Charts are powerful because we read them at a glance. That same speed makes them easy to bend. In 1954 the journalist Darrell Huff gathered many such tricks in a popular book, How to Lie with Statistics, and most still circulate today.

Truncated axes

Suppose two values are 52 and 54. On a bar chart whose vertical axis starts at zero, the bars look almost equal, which is honest: the difference is small. If the axis starts at 50, the second bar looks twice as tall as the first. Starting the axis well above zero can make small differences look much larger than they are. Because we judge bars by their length, bar charts should normally start at zero. Line charts are different: they show change over time, and zooming in can be reasonable, as long as the axis is clearly labeled.

Cherry-picked ranges

Cherry-picking means choosing start and end points, or a subset of data, that favor a conclusion. A figure that rises and falls over decades can be shown as “soaring” or “collapsing” depending on which two years are chosen. The remedy is to ask to see the longer series and to wonder why a chart starts where it does.

Area and 3D distortions

Pictures can mislead through area. If a chart shows money as a picture of a moneybag and doubles both the height and width of the bag to show twice as much money, the area grows four times, and readers perceive a far bigger change than the data show. Three-dimensional effects and tilted pie charts add more distortion. The statistician Edward Tufte called the ratio between the size of an effect in a graphic and its size in the data the lie factor; ideally it should be close to 1.

Simpson's paradox

Sometimes a pattern in every group reverses when the groups are combined. This is Simpson's paradox. Consider two invented hospitals, each treating 1,000 patients.

• Hospital A treats 800 mild cases, of whom 760 survive (95 percent), and 200 severe cases, of whom 100 survive (50 percent). Overall, 860 of 1,000 survive: 86 percent.
• Hospital B treats 200 mild cases, of whom 196 survive (98 percent), and 800 severe cases, of whom 480 survive (60 percent). Overall, 676 of 1,000 survive: 67.6 percent.

Hospital B does better for mild cases and for severe cases, yet worse overall. The reason: Hospital A treats mostly mild cases, which have high survival, while Hospital B takes mostly severe ones. Combining groups that differ in their mix can reverse a comparison. The lesson is to ask whether the groups being compared are alike, and whether the breakdown tells a different story from the total.

Survivorship bias

Survivorship bias means drawing conclusions only from cases that made it through a selection process, while the failures are invisible. The best known story comes from the Second World War. The mathematician Abraham Wald, working with a statistics group at Columbia University, studied how to protect bombers using the damage seen on aircraft that returned from missions. The naive reading is to add armor where returning planes show the most holes. Wald's insight was the opposite: the planes that were hit in other places, such as the engines, were the ones that did not come back. Areas with few holes on returning planes were the vulnerable ones, because planes hit there were less likely to return at all.

The story is often embellished. The familiar picture of a plane covered in red dots is a modern illustration, not a wartime document, and it is not clear exactly how Wald's advice was used. But his reasoning, set out in wartime reports, is real. Survivorship bias appears everywhere: studying only successful companies, admiring old buildings (the poorly built ones fell down long ago), or judging an investment fund family only by the funds that have not been closed.

Worked example: fixing a misleading chart

A town library's annual report shows two bars: 9,800 visits last year and 10,200 visits this year. The vertical axis starts at 9,700.

What the eye sees. The first bar rises 100 units above the axis (9,800 minus 9,700). The second rises 500 units (10,200 minus 9,700). So the second bar looks five times as tall as the first, suggesting visits quintupled.

What actually happened. Visits rose by 400. As a percentage, 400 divided by 9,800 is about 0.041, or about 4 percent. A modest, welcome increase.

The fix. Redraw the bars with the axis starting at zero. The bars now look nearly equal, which matches the data. If the change itself is the point, a clear label (“up 4 percent”) does the job honestly.

Check the range. The report shows only two years. Suppose the full record over ten years shows visits of about 12,000 a decade ago, falling to 9,800 last year. Then this year's figure is a small recovery within a long decline. Showing only two years is a form of cherry-picking, whether intended or not.

Check the comparison. If the town's population grew 5 percent this year, visits per resident actually fell slightly, because visits grew about 4 percent while the population grew 5 percent.

A single honest chart would show the ten-year series, start the axis at zero or label it clearly, and perhaps show visits per 1,000 residents. Each step brings the picture closer to what really happened.