The problem of judging angles


A number of visualization techniques invite the reader to use angle judgments in order to decode the data.  The most common example of this is the pie chart.

Here is a simple data set of 5 numbers:

Category 1: 23

Category 2: 19.5

Category 3: 18

Category 4: 17.5

Category 5: 22

Note how difficult it is to judge the relative sizes of these categories when portrayed as segments within a pie chart:




Replacing the angle judgment by position along a common axis (as in this bar chart) enables much more accurate visual decoding.



There is much experimental evidence to show that the human brain has difficulties reading angles.  A simple illustration of this point is the Poggendorff illusion, which dates back to 1860.


Most readers perceive the right-hand line at the bottom of the rectangle to be the continuation of the line coming in from the top: in fact, it is the left-hand line.  An explanation for this optical illusion is that we have a natural tendency to misread the size of angles formed where a horizontal line and a sloping line meet.

A Russian psychologist called Yarbus studied eye movements in the 1950s and 1960s using a somewhat bizarre suction cup apparatus attached to his subject’s eyeball. The picture below shows the result of one of his experiments.  His subject was shown the uppermost picture and asked to trace the horizontal line from left to right and then trace the diagonal line from left to right, extending this along an imaginary straight line below the horizontal line.  An example of the resulting trace is shown in the lower picture.


The key finding of this experiment is that the subjects of this experiment tended to overestimate the angle between the horizontal line and the line to be followed, which is the phenomenon experienced in the Poggendorff illusion.

A further problem is that the judgement of angle size also depends on the orientation of the angle. Maclean and Stacey found that an angle with a horizontal bisector is generally perceived as larger than the same angle presented with its bisector in a vertical orientation.  This is illustrated in the examples below.  There is a tendency to perceive the first chart as having a larger orange segment than second chart, although both are exactly the same size.



One of the lessons of this evidence is that pie charts should be used with caution.  There are also implications for slope judgments that we will return to in a later post to this blog.

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