How to Lie With Negative Numbers

How Not to Be Wrong
The power of mathematical thinking.
June 5 2014 8:32 AM

How to Lie With Negative Numbers

Donald Trump.
About those 1 percenters ...

Photo by Jewel Samad/AFP/Getty Images

A recent working paper by economists Michael Spence and Sandile Hlatshwayo painted a striking picture of job growth in the United States. It’s traditional and pleasant to think of America as an industrial colossus, whose factories run furiously night and day producing the goods the world demands. Contemporary reality is rather different. Between 1990 and 2008, the U.S. economy gained a net 27.3 million jobs. Of those, 26.7 million, or 98 percent, came from the “nontradable sector”—the part of the economy including things like government, health care, retail, and food service, which can’t be outsourced and which don’t produce goods to be shipped overseas.

That number tells a powerful story about recent American industrial history, and it was repeated everywhere from the Economist to Bill Clinton’s latest book. But you have to be careful about what it means. Ninety-eight percent is really, really close to 100 percent. So does the study say growth is as concentrated in the nontradable part of the economy as it could possibly be? That’s what it sounds like—but that’s not quite right.

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Jobs in the tradable sector grew by a mere 620,000 between 1990 and 2008. But it could have been worse—they could have declined! That’s what happened between 2000 and 2008; the tradable sector lost about 3 million jobs, while the nontradable sector added 7 million. So the nontradable sector accounted for 7 million jobs out of the total gain of 4 million, or 175 percent!

The slogan to live by here is: Don’t talk about percentages of numbers when the numbers might be negative.

This may seem overly cautious. Negative numbers are numbers, and as such they can be multiplied and divided like any others. But even this is not as trivial as it first appears. To our mathematical predecessors, it wasn’t even clear negative numbers were numbers at all—they do not, after all, represent quantities in exactly the same way as positive numbers do! I can have seven apples in my hand, but not negative seven.

The great 16th-century algebraists, like Cardano and Vieta, argued furiously about whether a negative times a negative equaled a positive. Rather, they understood that consistency seemed to demand that this be so, but there was real dissent about whether this had been proved factual or was only a notational expedient. Cardano, when an equation he was studying had a negative number among its solutions, had the habit of calling the offending solution ficta, or fake.

The mathematical arguments of the Italian Renaissance can at times seem as recondite and irrelevant to us as their theology. But they weren’t wrong that there’s something about the combination of negative quantities and arithmetic operations like percentage that short-circuits one’s intuition. When you disobey the slogan I gave you, all sorts of weird incongruities start to bubble up.

For example, say I run a coffee shop. And people, sad to say, are not buying my coffee—last month, I lost $500 on that part of my business. Fortunately, I had the prescience to install a pastry case and a CD rack, and those two operations made a $750 profit each.

In all, I made $1,000 this month. Seventy-five percent of that amount came from my pastry case, which sounds like the pastry case is what’s really moving my business right now; almost all my profit is croissant-driven. Except that it’s just as correct to say that 75 percent of my profits came from the CD rack. And imagine if I’d lost $1,000 more on coffee—then my total profits would be zero, infinity percent of which would be coming from pastry! Seventy-five percent sounds like it means “almost all,” but when you’re dealing with numbers that could be either positive or negative, like profits, it might mean something very different.

This problem never arises when you study numbers that are constrained to be positive, like expenses, revenues, or populations. If 75 percent of Americans think Paul McCartney was the cutest Beatle, then it’s not possible that another 75 percent give the nod to Ringo Starr. Ringo, George, and John have to split the remaining 25 percent between them.

One can’t object very much to what Spence and Hlatshwayo wrote. It’s true, the total job growth in an aggregate of hundreds of industries can be negative, but in a normal economic context over a reasonably long time interval, it's extremely likely to be positive. The population keeps growing, after all, and, absent total disaster, that tends to drag the absolute number of jobs along with it.

But other percentage-flingers are not so careful. In June 2011, the Republican Party of Wisconsin issued a news release touting the job-creating record of Gov. Scott Walker. It had been another weak month for the U.S. economy as a whole, which added only 18,000 jobs nationally. But the state employment numbers looked much better: a net increase of 9,500 jobs. “Today,” the statement read, “we learned that over 50 percent of U.S. job growth in June came from our state.” The talking point was picked up and distributed by GOP politicians, like Rep. Jim Sensenbrenner, who told an audience in a Milwaukee suburb, “The labor report that came out last week had an anemic 18,000 created in this country, but half of them came here in Wisconsin. Something we are doing here must be working."

This is a perfect example of the soup you get into when you start reporting percentages of numbers, like net job gains, that might be either positive or negative. Wisconsin added 9,500 jobs, which is good; but neighboring Minnesota, under Democratic Gov. Mark Dayton, added more than 13,000 in the same month. Texas, California, Michigan, and Massachusetts also outpaced Wisconsin’s job gains. Wisconsin had a good month, that’s true, but it didn’t contribute as many jobs as the rest of the country put together, as the Republican messaging suggested. In reality, job losses in other states almost exactly balanced out the jobs created in places like Wisconsin, Massachusetts, and Texas. That’s how Wisconsin’s governor could claim his state accounted for half the nation’s job growth, and Minnesota’s governor, if he’d cared to, could have said that his own state was responsible for 70 percent of it, and they could both, in this technically correct but fundamentally misleading way, be right.

Percentages of negative numbers are especially perilous when you start thinking about inequality, as Felix Salmon explained in an April article about why it makes no sense to say the 85 richest people on Earth hold 50 percent of the world’s wealth. Or take this 2012 New York Times op-ed by Steven Rattner, which used the work of economists Thomas Piketty and Emmanuel Saez to argue that the current economic recovery is unequally distributed among Americans:

New statistics show an ever-more-startling divergence between the fortunes of the wealthy and everybody else—and the desperate need to address this wrenching problem. Even in a country that sometimes seems inured to income inequality, these takeaways are truly stunning.
In 2010, as the nation continued to recover from the recession, a dizzying 93 percent of the additional income created in the country that year, compared to 2009— $288 billion—went to the top 1 percent of taxpayers, those with at least $352,000 in income. … The bottom 99 percent received a microscopic $80 increase in pay per person in 2010, after adjusting for inflation. The top 1 percent, whose average income is $1,019,089, had an 11.6 percent increase in income.

The article comes packaged with a handsome infographic that breaks the income gains up even further: 37 percent to the ultra-rich members of the top 0.01 percent, with 56 percent to the rest of the top 1 percent, leaving a meager 7 percent for the remaining 99 percent of the population. You can make a little pie chart:

Recall that the top 1 percent scored 93 percent of income gains. Now let’s slice the pie one more time, and ask about the people who are in the top 10 percent, but not the top 1 percent. Here you’ve got the family doctors, the non-elite lawyers, the engineers, and the upper-middle managers. How big is their slice? You can get this from Piketty and Saez’s data, which they’ve helpfully put online. (Warning: Clicking that link will download a large spreadsheet.) And you find something curious. This group of Americans had an average income of about $159,000 in 2009, which increased to a little more than $161,000 in 2010. That’s a modest gain compared with what the richest percentile racked up, but it still accounts for 17 percent of the total income gained between 2010 and 2011.

wealth pie chart

Try to fit a 17 percent slice of the pie in with the 93 percent share held by the 1 percenters and you find you’ve got more pie than plate.

Ninety-three percent and 17 percent add up to more than 100 percent; how does this make sense? It makes sense because the bottom 90 percent actually had lower average income in 2011 than they did in 2010, recovery or no recovery. Put negative numbers in the mix, and percentages get wonky.

None of which is to deny that morning in America comes a little earlier in the day for the richest Americans than it does for the middle class. But it does put a slightly different spin on the story. It’s not that the 1 percent are benefiting while the rest of America languishes. The people in the top 10 percent but not the top 1 percent—a group that includes, not to put too fine a point on it, many readers of the New York Times opinion page—are doing fine, too, capturing more than twice as much as the 7 percent share that the pie chart appears to allow them. It’s the other 90 percent of the country whose tunnel still looks dark at the end.

Jordan Ellenberg is a professor of mathematics at the University of Wisconsin and the author of How Not to Be Wrong. He blogs at Quomodocumque.

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