Michael Gastner, Cosma Shalizi, and Mark Newman at The University of Michican have put together a great collection of maps and cartograms analyzing the 2004 election results by state and county.

Election results by state

On election night and in the months since then, we have seen many maps that look like this:

The (contiguous 48) states of the country are colored red or blue to indicate whether a majority of their voters voted for the Republican candidate (George W. Bush) or the Democratic candidate (John F. Kerry) respectively. The map gives the superficial impression that the “red states” dominate the country, since they cover far more area than the blue ones. However, as pointed out by many others, this is misleading because it fails to take into account the fact that most of the red states have small populations, whereas most of the blue states have large ones. The blue may be small in area, but they are large in terms of numbers of people, which is what matters in an election.

We can correct for this by making use of a cartogram, a map in which the sizes of states have been rescaled according to their population. That is, states are drawn with a size proportional not to their sheer topographic acreage — which has little to do with politics — but to the number of their inhabitants, states with more people appearing larger than states with fewer, regardless of their actual area on the ground. Thus, on such a map, the state of Rhode Island, with its 1.1 million inhabitants, would appear about twice the size of Wyoming, which has half a million, even though Wyoming has 60 times the acreage of Rhode Island.

Here are the 2004 presidential election results on a population cartogram of this type:

The cartogram was made using the diffusion method of Gastner and Newman. Population data was taken from the 2000 US Census.

The cartogram reveals what we know already from the news: that the country was actually very evenly divided by the vote, rather than being dominated by one side or the other.
The presidential election is not decided on the basis of the number of people who vote each way, however, but on the basis of the electoral college. Each state contributes a certain number of electors to the electoral college, who vote according to the majority in their state. The candidate receiving a majority of the votes in the electoral college wins the election. The electoral votes are apportioned roughly according to states’ populations, as measured by the census, but with a small but deliberate bias in favor of smaller states.

We can represent the effects of the electoral college by scaling the sizes of states to be proportional to their number of electoral votes, which gives a map that looks like this:

This cartogram looks very similar to the one above it, but it is not identical. Wyoming, for instance, has approximately doubled in size, precisely because of the bias in favor of small states.

The areas of red and blue on the cartogram are now proportional to the actual numbers of electoral votes won by each candidate. Thus this map shows at a glance both which states went to which candidate and which candidate won more votes — something that you cannot tell easily from the normal election-night red and blue map.
Election results by county
But we can go further. We can do the same thing also with the county-level election results and the images are even more striking. Here is a map of US counties, again colored red and blue to indicate Republican and Democratic majorities respectively:

Similar maps have appeared in the press, for example in USA Today, and have been cited as evidence that the Republican party has wide support. Again, however, a cartogram gives a more accurate picture. Here is what the cartogram looks like for the county-level election returns:

Again, the blue areas are much magnified, and areas of blue and red are now nearly equal. However, there is in fact still more red than blue on this map, even after allowing for population sizes. Of course, we know that nationwide the percentages of voters voting for either candidate were almost identical, so what is going on here?

The answer seems to be that the amount of red on the map is skewed because there are a lot of counties in which only a slim majority voted Republican. One possible way to allow for this, suggested by Robert Vanderbei at Princeton University, is to use not just two colors on the map, red and blue, but instead to use red, blue, and shades of purple to indicate percentages of voters. Here is what the normal map looks like if you do this:

And here’s what the cartogram looks like:

In this map, it appears that only a rather small area is taken up by true red counties, the rest being mostly shades of purple with patches of blue in the urban areas.

A slight variation on the same idea is to use a nonlinear color scale like this:

These maps use a color scale that ranges from red for 70% Republican or more, to blue for 70% Democrat or more. This is sort of practical, since there aren’t many counties outside that range anyway, but to some extent it also obscures the true balance of red and blue.

Maps and cartograms above from http://www-personal.umich.edu/~mejn/election