Deceitful Words: Fast Growing

This is my second post in what I hope will be a series about words and phrases that should warn you that someone is trying to get something past you. (First one here.)

[Update: Since originally publishing this, I’ve rewritten parts of it to reduce confusion, adding context to the first example, and appending a few cautions.]

Today I’d like to talk about the phrase “growing faster,” and why it can be a sign that someone is maybe trying to trick you. I’ve been meaning to bring this one up for a while, and I finally decided to do it after a random observation by Scott Greenfield about the Guiding Vision and Definition of Principles document for the Women’s March on Washington, which includes this statement (emphasis mine):

We believe it is our moral imperative to dismantle the gender and racial inequities within the criminal justice system. The rate of imprisonment has grown faster for women than men, increasing by 700% since 1980

(The statement goes on to make some good points about the special issues affecting women in prison, but I’d like to focus on just this section.)

It seems odd to talk about wanting to “dismantle the gender…inequalities” of prison by focusing on the rate of imprisonment of women, since there are far more men in prison than women. It’s not even close. By the numbers, incarceration is a men’s problem — and disproportionately a problem for men of color — with a relatively small group of women also being incarcerated.

That’s not to say that women don’t have it bad in prison, and it’s not to say that women don’t face special problems in prison — the Guiding Vision statement mentions sexual assault, pregnancy, and health care in general — and in any case there certainly isn’t anything wrong with protesting the conditions in women’s prisons or arguing that we need to pay more attention to them. But that wasn’t enough for the authors of this document. They apparently wanted to make an issue of the the rate of imprisonment of women.

The problem is, women aren’t imprisoned at a very high rate by U.S. standards. Using Bureau of Justice Statistics figures, 64 out of every 100,000 women were in prison in 2015, compared to 863 men per 100,000. Back in 1980, the figure for men was 274 (four times higher than the current rate for women), compared to only 11 for women, so imprisoned women have always been a relative rarity. But by choosing to look at the rate of change, the authors can cast this as a women’s problem, since the rate of imprisonment for men only tripled, whereas the rate for women increased nearly sixfold.

(I’m not sure why the Guiding Vision document indicates a 700% increase. Perhaps they are using a source that includes people held pending trail or something. I don’t think it changes my basic point.)

That’s a rarely mild example, and it might not even be intentional: Advocates for women’s issues may be genuinely alarmed at the increase in incarceration rates among women. But given that men are incarcerated at a much higher rate, the numbers do not show that women are suffering from inequality. They are at an advantage, and spending more time locked in a cage reduces the size of the inequality with men.

This is just one of many examples — and not a particularly pernicious one — in which advocates talk about the rate of growth of some statistic, expressed as a percentage or ratio, to mask the fact that current values are relatively small. You see this a lot in sensationalist crime stories in which some law enforcement spokesperson tells us about a new crime that is “the fastest growing crime” in their territory. This is especially common with newly-defined crimes.

For example, the first iPhones were sold in 2007, and the number of iPhones in the world grew very fast. And so too, presumably, did the number of iPhones that were stolen. This could easily be the basis of a news story: If in some law enforcement agency’s territory there were 25 iPhones stolen in 2007, and 75 iPhones stolen in 2008, a publicity-seeking top cop could put out a press release drawing attention to what his department was doing about the “shocking 200% increase in iPhone theft.” This trick works with almost anything new. Whenever a new product enters the marketplace, you’ll see stories about the “fastest-growing brand” or the “fastest-growing trend” as industry marketers and journalists collude to hype stories.

New things can grow so fast because they start so small. If you sell 1 million items the first year, you’ll have to sell 2 million items the second year to claim 100% year-over-year growth. But if you sell only 10 items the first year, then you only need to sell 20 items the second year to claim the same 100% year-over-year growth rate. And selling 20 items is much easier than selling 2 million items.

It could even happen by chance: If the numbers are small, the thing doesn’t even have to be new to have a high rate of change, because random chance will do the job. For example, a quiet town might have a 1-in-4 chance of having a murder in any given month. On average, that will be about 3 murders per year, but the murders aren’t going to tick off like clockwork at the rate of exactly 3 per year. That’s just not how murders happen. Some years there will be 3 murders, some years there will be fewer, and some years there will be more.

I generated a random sample of 10 years of murder rates based on a 1-in-4 chance of a murder in any given month, and I got this sequence:

YearMurders
12
25
33
 43
 54
 61
 72
 83
 96
 102

That is a completely random sequence, with nothing changing in the model from year to year, but in the first two years the murder rate more than doubles. And look at the sequence from years 6 to 9: The murder rate doubles, then goes up 50%, then doubles again. Imagine being the police chief in this town in the 9th year and having to explain to the City Council why the murder rate has gone up 500% in the last three years. Then in the 10th year, your replacement gets credit for a 67% drop in murders. And it’s all just random events. Nothing material has changed.

The effects of random variations aren’t as extreme in larger populations. The numbers still go up and down, but the random variations tend to cancel each other out. Maybe some neighborhoods will get more violent, while others will quiet down, leading to a more or less steady murder rate. In absolute number of murders, the year-to-year variations in a large city are much larger than in a small town, but expressed as a percentage or ratio they will be smaller. As a general rule, all other things being equal, the larger the numbers involved, the smaller the variability from year to year when expressed as a ratio or percentage.

For that reason, people who want to convince you that big changes are happening will often talk only about a small subset of the data. Someone looking to exaggerate the crime rate may find to their disappointment that the U.S. murder has gone down. But if they look at the rates in the major cities, they’ll find that some have gone down and, crucial to their agenda, some have gone up. Lazy journalists do this all the time. One year they’ll write that “The murder rate in Denver, Chicago, and St. Louis has gone up a shocking 22 percent!” Then the next year, when those cities have quieted down, they’ll write about the shocking increase in murders in three different cities.

Here’s another example from the new White House page on law enforcement:

In 2015, homicides increased by 17% in America’s fifty largest cities. That’s the largest increase in 25 years. In our nation’s capital, killings have risen by 50 percent.

The 50% increase in killings in the capital is a classic case of what I was just talking about, picking a small region to isolate a large increase. And the sentence before that is even more amazing, talking about “the largest increase in 25 years.” Crime has been decreasing for decades, so any increase is going to be one of the largest. By comparing increases, the author is not just talking about the rate of change, but the rate of change of the rate of change. It’s nonsense squared.

Even when talking about non-random growth in large numbers, it’s still possible for “fast growing” to be deceptive if the quantity under consideration is starting unusually low and growing toward a limit. President Obama’s fans have been pulling this trick for years on social media, posting statistics about the improving economy under his administration. But Obama took office in the depths of the deepest recession since the 1930s. All of the economic indicators were terrible. Since then, they’ve been returning to normal, catching up to where they would have been if U.S. production had remained at full capacity, which is not as difficult as increasing the capacity itself. It’s like bragging about quadrupling your running speed as you recover from knee surgery. The gains are real and important, but it doesn’t make you an Olympian.

When someone tries to make a point by telling you that something is growing fast, consider whether any of these possibilities apply:

  • Is the thing in question growing fast because it’s starting so small, so that a small change produces a large ratio?
  • If it’s small, could the growth just be random variations?
  • Are the numbers drawn from a small subset of a larger population, perhaps one chosen because it happens to have a high growth rate?
  • Is it growing because it’s recovering from a problem and returning to normal values?

None of these things are definitive proof that someone is trying to trick you. It’s possible that they’re using the only data available, or that they’re trying to make a point in which the growth rate really is important. But you should think about it carefully before accepting their conclusions.

A few caveats:

First, a small problem that’s growing fast is much more of a concern if the current value is an input into the growth rate of the problem. With a contagious disease, the number of people who are exposed increases with the number of people who are already infected, so that growth rate will accelerate and could reach huge numbers very fast. Thus it makes sense to pay attention to sudden contagious disease flare-ups, even when they are small.

Second, there’s a difference between a small problem, and a small input to a problem. There doesn’t have to be a lot of botulinum toxin in a city’s water supply to cause a lot of deaths, and there doesn’t have to be a lot of carbon dioxide in the atmosphere to cause global warming.

Third, whenever something is growing, it pays to watch out for threshold effects. A one-inch rise in sea level is not a disaster, unless it’s one inch higher than the top of the seawall.

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