GEOG 486
Cartography and Visualization

Graduated and Proportional Symbol Maps


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, below).

Minard's proportional symbol map
Figure In this early proportional symbol map, Minard showed the proportion of different types of meat that were sent to Paris butcheries from different regions in France.
Credit: Minard 1858

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, 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.

Two maps, side by side. On the left is an example of the use of graduated symbols, and on the right is an example of the use of mimetic symbols.
Figure Although some map makers may find pictographic symbols more interesting (e.g., the cars in the map at the right), such symbols have complicated shapes that make their areas more difficult to estimate and create greater problems with symbol overlap than more abstract, geometric shapes (e.g., the circles in the map on the left).
Credit: University of Minnesota

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, below).

the Ebbinghaus optical illusion explained in caption
Figure Which central circle appears larger to you? Both circles are the same size, but the context in which you see them alters your perception of their sizes.
Credit: Adrienne Gruver

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 below).

Crater map, explained in caption and text
Figure This map uses proportional symbols to represent the largest known craters to have marked our Earth. The legend shows two sizes that are representative of symbol sizes on the map. The largest crater is 99.4 miles in diameter (in South Africa) and, appropriately, the largest symbol size used in the legend represents a crater with a 99 mile diameter. The smaller symbol in the legend represents a 20 mile diameter, and although there are craters with smaller diameters on the map, it is easier to visually see the differences with smaller symbol sizes. The earth is marked with about 180 named craers that are scars from previous run-ins with asteroids like the one that exploded over Russia recently. This map was designed to be interactive, as the map user can hover over the symbols and see the particular size, place and age of the craters. See this link for the interactive version of the map.
Credit: The Washington Post

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, below).

A series of maps with varying scaling factors. The map in the upper left shows a scaling factor that is too small, the map in the upper right shows a scaling factor that's too big, and the map at lower left shows a scaling factor that is just right.
Figure If you choose a scaling factor that is too small, it will be more difficult for the map reader to see patterns in the data (top left), while if the scaling factor is too large, s/he will be presented with many overlapping symbols, which will also make it difficult to see patterns in the data (top right). A better choice of scaling factor will create a slight overlap between symbols in the most crowded area of the map (bottom), and will help the reader identify patterns in the data without obscuring other symbols.

Want more examples of graduated/proportional symbol maps?

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.