Misleading Graphs

Show your work for each problem.

1.
The vertical axis of a bar graph starts at 90 instead of 0. What is the main danger?
1. a)Differences look smaller than they really are
2. b)The values can no longer be read
3. c)Differences look bigger than they really are
4. d)The bars swap their order
Answer: Differences look bigger than they really are
Only the part above 90 is drawn, so the bars show just the tops of the values. Small differences between the values then look much bigger than they are.
2.
A line graph shows that sales grew from 200 to 230. What is the real percent increase? Write only the number (for 12% write 12).
Answer: 15
The change is $230 - 200 = 30$ , and $\frac{30}{200} \times 100\% = 15\%$ .
3.
A pie chart shows four slices labelled 35%, 30%, 25% and 20%. By how many percentage points is the total too big?
Answer: 10
1. What is $35 + 30$ ? 65
2. What is the total of all four slices? 110
3. How much more than 100 is the total? 10
4.
A pie chart has four slices. Three are labelled 45%, 25% and 20%; the fourth has no label. What percent should the fourth slice be? Write only the number.
Answer: 10
$45 + 25 + 20 = 90$ , so the missing slice is $100 - 90 = 10$ , that is 10%. A missing label is a warning sign: always check that the parts add up to 100%.
5.
A line graph with a squashed scale shows a price falling from 80 to 60, and the line looks almost flat. What is the real percent change? (A decrease is negative.)
Answer: -25
1. What is the change $60 - 80$ ? -20
2. Divide the change by the old value: what is $\frac{-20}{80}$ ? -0.25
3. Multiply by 100: what is the percent change? -25
6.
Which of these graphs is drawn fairly?
1. a)A picture graph where twice the amount gets an icon twice as tall and twice as wide
2. b)A bar graph whose axis starts at 95 to show small differences
3. c)A bar graph whose axis starts at 0 with equal steps
4. d)A pie chart whose slices add up to 120%
Answer: A bar graph whose axis starts at 0 with equal steps
A bar graph with an axis from 0 and equal steps shows the bars in their true proportions. The other graphs use a truncated axis, a broken pie chart or icons that grow in two directions.
7.
In a picture graph, one coin icon stands for last year's savings. This year's savings are twice as large, and the artist draws the coin twice as tall and twice as wide. How many times bigger does the new coin look (by area)?
1. a)8 times
2. b)2 times
3. c)3 times
4. d)4 times
Answer: 4 times
The area is multiplied by $2 \times 2 = 4$ , so the coin looks 4 times bigger although the savings only doubled.
8.
A bar graph compares two shops: shop A sold 66 bikes and shop B sold 60 bikes. The vertical axis starts at 57. How many times taller does bar A look, and what is the real percent difference?
Answer: 10
1. How tall is bar A drawn above the start of the axis? 9
2. How tall is bar B drawn above the start of the axis? 3
3. How many times taller does bar A look? 3
4. What is the real percent difference $\frac{66 - 60}{60} \times 100$ ? 10
9.
A stretched line graph shows a town's electricity use rising from 1200 to 1260 units. The line shoots up steeply. What is the real percent increase?
Answer: 5
1. What is the change $1260 - 1200$ ? 60
2. What is $\frac{60}{1200}$ as a decimal? 0.05
3. What is the percent increase? 5
10.
A 3D graph shows car sales as cubes. The second cube has every edge 2 times as long as the first. How many times bigger is the volume of the second cube?
Answer: 8
All three dimensions double, so the volume is multiplied by $2 \times 2 \times 2 = 2^3 = 8$ . If sales only doubled, the picture exaggerates 4 times.
11.
A line graph puts the years 2000, 2010, 2020, 2021 and 2022 at equal distances on the horizontal axis. What is misleading about it?
1. a)The time gaps are not equal, so the slopes cannot be compared
2. b)The axis should start at year 0
3. c)The values on the vertical axis must be wrong
4. d)Line graphs cannot show years
Answer: The time gaps are not equal, so the slopes cannot be compared
The first gaps are 10 years, the last ones only 1 year, but they take the same space. So the slopes of the line pieces cannot be compared: a change over 10 years and a change over 1 year look alike.
12.
A graph shows a tree's height: 30 cm in 2000, 50 cm in 2010 and 60 cm in 2015, with the three years at equal distances. The second piece of line looks flatter. Find the growth per year in each period.
Answer: 2
1. How many cm did the tree grow from 2000 to 2010? 20
2. How many cm per year is that (10 years)? 2
3. How many cm did the tree grow from 2010 to 2015? 10
4. How many cm per year is that (5 years)? 2
13.
A graph of monthly rent uses a vertical axis from 0 to 10000 dollars, so the line looks flat. The rent actually rose from 800 to 1000 dollars. What is the real percent increase? Write only the number.
Answer: 25
$\frac{1000 - 800}{800} = \frac{200}{800} = 0.25$ , so the rent rose by 25%. A squashed scale hides a big increase.
14.
On a bar graph, bar A looks exactly twice as tall as bar B. Later you notice that the axis starts at 40 and that B = 50. What is the value of A, and how many percent more than B is it?
Answer: 20
1. How tall is bar B drawn above the start of the axis? 10
2. Bar A looks twice as tall. How tall is it drawn? 20
3. What is the real value of A? 60
4. How many percent more than B is A? 20
15.
A share price was 120 in 2019, 60 in 2022 and 75 in 2024. An advert shows only 2022 to 2024. Find the percent change the advert shows, then the percent change over the whole time from 2019 to 2024.
Answer: -37.5
1. What is the change from 2022 to 2024? 15
2. What percent change does the advert show? 25
3. What is the change from 2019 to 2024? -45
4. What is the percent change over the whole time? -37.5
16.
On a bar graph with a truncated axis, a bar for the value 50 is drawn 5 cm tall and a bar for the value 60 is drawn 15 cm tall. At what value does the axis start?
Answer: 45
The values differ by $60 - 50 = 10$ and the bars by $15 - 5 = 10$ cm, so 1 cm stands for 1. The bar for 50 is 5 cm tall, so the axis starts at $50 - 5 = 45$ . Check: $60 - 45 = 15$ .
17.
A headline says "Sales doubled!". The bar graph shows 600 for 2023 and 700 for 2024, and its axis starts at 500, so the 2024 bar is drawn twice as tall. What is the real ratio of 2024 sales to 2023 sales?
1. a)2 times
2. b)7/6, about 1.17 times
3. c)1.5 times
4. d)4 times
Answer: 7/6, about 1.17 times
The drawn heights are $600 - 500 = 100$ and $700 - 500 = 200$ , which is why the bar looks doubled. The real ratio is $\frac{700}{600} = \frac{7}{6} \approx 1.17$ , an increase of only about 17%.
18.
A country recycled 2 million tons in 2015 and 8 million tons in 2025. An infographic makes the 2025 bottle icon 4 times as tall and 4 times as wide as the 2015 icon. Find how much the picture exaggerates, and the fair scale for each side.
Answer: 2
1. What is the real ratio $\frac{8}{2}$ ? 4
2. By what factor does the area of the icon grow? 16
3. How many times too big does the picture look? 4
4. For an area factor of 4, by what factor should each side grow? 2