How to lie with a graph
Tonight's idea
Every graph makes a claim with its shape. A tall bar says "big". A steep line says "fast". The tricks below all work by making the shape say something the numbers do not.
The chopped axis. Start the vertical axis at 50 instead of 0, and a change from 51 to 52 looks like doubling. The two charts below show the same data.
The chosen window. Show only the three months when something went up and hide the two years when it went down.
The inflated picture. A pictogram of a coin that is twice as tall is also twice as wide, so it has four times the area. Your eye reads area. "Sales doubled" becomes "sales quadrupled".
Two axes. Put two different scales on the left and right, and you can make any two lines cross wherever you like.
The total that only goes up. A graph of "total sales ever" rises even in a terrible year. Only the slope tells you anything.
The honest rules: bars start at zero; axes are labelled with units; the whole relevant range is shown; pictures scale in one direction only.
Watch one: the pictogram
A chart shows last year's profit as a money bag 2 cm tall and this year's as a bag 3 cm tall, "because profit rose 50%".
Heights are in the ratio \(3 : 2 = 1.5\), fine. But the bags are drawn in proportion, so widths are also \(1.5\times\), and areas are \(1.5 \times 1.5 = 2.25\times\). The eye sees profit more than doubling. If the artist drew a 3D bag, volume is \(1.5^3 \approx 3.4\times\).
Your turn
- A news chart shows a tax rate going from 35% to 39.6% with the vertical axis running from 34% to 42%. Roughly how many times taller is the second bar than the first? What is the real ratio?
- Here are a shop's monthly sales for a year: 40, 42, 38, 35, 33, 30, 28, 31, 36, 41, 45, 48. The owner shows a chart of the last four months only, titled "Sales soaring". Is anything false? Is anything misleading? Which months would a rival show?
- A pictogram uses a house symbol for homes built. One year's house is drawn 4 cm tall, the next 6 cm. By what factor did the number of homes rise, by what factor does the area rise, and what should the artist have done?
- Redraw the sales data from question 2 honestly in the space below: draw your own axes, vertical axis from 0 to 50 with a label, and twelve bars.
- A headline says "Average pay at the company is $95,000". Ninety workers earn $40,000 and ten managers earn $590,000. Which average did they use, and what would the other one say? (See Lying with averages.)
- A chart plots "total number of users since launch" and rises smoothly. The company says "growth has never been stronger". What would you need to see to check that?
Stretch: the log scale
Some charts mark the vertical axis 1, 10, 100, 1,000 at equal spacing. That is a logarithmic scale (see Counting digits). It is the honest choice for things that grow by multiplying, like a population doubling every year, because steady doubling becomes a straight line. But it also flattens drama: on a log scale, a rise from 1,000 to 10,000 looks exactly like a rise from 1 to 10. When might a company prefer a log scale for its losses, and when might a scientist insist on one for an epidemic?
Answers
- Above the 34% baseline the bars are 1 and 5.6 units tall, so the second looks about 5.6 times taller. The real ratio is \(39.6 / 35 \approx 1.13\), a 13% rise.
- Nothing is false: sales did rise from 36 to 48 over four months. It is misleading because the year as a whole started at 40 and dipped to 28. A rival would show January to July (40 down to 28) titled "Sales collapsing". Both are cherry-picked windows.
- Heights \(6/4 = 1.5\), so homes rose by 50%. Area rises by \(1.5^2 = 2.25\). The artist should have used 1.5 houses of the same size, or a bar.
- Twelve bars from 28 up to 48 on an axis starting at 0. The honest picture is a shallow dip and recovery, not a surge.
- The mean: \((90 \times 40{,}000 + 10 \times 590{,}000)/100 = 95{,}000\). The median is $40,000, which is what a typical worker earns.
- Users per month (new sign-ups), not the running total. The total can only rise; "strongest growth ever" needs the slope to be steeper than before, which the smooth curve does not show.
Stretch. A company with losses growing from 1 million to 100 million would love a log scale, because the line looks gentle. A scientist tracking an epidemic insists on one because case counts multiply, and on a log scale a straight line means steady doubling and a bend tells you the doubling rate has changed: bending downward means growth is slowing relative to its size, bending upward means it is speeding up. On an ordinary axis the same data just looks like a curve getting steeper, and it is hard to tell steady doubling from doubling that is slowing down.