Today on the radio, I heard an announcer say, "Chicago has a higher murder rate than New York and Los Angeles combined." The compassionate human being in me cringed, and the statistical pedant in me also cringed. What does that mean?
When I heard, "New York and Los Angeles combined," I intuitively thought of combining the populations of New York and Los Angeles and their murders to get the murder rate of some bicoastal megalopolis. Of course, instead of adding the two murder rates together, this equivalent to taking a weighted average of the rates, yielding a number somewhere in between New York's 5.1 murders per 100,000 people and Los Angeles's 7.8 murders per 100,000 people. (Chicago's murder rate in 2012, the year whose data is in this table I'm using from Wikipedia, was 18.5 per 100,000 people.)
Intuitively, it's not easy to imagine what it means to combine murder rates. It's easy to understand the statement, "Chicago's murder rate was more than twice as high as the murder rate in Los Angeles." For every person murdered in Los Angeles, I imagine two people murdered instead, and then add even more. It's gruesome but not mentally difficult. But it's hard for me to understand adding the murder rates for the two cities. If I combine them into one giant city, do I imagine each murder counting double? That doesn't really work. Do I imagine a city the size of Los Angeles with a murder rate that is not quite double the current murder rate? Do I imagine a city the size of New York with a murder rate that is a little more than double the current murder rate? Those last two options are accurate but fail to resonate with me.
New York has a pretty low murder rate, or at least it did in 2012, one of its record low years. Why would the reporter not choose two cities whose murder rates come closer to adding up to Chicago's? "Chicago has a higher murder rate than Long Beach, California and Indianapolis, Indiana combined!" "Chicago has a higher murder rate than Fort Worth, Texas and Mobile, Alabama combined!" Of course, those sentences would sound bizarre on the radio. New York and Los Angeles are much more like Chicago, and we have a visceral reaction to them as dangerous big cities. But "Chicago has a higher murder rate than Fort Worth, Texas and Mobile, Alabama combined" is just as meaningful as "Chicago has a higher murder rate than New York and Los Angeles combined."
Taking out the gore, what we're doing is analogous to saying, "that car was traveling as fast as two other cars combined" or, from the memorable "Superpowers" episode of This American Life, the failed superhero 3D-Man was "as fast as three people combined." It makes intuitive sense for a person to be as tall as three other people combined. Just stand the three people on top of each other. Wow, that's a tall person! But to be as fast as three people? Is one person running on the surface of the earth, one running in the inertial frame of the first person, and the third running in the inertial frame of the second person? Do we have an odometer that is linked to all three pairs of shoes, counting up the total miles they log? I don't think either of those help me get a feel for 3D-Man's speed.
When we combine rates, we're adding up ratios of other numbers, so intuitively we want to add the numerators and/or denominators, which are the things we have more of a feel for. But as all elementary school math teachers—and many elementary school math students—will tell you, adding numerators and denominators is not the correct way to add fractions.
My feeling is that describing Chicago as having a murder rate that is higher than New York and Los Angeles combined is technically correct but practically meaningless. What do you think? Am I just a pedantic scold? Does it ever make sense to add rates like this?