The Probability of Zero

Good news: Ogden has had zero homicides so far in 2011 (probably).

Bad news: Journalists don’t understand statistics (still).

“Killings down in Ogden,” proclaimed the headline across the top of Sunday’s front page, with a great big zero on one side. Pending a final ruling on whether a fatal July shooting was accidental, Ogden has probably gone for nine months without a murder or automobile homicide. This isn’t just great news; it’s historic.

The article falls short, though, in discussing the possible causes of this unprecedented drop in killings. Relying entirely on statements from the police chief and the county attorney, the article mentions three possible contributing factors: a new police “Crime Reduction Unit” created four years ago; a year-old injunction against the city’s oldest street gang; and a shift over the last several years toward handling gun-related crimes in federal court.

Of course, all of these factors could very well be contributing to a long-term reduction in crime, and the lack of recent homicides could very well be part of that long-term trend. But statistically, you just can’t tell.

You see, Ogden’s homicide rate was already pretty low. According to a data table printed on page 5, Ogden hasn’t had more than four homicides in a calendar year since 2001. During the last nine years, the average number in any nine-month period was only two and a half.

With this data and a simple formula from elementary statistics, we can answer the obvious question: Given this average rate of homicides, what’s the probability of getting zero homicides in any given nine-month period? The answer is one in e^2.5, where e is the famous mathematical constant 2.718 (approximately). Do the math and you find that the probability is about one in 12. (Note that e^2.5 means e times e times the square root of e, or about 2.7 times 2.7 times 1.6.)

I’m assuming, though, that each homicide is an independent event. In fact, some homicides occur in related groups. If the average number of independent homicide groups during any nine-month period is only 2.0, then the probability of getting zero in such a period is one in e², or about one in 7.

If these probabilities still seem rather low, remember that the zero didn’t have to occur this year. Now that the average homicide rate has been at this level for about a decade, we’ve had ten one-in-seven chances so far to get zero homicides during the first nine months of a year. In other words, we were over-due.

It’s understandable that the police chief and county attorney would attribute the lack of homicides to their own efforts. It’s also human nature to look for simple cause-effect relationships. But at this point, the most natural explanation for Ogden’s zero homicides in 2011 (so far) is a mere statistical fluctuation. The article doesn’t even mention this possibility, and it should.

What can’t be explained by mere statistics is the long-term trend. Homicide rates across the U.S. have steadily declined for the last two decades, and Ogden appears to be following this trend. Social scientists have proposed a host of possible reasons for the decline, including better policing and increased incarceration rates, but also including our aging population, changes in immigration, shifts in the illegal drug trade, and availability of abortions (resulting in fewer unwanted children). The even more striking decline over the very long term is probably a result of improving economic conditions, gradually changing attitudes toward killing, and/or increased acceptance of government as the enforcer of laws.

Let’s hope these long-term trends continue, but let’s not jump to conclusions based on local short-term fluctuations.