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A Higher Murder Rate than New York and Los Angeles Combined

The views expressed are those of the author and are not necessarily those of Scientific American.

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Non-Violence, a sculpture by Carl Fredrik Reuterswärd in Malmö, Sweden. Image: Francois Polito, via Wikimedia Commons.

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 teachersand many elementary school math studentswill 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?

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.

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  1. 1. llirbo 7:49 pm 05/27/2014

    Nothing pedantic here unless accuracy and meaningfulness are disregarded.

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  2. 2. larkalt 5:40 pm 05/28/2014

    There’s nothing even vaguely puzzling about adding murder rates. So NY has a rate of x per 100,000 per year. LA has a rate of y per 100,000 per year. Chicago has a murder rate more than x+y per 100,000 per year.
    It’s a clear and intuitive concept.
    There are many much more puzzling and interesting math things to write about.

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  3. 3. evelyn haskins 7:02 pm 05/28/2014

    I would only interpret this as wherever having a murder “rate” higher than 13.9 per 100 000 people — which is therefore correct.

    You are right about adding fractions, but in the instance given, the ‘denominator’ is constant. Therefore you simply add the numerators.

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  4. 4. Rich102 11:19 pm 05/28/2014

    I share the author’s discomfort at the combination of rates. I have celiac disease and I run across this all the time because of the government’s rule that in order to be labeled “gluten free”, a food product must be under 20 parts per million (ppm) gluten, a rate which has been determined to be “safe” for most people who have a problem with gluten. I’ve actually seen someone contend that if you consume one food with, say, 20ppm of gluten and another with 15ppm, that you’re getting 35ppm (I don’t recall the exact numbers, but the combining of the rates stuck with me).

    Some people seem to think that one crumb of a bread which tests at 200ppm gluten would be worse for a celiac than a whole slice of 15ppm bread. The confusion of rates and amounts is maddening and potentially dangerous.

    “…a pedantic scold?” I don’t think so. More someone who is fighting the good fight in an innumerate world.

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  5. 5. thomasjj70 4:09 pm 05/30/2014

    I stopped reading at “in this table I’m using from Wikipedia” Hilarious…..

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  6. 6. semigroup 12:59 am 05/31/2014

    Thomas, your objection is ridiculous. The data in the Wikipedia article is from the FBI Uniform Crime Report, and the Wikipedia article gives a direct link to the data on the FBI website. Moreover, Dr. Lamb was only using the statistics to illustrate a mathematical question, not to comment on crime rates.

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  7. 7. thomasjj70 10:07 am 05/31/2014

    semigroup, Since when is Wikipedia ever an acceptable source? Example from University of Harvard, “When you’re doing academic research, you should be extremely cautious about using Wikipedia. As its own disclaimer states, information on Wikipedia is contributed by anyone who wants to post material, and the expertise of the posters is not taken into consideration. Users may be reading information that is outdated or that has been posted by someone who is not an expert in the field or by someone who wishes to provide misinformation”

    I have been educated at various college institutions across the United States, and I have never been permitted to use Wikipedia for obvious reasons, certainly a great start on hour one in one’s research to obtaining additional resources, but that is it.. I understand this is just a blog, hardly the level of academic research, but I still think there is no excuse for it.. This blog may be the most accurate post in the history of the internet, yet it could be the most inaccurate one as well… My point, Dr. Lamb should have obtained her information directly from the FBI, which is easy enough.. I think I will spend my morning “adjusting” the data on Wikipedia, even though I have no practical knowledge on the topic.. Good Day.

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  8. 8. KBCash 10:08 am 05/31/2014

    Fascinating that an article on something as simple as adding “rates” cannot be understood by SA fans.

    Emotion trumps science every time.

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  9. 9. zhimbo 10:21 am 05/31/2014

    It’s more interesting that they choose New York, which as you mention doesn’t have an especially high murder rate, yet still screams “big city danger” to the uninformed. It’s an emotional argument, not a mathematical one.

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  10. 10. Sylvaner 12:00 pm 05/31/2014

    @ thomasjj70 : once again, this is math : who cares about the data origin ? it’s actually amusing that you should launch a crusade against wikipedia from an article concluded with the words “pedantic scold”…

    On topic, I agree that the initial “combined rate” comparison is goofy : the journalist should have compared raw numbers, if anything.
    It’s like adding concentrations in chemistry…

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  11. 11. larkalt 4:40 pm 06/1/2014

    “the initial “combined rate” comparison is goofy”

    Not really. The intuitive message is “if you live in Chicago your risk of getting murdered is more than what your risk would be if you lived in NY, PLUS what your risk would be if you lived in Los Angeles”.
    Adding risks is natural. We balance one risk against another all the time, and we can (roughly) add them too.

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  12. 12. hkraznodar 12:36 pm 06/13/2014

    @thomasjj70: Wikipedia has the disclaimer but has also put a variety of controls in place to eliminate contributors that post significant misinformation or make inflammatory or biased content. There have been studies done by a variety of colleges that have proven that Wikipedia is more accurate and reliable than most encyclopedias or K – 12 textbooks. Considering the number of errors the professors pointed out in my son’s textbooks that may apply to them as well.

    If you are talking about comparing a peer reviewed article in a science journal with Wikipedia then I’m pretty confident that the peer reviewed article will win, although there seem to be quite a few of those articles that get retracted or proven wrong.

    When discussing terminology used by journalists it is idiotic to insist on peer reviewed articles as a source. With many of them anything beyond “see spot run” is too demanding.

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  13. 13. BensonBear 6:42 am 06/15/2014

    I think the principle for when it makes sense to add rates requires that all the rates involved be associated with processes that can be see as collectively acting in parallel on the very same thing. So the city example really does not make sense, nor does the running example, or even the height example (since people standing on top of one another to get height is a contrived “thing”).

    Where it would make sense would be, say, painting speed: you paint at the rate of Alice, Bill, and Charles combined, where the assumption is that when combined they would each paint a portion of the area required to paint. Or, if must be, for deaths in cities, you could say for example, guns kill people at the rate of knifes and bricks combined (for a given city). It’s a bit trickier to get a net rate for the deaths, given the possibility of multiple causes of death. But actually, for painting the problem also arises, if you cannot make the simplifying assumption that each painter does not interfere with or aid the others, and there is no overhead in coordinating what subareas to paint.

    So this is the sort of case where it makes sense to “combine” (the term in the article does not say they are “added”, although it seems like that is the “combination” it had in mind) rates, although the function to do so is more complex than “adding”, and also may not be purely specifiable as a mathematical function without idealizing assumptions.

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