When a Graph Lies Without Changing a Number

A politician presents two bars. One is twice as tall as the other. The figures represented are 51% and 49%. The graphic is not necessarily numerically false. But if the vertical axis starts at 48%, it could easily leave an impression that the data do not justify.

That is the fundamental difficulty with statistical graphics: the picture is also an argument.

First, find the denominator

Suppose one town reports 400 crashes and another reports 200. The first has more recorded crashes. But suppose the first town has ten times the residents and considerably more road travel. Counts and rates answer different questions.

Before comparing groups, identify the units: crashes per resident, per mile driven, per licensed driver, or simply total events. None of these denominators is intrinsically correct for every investigation.

Scales change the impression

In a bar chart, bar length normally represents an amount. Starting the baseline above zero can exaggerate small absolute differences. In a line chart, however, a truncated axis may legitimately reveal a small change over time, provided the scale is explicit and the interpretation does not falsely imply a dramatic absolute difference.

Logarithmic axes are useful for multiplicative change: a move from 10 to 100 and from 100 to 1,000 occupies the same vertical distance on a base-ten logarithmic scale. Do not use such an axis without explaining it, and do not interpret equal visual increments as equal arithmetic differences.

Two-dimensional symbols can be even worse. Doubling both a circle’s diameter and its height quadruples its area; a visually enlarged illustration can overstate the magnitude of the data.

Context the chart cannot supply

Every useful graphic should answer four questions. What was measured? How was it collected? Which comparisons does the graphic support? And which uncertainties remain?

If it displays estimates, show uncertainty where relevant. A graph that compares two estimates without acknowledging their precision invites overconfidence. If observations are incomplete, show the missingness or state what was excluded.

The data should remain inspectable. A table and downloadable source often complement the picture.

Worked comparison

A chart compares 98 and 102 on a baseline of 95. The displayed heights are 3 and 7, making one bar appear over twice the other. But the values themselves differ by only 4/98 ≈ 4.1% relative to the smaller value. For an ordinary quantity bar chart, restoring a zero baseline removes that visual amplification.

This does not tell us whether the 4-unit difference matters scientifically. That requires measurement, study design, context, and perhaps uncertainty intervals.

Check yourself

1. Which denominator would help compare traffic injuries between two cities with different populations? Could you justify a different one?
2. Why is a zero baseline especially important for ordinary bar heights?
3. A time-series graph uses a logarithmic scale. What must the caption explain?

Worked responses

1. Injuries per resident might be a starting point; injuries per vehicle-mile may be more appropriate for exposure to travel. The research question determines the denominator.

2. Without it, the visible height is not proportional to the underlying quantity.

3. Equal vertical movement represents an equal multiplicative factor, not an equal arithmetic difference.

Further reading

ASA GAISE College recommendations emphasize context, critical communication, and real data.

Audit the mapping from numbers to ink

For a bar chart, length from the baseline encodes magnitude, so a truncated baseline can magnify a small difference. A line chart can use a narrower vertical range to reveal variation, but should label that range and avoid presenting the visual slope as a unit-free effect. Axis choice is a claim about emphasis that readers should be able to inspect.

If a circular symbol’s radius is doubled to represent a doubling in count, its area quadruples. To make area proportional to count, radius must scale with the square root of count. For values 25 and 100, radii in ratio 1:2 give areas in ratio 1:4. State what a symbol encodes rather than allowing width, area, and volume to compete silently.

An uncertainty display must name its interval: sample standard deviation, standard error, confidence interval, and prediction interval convey different quantities. Two overlapping 95% intervals for separate means do not by themselves establish that a difference is nonsignificant; the uncertainty of the difference depends on both variances and their dependence. Plot the relevant contrast and its own interval where that is the question.

Further practice

1. If area represents count, what radius ratio represents a count ratio of nine?
2. A chart reports a rise from 98 to 102. Calculate absolute and relative changes.
3. Why should a caption identify the denominator and missing records?

Worked further responses

1. Radius ratio three, since area scales as radius squared.
2. Absolute change 4; relative change $4/98\approx4.0816\%$.
3. Percentages and comparisons depend on who is included. Missingness or changing eligibility can alter the apparent trend without a corresponding change in the intended population.