A Bar Graph That Does Not Start at 0

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?

Find the drawn heights above the start of the axis: A: $52 - 45 = 7$, B: $48 - 45 = 3$ = Drawn heights 7 and 3

Compare the drawn heights: $\frac{7}{3} \approx 2.33$ = Bar A looks about 2.3 times as tall

Compare the real values: $\frac{52}{48} \approx 1.08$ = A is really about 1.08 times B

Write the real difference as a percent: $\frac{52 - 48}{48} \times 100\% \approx 8.3\%$ = About 8.3% more votes

Answer: Bar A looks more than twice as tall, but Juice A got only about 8.3% more votes than Juice B. Compare the two bar graphs above: with the axis starting at 45, bar A towers over bar B; with the axis from 0, the bars are almost the same height.

A Stretched Line Graph

The first line graph above shows website visitors from January (1000) to May (1050). Its axis starts at 980, so the line climbs steeply. What is the real percent change from January to May?

Read the old and the new value: January: 1000, May: 1050 = Old = 1000, new = 1050

Find the change: $1050 - 1000 = 50$ = 50 more visitors

Divide by the old value: $\frac{50}{1000} = 0.05$ = 0.05

Write it as a percent: $0.05 \times 100\% = 5\%$ = 5%

Answer: The number of visitors grew by only 5%. On the second graph, where the axis starts at 0, the same line is almost flat.

A Picture Graph With Growing Icons

A company sold 2 million phones in 2020 and 4 million in 2024. Its report draws one phone icon for each year, and the 2024 icon is twice as tall and twice as wide. How many times bigger does the 2024 icon look?

Find the real ratio: $\frac{4}{2} = 2$ = Sales doubled

Find the area of the bigger icon: Area $= 2 \times 2 = 4$ times the small icon = 4 times the area

Compare picture and reality: $\frac{4}{2} = 2$ = The picture exaggerates by a factor of 2

Find a fair scale for the icon: Each side times $\sqrt{2} \approx 1.41$, so the area is $2$ times = Sides about 1.41 times longer

Answer: The 2024 icon has 4 times the area, so it looks like sales grew 4 times when they only doubled. A fair picture graph uses two icons of the same size, or scales each side by about 1.41.

A Pie Chart That Adds Up to 135%

A class survey asked: "Which sports do you play?" Students could pick more than one. The pie chart above shows Football 60%, Swimming 45%, Tennis 30%. What is wrong?

Add all the percentages: $60 + 45 + 30 = 135$ = 135%

Compare with the whole: A full circle is $100\%$, and $135 - 100 = 35$ = 35 percentage points too many

Look at the slices: The slice labelled 60% fills only $\frac{60}{135} \approx 44\%$ of the circle = The slices do not match the labels

Choose a better graph: Each percent is its own question with yes or no answers, so the parts do not make one whole = Use a bar graph

Answer: A pie chart only works when the parts add up to exactly 100%. Here students could give several answers, so the percentages add up to 135% and a bar graph is the honest choice.

Uneven Intervals on the Time Axis

A town's population (in thousands) is shown for the years 2000, 2010, 2020, 2022 and 2024: 50, 60, 70, 72, 74. The years are drawn at equal distances (see the graph above). The headline says: "Growth has almost stopped!" Is that true?

Find the gaps between the years: 10, 10, 2 and 2 years = The intervals are not equal

Find the growth per year from 2000 to 2020: $\frac{70 - 50}{20} = 1$ = 1 thousand people per year

Find the growth per year from 2020 to 2024: $\frac{74 - 70}{4} = 1$ = 1 thousand people per year

Compare: $1 = 1$ = The growth rate did not change

Answer: The headline is false. The town grows by 1 thousand people every year the whole time. The line only looks flatter because 2 years take the same space on the axis as 10 years.

Mistake: Judging the size of a difference by the bar heights alone

Why: 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: $\frac{52}{48} \approx 1.08$, only about 8% more.

Mistake: Dividing by the new value when computing percent change

Why: 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: $\frac{1050 - 1000}{1000} \times 100\% = 5\%$.

Mistake: Thinking a steep line always means a big change

Why: 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.

Mistake: Comparing picture or 3D graphs by height only

Why: 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.

Adverts and Product Claims

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

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.

Investment Funds and Cherry-Picked Time Ranges

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

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%!"

Infographics With Pictures

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

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.

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

A: 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.

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

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

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

A: 3D effects add depth, so our eyes compare volumes. A box with every edge twice as long has $2^3 = 8$ times the volume. 3D pie charts also tilt the circle, so front slices look bigger than back slices.

Q: Can a misleading graph contain only true numbers?

A: 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.