A bar chart seems to need no explanation: taller bar, bigger number. In the language of chapter 2, it is an icon: the height resembles the quantity. But behind every chart there is a convention the eye does not read: the scale. And when the convention changes, the icon goes on speaking with the same confidence, now saying something else.

The cut axis

A company’s sales were $96$, $98$, $101$ and $104$ in four consecutive years. Modest growth: $8.3\%$ in three years. Now draw the bars with the vertical axis starting at $95$ instead of $0$. The first bar has height $1$; the last, height $9$. The last one looks nine times larger than the first.

The drawing does not lie about any number: they are all written there. It lies about the relation between them, which is precisely what a bar chart promises to show. In 1954 the American journalist Darrell Huff called this trick the gee-whiz graph in his book How to Lie with Statistics, still required reading in statistics courses today.

The rule for bars is simple: they compare lengths, and therefore must start at zero. A line chart is different: it shows changes, and starting the axis near the data can be legitimate, as long as the scale is written and the height of the line is not used as a measure of size.

The logarithmic scale

Some scales distort on purpose, and for the good. A quantity that doubles in each time interval, such as a bacterial population or the number of cases at the start of an epidemic, grows so explosively that, on an ordinary chart, the first values stick to the floor and the last ones leave the page.

On a logarithmic scale, the marks on the vertical axis are not $0, 10, 20, 30$ but $1, 10, 100, 1000$: each step of the same size multiplies by 10. Exponential growth then becomes a straight line, and the slope of the line tells the speed of the multiplication. In 2020 many newspapers showed the pandemic curves on both scales, and much of the public discovered that the same data could look like an explosion or a gentle ramp.

The logarithmic scale is a good example of a sign that has to be learned. The eye reads height as quantity; on a log scale it represents the order of magnitude. Whoever does not notice the change of convention reads the chart wrongly.

The aspect ratio

Even with the axis at zero and an ordinary scale, the shape of the chart’s box changes the impression. The same data series, drawn in a wide, low rectangle, looks stable; in a narrow, tall one, it seems to shoot up. The statistician William Cleveland proposed a practical rule: choose the aspect ratio so that the important slopes are close to $45°$, the angle at which the eye best tells differences in slope apart.

Exercise: the exaggeration of the axis

With the axis starting at $90$, how many times larger does the $104$ bar look than the $96$ bar? And starting at $100$?

With the axis at $90$, the visible heights are $104-90=14$ and $96-90=6$: the last looks $\frac{14}{6}\approx2.3$ times larger. With the axis at $100$, the $96$ bar does not even appear, and the $104$ bar has height $4$: the chart stops making sense. The true ratio is always $\frac{104}{96}\approx1.08$.

Exercise: a straight line on a log scale

A population doubles every day: $1, 2, 4, 8, 16, \ldots$ On a base-$10$ logarithmic scale, how much does the height rise per day?

The height is $\log_{10}$ of the value, and $\log_{10}(2^n)=n\log_{10}2\approx0.301\,n$. Each day the height rises by $0.301$ units: always the same amount, which is why the drawing is a straight line. In about $3.3$ days it rises a whole unit, which means multiplying by 10.

Chapter index · ← Charts that saved lives: Playfair, Snow, Nightingale and Minard · Argand’s plane: the imaginary number that found a place →