Graduated and proportional symbol maps are a class of maps that use the visual variable of size to represent differences in the magnitude of a discrete, abruptly changing phenomenon, e.g. counts of people. Like choropleth maps, you can create classed or unclassed versions of these maps. The classed ones are known as *range-graded* or *graduated* symbols, and the unclassed are called *proportional* symbols, where the area of the symbols are proportional to the values of the attribute being mapped.

Although graduated/proportional symbols were used earlier in statistical graphics, they were first used on a map in the 1850s in France by Charles Joseph Minard (see Figure 5.cg.1, below).

Graduated and proportional symbol maps have been created with many different types of symbols, ranging from abstract, geometric symbols (e.g., bars, circles, or squares) to mimetic, pictographic symbols (e.g., almost any object you can think of; see Figure 5.cg.2, below). Keep in mind that while designing graduated symbol maps, you really want the map reader to be able to estimate the value of a symbol. This is most easily accomplished with geometric symbols. And, although it is possible to use three-dimensional (i.e., volumetric) symbols, map readers have a more difficult time estimating volumes than they do areas.

Map readers' estimation of values from symbols has been the subject of extensive research by cartographers. These studies have generally found that readers make the most accurate estimates from bars and squares and are generally not as good at estimating values from circles, whose areas they tend to underestimate, especially for larger symbol sizes. Although bars and squares are more easily estimated, circles tend to be a more popular choice, as they are a very compact symbol, and cause fewer visual problems on maps (e.g., they don't have the problem of running off the page with large values, as bars might).

Through their research on value estimation, cartographers measured the average amount that map readers underestimated the area of circles for different symbol sizes. Flannery (1971) calculated a scaling factor, which he proposed could be used to "correct" the sizes of circles so that map readers would correctly estimate values from symbols. However, this correction may not actually be very effective, as the task he used in the experiment in which he calculated this factor may not match common real-world uses of graduated symbol maps. Moreover, there was a large amount of variation in individual map users' abilities to estimate values, so while applying the correction factor may help some map readers estimate values more correctly, others (with better estimation abilities) may actually be less able to estimate values correctly from modified symbols. Finally, the Flannery correction does not take into account the effect of map context on map readers' size estimation abilities. One well known phenomenon related to map context is the Ebbinghaus illusion, a situation in which two identical circles can appear to be different sizes, depending on the symbols that surround the central circles (see Figure 5.cg.3, below).

So if the Flannery correction doesn't work that well, is there anything that cartographers can do to help map readers estimate values more accurately? One solution lies in good legend design. Map readers are generally better at interpolating between two sizes displayed in the legend than in extrapolating beyond the largest symbol. This suggests that your legend design should include examples that are similar in size to the smallest and largest values present in the map, as well as some at intermediate values (see Figure 5.cg.4 below).

A second alternative is to use range-graded symbols (i.e., classify the data), and avoid the estimation problem entirely. Once you have decided whether you want to classify your data or proportionally scale it, it is still necessary to make decisions about the sizes of symbols you would like to use. In the case of proportional symbols, this means choosing a scaling factor, while in the case of range-graded symbols, you will need to choose a set of symbols that are clearly differentiable from each other (i.e., choose symbol sizes that will not be confused with each other). One general guideline is to choose symbol sizes so that there is a slight overlap in symbols in the most crowded areas of the map. This will help enhance the map reader's perception of the map pattern without making the map illegible (see Figure 5.cg.5, below).

#### Want more examples of graduated/proportional symbol maps?

- Map of U.S. risk to natural disasters using graduated symbols to represent metro area populations (and hue to represent risk)
- Map of Hurricane Katrina's diaspora using proportional circles to represent number of FEMA assistance applications after the hurricane, by zip code Note the compact and efficient legend providing five example symbol sizes stacked on top of each other.
- Map using proportional circles to represent London Summer Olympic medals won by country Interact with the map to see results from other years, back to 1896.
- Animated map of jobs lost and gained (in the contiguous U.S.) from January 2004 through December 2012

If you know of other interesting graduated/proportional symbol maps, please post them in the Lesson 5 discussion forum in ANGEL.

#### Recommended Readings

If you are interested in investigating this subject further, I recommend the following:

- Brewer, C.A. and A.J. Campbell. 1998. "Beyond graduated circles: Varied point symbols for representing quantitative data on maps." Cartographic Perspectives. 29: 6-25.