An accurate chart can still mislead
In a vote between two candidates, A wins 52% of the votes and B wins 48%. That is a close result: A is ahead by 4 percentage points, and , so A’s share is only about 8% larger than B’s.
A bar chart drawn with its scale starting at 0 shows it as close. Each bar is as long as its value, so the ratio of the heights, 52 : 48, is the ratio of the values, and the two bars look almost the same.
Drawn from 0, the bars for 52 and 48 are almost the same height, as the result is close.
A truncated axis
Now draw the same two numbers with the vertical axis starting at 40 instead of 0. A truncated axis like this cuts off the bottom of every bar. The bar for A is drawn from 40 to 52, which is 52 − 40 = 12 units tall, and the bar for B from 40 to 48, which is 48 − 40 = 8 units.
A’s bar now looks 12 ÷ 8 = 1.5 times as tall as B’s, and the close result looks like a landslide. Nothing in the data changed and every number on the chart is correct. What changed is the baseline: the bars no longer start from a common zero, so their heights are no longer in the ratio of the values.
The honest bar chart starts at 0. When reading any chart, check the number at the bottom of the vertical axis before comparing heights, and work out the real ratio from the values themselves. A line graph may start above 0, because it shows change rather than size, but its axis should show a break so the reader knows the bottom is missing.
The same two values, 52 and 48. The dashed line marks 40 on the chart from 0, and the chart cut at 40 is everything above that line: its bars are 12 and 8 units tall, so A looks one and a half times B.
Pictures that grow in every direction
A farm sold 1 thousand crates of apples last year and 2 thousand this year, twice as many. A poster shows each year as a drawing of a crate, and draws this year’s crate twice as tall. To keep its shape, the drawing is also made twice as wide and twice as deep.
The front of the new crate covers 2 × 2 = 4 times the area of the old one, and the whole crate holds 2 × 2 × 2 = 8 times as much. The eye reads the size of the whole drawing, so a doubling looks like four or eight times as much.
The honest picture changes one length only: a bar twice as long, or two crates of the same size in place of one.
Every length of this year’s crate is doubled. Its shaded front face is 4 of last year’s faces, and it holds 8 of last year’s crates.
A chart drawn in 3D
A pie chart drawn as a tilted 3D disc shows the sectors at the front closer to the reader, with the thickness of the disc showing below them, while the sectors at the back are squashed by the tilt. Two sectors with the same angle then look different sizes, and the front one looks bigger.
Bars drawn as 3D blocks have the same fault: their depth adds area that stands for nothing, and it is hard to tell which edge of the block to read against the scale. The honest chart is flat, so every sector is measured by its angle and every bar by its height.
Unequal steps on an axis
A town had 10 thousand people in 2000, 20 thousand in 2010, 25 thousand in 2015 and 30 thousand in 2020. The population grew by 1 thousand every year: 10 thousand in the 10 years to 2010, then 5 thousand in each 5 years after.
Suppose a line graph puts the four years at equal spacing along the time axis. The step from 2000 to 2010 is 10 years, but it is drawn as long as each 5-year step after it. The graph then rises steeply at first and flattens out, as if the growth had slowed, although it never changed.
The honest graph spaces the years by the time between them. Plotted every 5 years, 2000, 2005, 2010, 2015 and 2020, the populations are 10, 15, 20, 25 and 30 thousand, and the points lie on a straight line: steady growth. The same fault appears in bars of different widths. A wide bar looks bigger because it covers more area, which is why a histogram with classes of different widths is drawn with frequency density.
Misleading: 10 years and 5 years are drawn the same length, so steady growth looks as if it slowed after 2010.
Honest: every step is 5 years, and the population rises by 5 thousand in each, so the points lie on a straight line.
A cherry-picked range
A magazine sold 60 thousand copies in 2015, then fewer each year: 55, 51, 46, 40 and 36 thousand in 2020. Sales then rose to 38 thousand in 2021 and 41 thousand in 2022.
A chart of 2020 to 2022 alone shows only the rise, from 36 to 41 thousand, an increase of 5 thousand, and , so the headline says sales are up 14%. Every point on that chart is true. Choosing the start and end of the range to fit a story is called cherry-picking.
The honest chart shows the whole range available. From 2015 to 2022, sales fell from 60 to 41 thousand, a decrease of 19 thousand, and , so sales are about 32% lower than they were. When a chart covers only a few years, ask what happened before them.
Cherry-picked: from 2020 to 2022 alone, sales rise from 36 to 41 thousand.
The whole range: sales fell for five years, from 60 to 36 thousand, and the last two years recover only part of the fall.
Worked example: An Advertisement's Bar Chart Whose Axis Starts at 470
Question An advertisement shows a bar chart of the bottles of a drink sold, in thousands: 480 in 2024 and 504 in 2025. The vertical axis of the chart starts at 470, not at 0. (a) How many times as tall as the 2024 bar is the 2025 bar on this chart? (b) Find the true percentage increase in sales, and describe how the two bars compare when the axis starts at 0.
1.On the advertisement each bar starts at 470. The 2024 bar has height 480 − 470 = 10 and the 2025 bar has height 504 − 470 = 34.
Each bar starts at 470, so the bars are 480 − 470 = 10 and 504 − 470 = 34 high. 2.(a) 34 ÷ 10 = 3.4, so the 2025 bar is drawn 3.4 times as tall as the 2024 bar. Sales did not become 3.4 times as large.
(a) 34 ÷ 10 = 3.4, so the 2025 bar is drawn 3.4 times as tall. 3.Use the full values for the true change. The increase is 504 − 480 = 24 thousand bottles, and it is compared with the 2024 value, 480 thousand.
With the axis from 0 the bars show the full values. The increase is 504 − 480 = 24 thousand, measured against 480 thousand. 4.(b) 24480 = 0.05, so sales rose by 5%. On an axis that starts at 0 the bars have heights 480 and 504, so the 2025 bar is only 1.05 times as tall and the two bars look almost the same.
(b) 24480 = 5%. From 0 the two bars are almost the same height.
Answer: (a) 3.4 times as tall; (b) a true increase of 5%, and from 0 the two bars are almost the same height
Common mistakes
- Saying that sales more than tripled because one bar is more than three times as tall as the other. The bars on this chart show only the amounts above 470. The values themselves are 480 and 504, which differ by 5%.
- Dividing the increase by the new value, 24504. A percentage increase is measured from where the change started, so the 24 thousand is divided by the 2024 value, 480 thousand.