The image on this page is the earliest known use of a dot density cartogram to compare disease rates (Wallace 1926).
Wallace was studying smallpox death rates in California. As expected, most deaths occur in counties with large populations. The question of interest is whether the smallpox rates are approximately equal across the entire state. The customary technique, calculating rates for each county, is unsatisfactory because the number of deaths in each county is small, even zero, in many counties.
Grouping counties is also unsatisfactory because geographic detail is lost. Furthermore, one must make an arbitrary choice about how to group the counties. Such an arbitrary choice can bias the results of the analysis.
When the number of deaths is small, a superior method is to adjust the county boundaries so that every county has an area proportional to its population. Now the population density is uniform over the whole map, and one can easily pick out regions where the deaths (the dots) have greater than average density.