Why your brain struggles to read maps that use circles to show data
In short: Proportional symbol maps use varying shapes to represent quantities, but our eyes often betray us. While we intuitively grasp that a larger symbol means more of something, we consistently underestimate the area of circles. This quirk of human perception has forced cartographers to invent mathematical workarounds for centuries.
The proportional symbol map is a staple of cartography, designed to visualize quantitative variables like city populations or shipping tonnage. By scaling symbols—most commonly circles—to match the data, mapmakers create an intuitive link between size and magnitude. Jacques Bertin, in his seminal Semiology of Graphics, argued that size is the most effective visual variable because it is almost impossible to interpret as anything other than a measure of importance or quantity. When a reader sees a circle twice as large as another, they naturally assume the underlying value is twice as great.
However, this intuitive leap is flawed. In his 1956 dissertation, James J. Flannery demonstrated that human perception does not perfectly track mathematical area. Instead, readers consistently underestimate the size of larger circles relative to smaller ones. This phenomenon, rooted in Stevens's power law, led to the development of the Flannery Scaling Adjustment. By using an exponent of 0.5716 rather than the standard 0.5, cartographers can artificially inflate larger symbols to correct for our cognitive bias. While some argue that a clear legend renders such complex adjustments unnecessary, the debate highlights the tension between mathematical precision and human psychology.
The choice of symbol also complicates the task. While circles are favored for their compact form and ease of overlap, they are aesthetically uninteresting and prone to these perceptual errors. Squares are easier to judge accurately, yet they are less common. More complex symbols, such as spheres or cubes, are often misused; readers tend to judge them by their two-dimensional projected area rather than their three-dimensional volume. Ultimately, the goal is to balance readability with accuracy, often requiring a delicate trade-off between absolute proportionality and the practical need to avoid visual clutter in dense regions.
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Sources
- Summary by Intellect · source: Proportional symbol map
- Also: Wikidata — proportional symbol map · CC0
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