Practice Scale Distortion in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
A y-axis starts at 2020 instead of 00. Two bars read 2222 and 2626. What does the truncation do to the difference?

Example 2

hard
A pictogram shows revenue with stacks of coins. Company A has 55 coins; company B has 1010. Each coin is the same size. Is this honest?

Example 3

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.

Example 4

medium
A chart maps value vv to pixels as p=5(v−70)p = 5(v - 70). Two values are 8080 and 9090. Find the true and the displayed ratios.

Example 5

hard
A bar of true value vv is shown with pixel height h(v)=k(v−b)h(v) = k(v - b). For b>0b > 0 and v1=110v_1 = 110, v2=100v_2 = 100, the perceived ratio overstates the true ratio by what factor when b=90b = 90?

Example 6

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 7

easy
On a linear y-axis, the gap from 00 to 5050 is 100100 pixels. How many pixels is the gap from 5050 to 100100?

Example 8

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 9

medium
A bar chart shows two values 400400 and 800800 on a y-axis from 00 to 10001000. The true ratio is 22. What is the pixel-height ratio?

Example 10

easy
A bar chart's y-axis is truncated to start at 9090. True values are 9292 and 9696. What is the true percent difference?

Example 11

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

Example 12

hard
Two charts show the same data but on aspect ratios 4:14:1 and 1:41:4. Explain why the perceived 'volatility' of a line series differs even though the slopes are mathematically the same.

Example 13

challenge
A bar of true value vv is drawn on a linear axis with baseline b<vb < v, so pixel height h(v)=k(v−b)h(v) = k(v-b). Prove that the perceived ratio h(v1)/h(v2)h(v_1)/h(v_2) equals the true ratio v1/v2v_1/v_2 if and only if b=0b = 0.

Example 14

easy
If a pictogram doubles BOTH the height and the width of an icon to represent a 2×2\times change, by what factor does the icon's area grow?

Example 15

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 16

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 17

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 18

medium
A linear chart maps value vv to pixels by p=10vp = 10v. A distorted version truncates the baseline to b=80b=80: p′=10(v−80)p' = 10(v-80). For v1=90,v2=85v_1=90, v_2=85, find the true height ratio and the distorted height ratio.

Example 19

challenge
On a log10_{10} axis, the segment from value aa to value b>ab>a has the same pixel length anywhere on the axis as long as b/ab/a is constant. A designer wants to make a 10%10\% growth look as big as a doubling did on a linear chart. Explain why a log axis prevents this trick.

Example 20

hard
A truncated-axis chart starts at bb. Show that the displayed ratio (v1−b)/(v2−b)(v_1 - b)/(v_2 - b) overstates the true ratio v1/v2v_1/v_2 whenever v1>v2>b>0v_1 > v_2 > b > 0.
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Related Concepts

Misleading Graphs