Can a Graph Lie?

Here is what happens to the two bars as the bottom of the axis moves up. Every row shows the same two prices.

The axis starts at	Last week’s bar	This week’s bar	This week’s bar is
Rs 0	82	88	1.07 times as tall
Rs 40	42	48	1.14 times
Rs 60	22	28	1.27 times
Rs 70	12	18	1.5 times
Rs 76	6	12	2 times
Rs 79	3	9	3 times
Rs 81	1	7	7 times

Look at the two bar columns. Moving the axis up cuts the same piece off both bars, so the gap between them stays 6. But the shorter bar gets shorter and shorter, and the gap of 6 becomes huge compared with what is left of it.

That is how to solve the puzzle without trying. For this week’s bar to be twice last week’s, the gap of 6 must be as big as all that is left of last week’s bar: so only 6 may be left, and the axis must start at 82 − 6 = Rs 76. For three times, the gap must be twice what is left: 3 left, so the axis starts at Rs 79.

And if the axis started at Rs 82 itself? Last week’s bar would vanish altogether, and no number of times, however big, could say how much taller this week’s is. No chart does that, but some come close.

Why do our eyes fall for it? Because we read a bar chart by comparing the bars, not the numbers on the axis. Long before you have read the small numbers on the side, you have seen that one bar is much taller than the other. A bar chart says “this is how much” with the length of each bar, so the length must be in proportion to the number.

The statistician Edward Tufte gave a way to measure how much a chart exaggerates. Take the change as a fraction of the first value, once in the data and once in the drawing:

• In the data, the price rose by 6 out of 82.
• In the drawing with the axis at Rs 76, the bar rose by 6 out of 6: it doubled.

Tufte’s lie factor is the change shown in the drawing divided by the change in the data:

$$\text{lie factor} = \frac{\text{change shown} \div \text{first size shown}}{\text{change in data} \div \text{first value}} = \frac{6 \div 6}{6 \div 82} \approx 13.7$$

An honest chart has a lie factor of 1: its pictures change by the same fraction as its numbers. This one makes the rise look 13.7 times as big as it really is. Tufte said a chart is badly off once its lie factor is above 1.05 or below 0.95.

A rule you can prove

Look at the fraction once more. Whatever the axis start, the change is 6 in both the drawing and the data (both bars lose the same piece, so the gap stays 6). The 6s cancel, and what is left is

$$\text{lie factor} = \frac{82}{82 - \text{axis start}}.$$

That is: the whole of the shorter bar, divided by the part of it you still see. Cut away half of last week’s bar and every change looks 2 times as big; cut away three quarters and it looks 4 times as big; leave a fortieth and it looks 40 times as big. The rule doesn’t even depend on this week’s price: whatever the second number, cutting the bottom off the chart exaggerates the change from last week’s price by the same factor.

So the rule for bar charts is short: a bar chart’s axis starts at zero. Two scientists who made a course on spotting bad data, Carl Bergstrom and Jevin West, call it the principle of proportional ink: when a shaded area stands for a number, the area should be in proportion to the number.

A real one

In July 2012, a US business channel, Fox Business, showed what would happen to the top rate of income tax in the United States if some tax cuts ended: from 35% now to 39.6% on 1 January 2013. Its chart’s axis went from 34% to 42%, just as the widget below starts.

Some charts use pictures instead of bars: a coin for money, a sack for rice, a house for houses sold. To show twice as much, the artist draws the picture twice as big. But twice as big which way?

The eye judges a flat picture by its area and a picture of a solid by its volume. So a coin drawn twice as wide shows a change from 1 to 4 for a number that went from 1 to 2: in the data the change is 1 out of 1, in the picture 3 out of 1, a lie factor of 3. The box, from 1 to 8, has a lie factor of 7. The journalist Darrell Huff warned about this trick in 1954, in his book How to Lie with Statistics: a number has only one dimension, its size, but a picture has two or three.

The axis and the pictures are two tricks. A third is the choice of what to show. Here is a real record: Earth’s average temperature every year from 1880 to 2025, as measured by NASA, the US space agency. Nothing in it is made up. Try making it say three different things.

Choosing only the stretch of data that supports your story is called cherry-picking, like picking only the ripe cherries off a tree and saying the tree has no green ones. Its favorite move is visible in the widget: start at an unusual point. A record-hot year, a month when sales were very low, the day before a festival: whatever you compare with an unusual start looks like a change.

The axis can be stretched too. A line graph drawn tall and narrow makes a gentle slope look like a cliff; drawn short and wide, a cliff looks gentle. A chart with two axes, one on each side for two different things, is easier still to bend: a data journalist at The Economist found one of its own charts where a dog’s weight and its neck size seemed to fall together, line on line. But the left axis went from 21 down to 18 (a fall of 14%) and the right one from 45 down to 42 (a fall of 7%).

You might now think: always start every axis at zero. Here is where that rule breaks.

Press the second axis button in the widget again. That chart, with its axis from −20 to 40 ∘°C, copies a real one. In December 2015 a US magazine, National Review, posted a chart of Earth’s average temperature since 1880 on an axis running from −10 ∘°F (about −23 ∘°C) up to 100 ∘°F (about 38 ∘°C; some reports say 110 ∘°F), and called it “The only #climatechange chart you need to see.” On it, the warming looked like a flat line. Its axis was far from cut off. It was too tall.

The trouble is that the axis no longer showed the size of a change that matters. In all 146 years of the record, the whole Earth’s coldest year (1909) and its warmest (2024) are only 1.76 ∘°C apart, so for this record a change of one degree is huge, even though it is tiny next to the difference between a winter morning and a summer afternoon. Zero degrees Celsius isn’t “no temperature” either: it is only where water freezes.

Your body works the same way. A child’s temperature rising from 37 ∘°C to 39.6 ∘°C is a real fever: a rise of 2.6 degrees, 7% of 37. On an axis from 0 ∘°C (Figure 27.2, top) it takes up only 6.5% of the height and looks like nothing at all. On an axis from 36 ∘°C (bottom) it takes up 65% and anyone can see it. For a line graph the bottom of the axis isn’t the length of anything, and the lie factor doesn’t apply: what matters is that the range shows the changes that matter, and that its numbers are written clearly enough to read. One designer’s rule of thumb, quoted by The Economist: when a line chart doesn’t start at zero, leave at least a third of the chart empty below the line, so it doesn’t look like a fall to nothing.

So the full rule has two halves:

• Bars, areas and pictures: the length or area is the number, so start at zero and grow one way only.
• Lines and dots: the position shows the number, so choose a range that shows the changes that matter, label it, and show enough of the data that the start isn’t a cherry-pick.

• What Does an Average Hide? One number for a whole class, and the many shapes of data that give the same average.
• How Much Rice Does Nepal Eat in a Day? Estimating a big number from a few you know, and checking whether it is roughly right.
• Can a Treatment Win in Every Group but Lose Overall? (Grades 9–12) Data that tells opposite stories depending on how you group it.
• Does Ice Cream Cause Drowning? (Grades 9–12) Two lines that rise together, and why that isn’t proof that one causes the other.
• Charts that tell the truth (Grade 6 Computing) A chart’s title, axes and numbers, and why bars start at 0. Grade 8 Computing designs charts both ways in Charts: the same data, different stories.