baseline

b < v

, so pixel height

h(v) = k(v-b)

. Prove that the perceived ratio

h(v_1)/h(v_2)

equals the true ratio

v_1/v_2

if and only if

b = 0

Example 21

challenge

On a log

_{10}

axis, the segment from value

a

to value

b>a

has the same pixel length anywhere on the axis as long as

b/a

is constant. A designer wants to make a

10\%

growth look as big as a doubling did on a linear chart. Explain why a log axis prevents this trick.

Example 22

challenge

Two honest-looking line charts of the same series differ only in aspect ratio: one is wide-and-short, one is tall-and-narrow. Explain how aspect ratio alone changes perceived slope, and what 'banking to 45 degrees' recommends.

Example 23

easy

A bar chart's y-axis goes from

50

60

. Two bars read

54

and

56

. Does the chart make the difference look bigger or smaller than it actually is?

Example 24

easy

A line chart's x-axis skips from

2010

directly to

2018

in one tick. What kind of distortion can this create?

Example 25

easy

A 3D bar chart uses tilted prisms instead of flat bars. How can this distort the reader's perception?

Example 26

easy

True or false: a pie chart whose slices don't sum to

100\%

is honest as long as labels are correct.

Example 27

easy

A truncated y-axis chart shows values

98

and

102

but starts at

97

. What is the true percent difference between the values?

Example 28

easy

On a y-axis from

0

200

, two bars show

40

and

60

. By what percent do they differ truly, and how big is the visual gap as a fraction of total chart height?

Example 29

medium

Two lines on the same chart use different y-axes (left

0

–

100

, right

0

–

10

). When they cross, what should a careful reader conclude?

Example 30

medium

A chart maps value

v

to pixels as

p = 5(v - 70)

. Two values are

80

and

90

. Find the true and the displayed ratios.

Example 31

medium

On a log

_{10}

axis, a point at

100

and a point at

10{,}000

are how many tick units apart?

Example 32

medium

A chart claims unemployment 'exploded' from

4.1\%

4.4\%

using a y-axis from

4.0

4.5

. What is the actual percent change in unemployment?

Example 33

medium

A chart compresses 20 years of data into a strip 100 pixels wide. The same data in a 1000-pixel-wide strip looks much more variable. Which is more honest?

Example 34

medium

A pictogram uses 3D barrels whose volumes encode revenue. One company's barrel is twice as tall and twice as wide as another's. By what factor does the volume scale?

Example 35

medium

A bar chart has y-axis

0

–

100

but inserts a wavy 'break' at the bottom. What is the purpose of the break, and what is the risk?

Example 36

medium

A bar chart shows two values

400

and

800

on a y-axis from

0

1000

. The true ratio is

2

. What is the pixel-height ratio?

Example 37

hard

A chart's y-axis is reversed (high values at the bottom). Stock prices rise on screen but the labels say they're falling. What category of distortion is this?

Example 38

hard

A truncated-axis chart starts at

b

. Show that the displayed ratio

(v_1 - b)/(v_2 - b)

overstates the true ratio

v_1/v_2

whenever

v_1 > v_2 > b > 0

Example 39

hard

Two charts show the same data but on aspect ratios

4:1

and

1:4

. Explain why the perceived 'volatility' of a line series differs even though the slopes are mathematically the same.

Example 40

hard

A bar of true value

v

is shown with pixel height

h(v) = k(v - b)

. For

b > 0

and

v_1 = 110

v_2 = 100

, the perceived ratio overstates the true ratio by what factor when

b = 90

Example 41

hard

A pictogram shows revenue with stacks of coins. Company A has

5

coins; company B has

10

. Each coin is the same size. Is this honest?

Example 42

challenge

On a log

_{10}

y-axis a slope of

0.3

per unit

x

implies what kind of growth in

y

← Back to Scale Distortion Practice Mode

Related Concepts

Misleading Graphs

Background Knowledge

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

misleading graphs