GEOG 486
Cartography and Visualization

Dot Maps

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Dot maps best represent data consisting of discrete observations that change more or less smoothly over space. Look back again at Figure 4.cg.1 in the section on Surfaces to remind yourself of the different ways geographic phenomena can be conceptualized in space. In a dot map, each dot represents a fixed number of observations. The dots are not supposed to be counted, but provide a description of the numerousness of a phenomenon (i.e., the overall magnitude of the phenomenon we are mapping in a particular area), the density of the phenomenon (i.e., how closely spaced instances of the phenomenon are from each other), and how the presence and density of the phenomena change over the space mapped. Because the map reader's impression of dot density is dependent upon the size of the areas in which the dots are placed, it is important to use an equal-area map projection so that areas are not distorted. Also because the dots do not vary in their size or value, the size and value you choose are important factors in making this representation effective.

Dot map of the United States, explained in caption
Figure 5.cg.6 This basic dot map uses one dot to represent 25,000 acres of cropland harvested in 1949, and illustrates the density and distribution of the harvested cropland. Dot size is important. If the dots are too large, there would be more overlap and less variation in density would be noticeable. Dot value is also important. Using too high a value would create fewer dots in the map, resulting in less variation and less opportunity to read and understand spatial patterns. Too low a value would result in more dots and possibly too many areas appearing dense. You want the densest areas to appear dense, and the sparse areas to appear sparse with variation inbetween those two extremes, as shown in this example.
Credit: U.S. Dept of Commerce

The decision about dot size and how many objects a dot should represent is related to the spatial distribution and the abundance of the phenomenon itself (i.e., whether it is fairly evenly distributed across space or has areas with very sparse and very dense distributions). The size and number of dots on the map can give a map reader different impressions about the spatial distribution of the phenomenon being mapped. If the dot size is small and each dot represents a large number of objects, the map may give the impression that the area is more sparsely filled with the phenomenon than it really is. Conversely, if the dot size is large and each dot represents one or a small number of objects, the map may communicate that the phenomenon is more pervasive than it actually is.

Population dot maps of New York, explained in caption.
Figure 5.cg.7 In this example of population in New York, a dot size that is too small makes the distribution in the left map appear very sparse (with the exception of New York City). In the middle map, a dot size that is too large makes it difficult to pick out true centers of population such as Buffalo, Rochester and Syracuse. In the right map, the intermediate dot size allows the map reader to see local population centers without giving the impression that the rest of the state is totally empty.
Credit: Adrienne Gruver

Choose a dot size and value so that the dots just begin to coalesce in the most densely 'populated' portion of the map. Though experience and trial and error may serve you better, one guide that has been developed in an effort to aid cartographers in selecting dot size and value is the Mackay nomograph (see Figure 5.cg.8, below). There are a couple of ways to use the nomograph depending on what you are trying to determine, e.g. the dot size, or the dot value. For example, if you are trying to determine an estimate of your dot value, start by drawing a line from the origin (i.e. the lower left intersection of the two axes) to the desired diameter of the dots to be used (along the line that makes a right angle above and to the right of the 'zone of coalescing dots'). Then where that line crosses the 'zone of coalescing dots', draw a vertical line down to the x-axis to find the number of dots per square centimeter that should be used in the densest areas of your dot map. To find out the value for each dot, you need to specify the region on your map that has the highest density of dots and its map area (at the scale of the map, e.g. 2 sq cm). Then multiply that area by the number of dots per sq cm from the nomograph. That will give you the number of dots that should be located in that specific region. Divide the attribute value of all the dots together that would be in that region by the number of total dots supposed to be in that area to find the unit value for each dot.

The nomograph, which can be used to determine dot size and an appropriate unit value.
Figure 5.cg.8 The nomograph (originally created by MacKay (1949) can be used to determine dot size and/or an appropriate dot unit value.
Credit: after Robinson et al. 1996

Though the nomograph may provide a systematic method for selecting dot size and value, it may be somewhat unnecessarily complex. Once you understand the data you are working with and how a dot map should appear, it is easier to use trial and error to find dot size and a good value for your dots. In addition to portraying variation in the density of dots, make sure to choose a rounded number for the unit value (e.g., 'one dot equals five hundred people' instead of 'one dot equals 484 people'), as it will be easier for the map reader to generate a rough estimate of magnitudes in a particular region from a round number.

Want to learn more about the Mackay nomograph?

For an illustrated example of the nomograph being used in the manner described above (i.e. to determine dot value) and to read about some issues with the tool and Dr. Kimerling's subsequent work and discussion of probability theory to determine the best dot density, see Dr. Kimerling's presentation slides "Dotting the Dot Map, Revisited".

Although dot maps are often used to represent the spatial distribution of one particular phenomenon, using dots with different hues can create a multivariate dot map (see Figure 5.cg.9). This type of map may be particularly useful for comparing the distribution of related phenomena or related attributes of a phenomena.

The black lines on the map show chicago's official community areas, each dot represents 25 people.
Figure 5.cg.9 This multivariate dot map uses a distinct hue for each race/ethnicity, while each dot represents 25 people. Multivariate dot maps work well for variables or attributes that relate to each other and vary in their spatial distriution, as can be seen in this map. To see the map at its intended size go to the Radical Cartography website. Explore the links along the top of the Radical Cartography webpage to compare the spatial distribution of race/ethnicity in 2010 to 2000, and also to income.
Credit: Bill Rankin

Under ideal conditions, we would have information on the precise location for each dot (e.g., from a survey that used a GPS to record the position of each instance of a phenomenon or the centroid of a group of instances), but in practice the only available data are often counts within some enumeration unit (e.g., a county or some other census collection unit). This results in the problem of how individual dots are placed on the map. Current GIS software simply places dots randomly within each enumeration area. If the enumeration areas are small enough, this may work well enough. If your data are not at a small enough scale, then dot density should not be used to communicate your information (see Figure 5.cg.10).

A pair of maps to show that use of a smaller enumeration unit (right) results in a more realistic representation of New York's population than does use of a larger enumeration unit.
Figure 5.cg.10 Compare the results of two dot maps of New York population: the one on the left has the data at the scale of the state (which places random dots throughout New York), and the one on the right uses the same data, but enumerated by county. Clearly, using the smaller enumeration unit generates a more realistic depiction of New York's population surface.
Credit: Adrienne Gruver

To place dots more intelligently, cartographers can use ancillary data to restrict dot placement to particular portions of an enumeration unit. Ancillary data are additional data that can help us infer something about the spatial distribution of the attribute we are interested in mapping. Below, in Figure 5.cg.11, look at the Chicago census tract along the lake shoreline with the blue polygon surrounding it. The dots are restricted to appear outside of parks, highways and other areas known to not contain housing.

A multivariate dot map of Chicago, explained in caption and text
Figure 5.cg.11 A multivariate dot map showing how dot placements can be restricted based on ancillary data like parks and roadways. This mpa shows the distribution of children less than 5 years old in Cook, Lake, Kane, McHenry, Kamkakee, Kendell, and DuPage counties as reported by the 2010 U.S. Census. Go here to see full interactive version of this map: http://media.apps.chicagotribune.com/chicago-census/less-than-five.html
Credit: Chicago Tribune

A final important aspect of dot maps is what type of information should be contained in the legend. Just because the software you use to create your dot map may have a default for the legend, does not mean that is the most effective way to communicate the meaning of your symbols. The most important information to convey is the value of each dot, which can be represented with a simple verbal statement (e.g., "one dot represents 10 people"). But it can also be useful to convey the appearance that a particular density of dots has on the map. e.g. with exemplars of representative densities. These representative densities are called visual anchors and, if included, should show low, medium and high densities. Creating a legend with multiple visual anchors is useful, as we know that map readers tend to underestimate densities, particularly at higher densities (Olson 1975).

An example of a legend for a dot map.
Figure 5.cg.12 Example of a good legend showing multiple exemplars for a dot map.
Credit: Adrienne Gruver

Recommended Readings and Videos

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

Mapping Social Statistics - Race and Ethnicity in Chicago
Click for a transcript.

A taxonomy of Transitions: Racial and Ethnic Segregation in Chicago.

BILL RANKIN: Hello. My name is Bill Rankin. I'm an assistant professor at Yale University. I teach history of science. I also have a website about mapping at radicalcartography.net.

In this video, I'm going to talk to you about this map I made that shows racial and ethnic segregation in Chicago. I'm going to do three things. I'm first going to walk you through the map and explain the graphics. Then I'm going to talk about why I made the map, how it's different from other maps, and then what this map tells us about Chicago, about urban planning, and about different ways of understanding space. Let's begin.

Two Hundred Thousand Dots

Graphically, my map is relatively simple. It uses colored dots to show where people of different races and ethnicities live. Each dot represents 25 people. Colors are based on people's own racial self-identification on the United States census.

Pink dots show people who identify as white. Blue dots are African Americans. Orange is for Hispanics or Latinos and green shows Asians. The map is based on a lot of data. I used information from about 70,000 individual blocks. The result is a really detailed map that shows just how starkly divided Chicago is, but it doesn't overwhelm you with complicated graphics.

Why Dots?

Dot maps have been around for more than a century, but they're still relatively rare. Instead, almost all statistical maps show data using solid colors. We see these kinds of maps in all sorts of places. For example, here's a recent map from the New York Times showing ethnicities in Pakistan and Afghanistan, with different colors for different groups. Or here's a map from the 1960s showing languages in South America.

Maps like these make a strong visual argument. They make it seem that human beings are naturally divided into relatively homogeneous groups separated by sharply defined boundaries. These colored blobs look just like pastel-shaded countries in a school atlas, and by shading each blob based on which group holds the majority, they visually reinforce political ideals of national self-determination and ethnic homogeneity.

When the same graphics are used to show city neighborhoods, they end up making a similar argument. For example, here's a map from the Encyclopedia of Chicago showing ethnic neighborhoods. It makes it look like the Polish area is all Polish and the Mexican area all Mexican. Neighborhoods become like miniature territorial states.

A similar style is used in non-statistical maps as well, like in this poster where every neighborhood is shown as internally coherent and perfectly delimited.

By using dots instead of solid colors, my map immediately challenges the idea that neighborhoods are homogeneous areas with sharp boundaries. My map highlights diversity instead of majority rule, but this means that it also highlights segregation in a much more powerful way. Using dots makes it possible to ask what kinds of transitions actually exist in the city instead of making assumptions about clean edges.

What Kind of Boundaries?

Chicago is especially important in the history of urban statistics. Here's a detail from a map made in the 1920s as part of research at the University of Chicago. The result of this mapping, which again used hard line edges, was the creation of so-called community areas by the famous sociologist Ernest Burgess.

Here are his boundaries. His goal was to define natural areas of ethnic identity that could be used to tabulate census data and help to rationalize urban planning. These community areas have long been criticized as inaccurate since their boundaries have stayed the same while the city has changed. And comparing these boundaries to current patterns of segregation, it's quite clear that the two don't line up.

But what my map shows even more clearly is that accurate boundaries could never be drawn. There are some places, like here, where the transition is, in fact, quite abrupt. But there are two other kinds of transitions, as well. There are large gaps of infrastructure or industry, and there are gradients, such as this gradual blurring between white and Hispanic.

Given these examples, urban planning with hard edges starts to seem almost irresponsible. It assumes that racial divisions will always be hard and abrupt, and it makes it difficult to imagine or to encourage greater mixing.

The larger lesson of my project is that we need to think more carefully about how we map social statistics since the way we make maps influences the way we think and the possibilities for planning and design. Right now, almost all map planning is done using sharp boundaries, whether it's the boundaries of neighborhoods or the boundaries of countries. What would happen to our political understanding or to our political interventions if we made maps without solid colors? Could we have a cartography without boundaries? And that's my biggest goal here, not just to show segregation, but to provoke a different kind of mapping that shows a different kind of world. Thanks a lot.

Exit Slide Text Says

I first presented this map in the spring of 2010.

In September 2010, a computer programmer named Eric Fischer used the same graphics to map more than 100 American cities.

In December 2010, the New York Times emailed me to say they had adopted my dot mapping for an interactive feature on the 2010 Census.

Credit: Bill Rankin
  • Quitero, A. and Smith, D. 2008. "Dot Density Mapping with ArcMap: Part 1.". (Discusses restricting placement of dots based on ancillary data).
  • Rankin, B. 2011. "A Taxonomy of Transitions: Racial and Ethnic Segregation in Chicago." (Discusses merits of dot mapping for social statistics compared to choropleth mapping).
  • Fabrikant, S. 2001. "Dot maps." .