In all, I made $1000 this month, and 75% 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% of my profits came from the CD rack. And imagine if I’d lost $1000 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% of Americans think Paul McCartney was the cutest Beatle, then it’s not possible that another 75% give the nod to Ringo Starr; he, George,* and John have to split the remaining 25% between them.
You can see this phenomenon in the jobs data, too. Spence and Hlatshwayo might have pointed out that about 600,000 jobs were created in finance and insurance; that’s almost 100% of the total jobs created by the tradable sector as a whole. They didn’t point that out, because they weren’t trying to trick you into believing that no other part of the economy was growing over that time span. As you might remember, there was at least one other part of the U.S. economy that added a lot of jobs between 1990 and today: the sector classified as “computer systems design and related services,” which tripled its job numbers, adding more than a million jobs all by itself. The total jobs added by finance and computers were way over the 620,000 jobs added by the tradable sector as a whole; those gains were balanced out by big losses in manufacturing. The combination of positive and negative allows you, if you’re not careful, to tell a fake story, in which the whole work of job creation in the tradable sector was done by the financial industry.
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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 Governor Scott Walker. It had been another weak month for the U.S. economy as a whole, which added only eighteen thousand jobs nationally. But the state employment numbers looked much better: a net increase of ninety-five hundred 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 Representative Jim Sensenbrenner, who told an audience in a Milwaukee suburb, “The labor report that came out last week had an anemic eighteen thousand 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 ninety-five hundred jobs, which is good; but neighboring Minnesota, under Democratic governor Mark Dayton, added more than thirteen thousand 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 fact, what was going on is that 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% of it, and they could both, in this technically correct but fundamentally misleading way, be right.
Or take a recent 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% to the ultrarich members of the top 0.01%, with 56% to the rest of the top 1%, leaving a meager 7% for the remaining 99% of the population. You can make a little pie chart:
Now let’s slice the pie one more time, and ask about the people who are in the top 10%, but not the top 1%. 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. And you find something curious. This group of Americans had an average income of about $159,000 in 2009, which increased to a little over $161,000 in 2010. That’s a modest gain compared to what the richest percentile racked up, but it still accounts for 17% of the total income gained between 2010 and 2011.
Try to fit a 17% slice of the pie in with the 93% share held by the one-percenters and you find you’ve got more pie than plate.
93% and 17% add up to more than 100%; how does this make sense? It makes sense because the bottom 90% actually had lower average income in 2011 than they did in 2010, recovery or no recovery. Negative numbers in the mix make percentages act wonky.
Looking at the Piketty-Saez data for different years, you see the same pattern again and again. In 1992, 131% of the national gains in income were accrued by the top 1% of earners! That’s certainly an impressive figure, but one which clearly indicates that the percentage doesn’t mean quite what you’re used to it meaning. You can’t put 131% in a pie chart. Between 1982 and 1983, as another recession retreated into memory, 91% of the national income gain went to the 10%-but-not-1% group. Does that mean that the recovery was captured by the reasonably wealthy professionals, leaving the middle class and the very rich behind? Nope—the top 1% saw a healthy increase that year too, accounting for 63% of the national income gain all by themselves. What was really going on then, as now, was that the bottom 90% continued to lose ground while the situation brightened for everybody else.
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% are benefitting while the rest of America languishes. The people in the top 10% but not the top 1%—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% share that the pie chart appears to allow them. It’s the other 90% of the country whose tunnel still looks dark at the end.
Even when the numbers involved happen to be positive, there’s room for spinners to tell a misleading story about percentages. In April 2012, Mitt Romney’s presidential campaign, facing poor poll numbers among women voters, released a statement asserting, “The Obama administration has brought hard times to American women. Under President Obama, more women have struggled to find work than at any other time in recorded history. Women account for 92.3% of all jobs lost under Obama.”
That statement is, in a manner of speaking, correct. According to the Bureau of Labor Statistics, total employment in January 2009 was 133,561,000, and in March 2012, just 132,821,000: a net loss of 740,000 jobs. Among women, the numbers were 66,122,000 and 65,439,000; so 683,000 fewer women were employed in March 2012 than in January 2009, when Obama took office. Divide the second number by the first and you get the 92% figure. It’s almost as if President Obama had been going around ordering businesses to fire all the women.
But no. Those numbers are net job losses. We have no idea how many jobs were created and how many destroyed over the three-year period; only that the difference of those two numbers is 740,000. The net job loss is positive sometimes, and negative other times, which is why taking percentages of it is a dangerous business. Just imagine what would have happened if the Romney campaign had started their count one month later, in February 2009.* At that point, another brutal month into the recession, total employment was down to 132,837,000. Between then and March 2012, the economy suffered a net loss of just 16,000 jobs. Among women alone, the jobs lost were 484,000 (balanced, of course, by a corresponding gain for men). What a missed opportunity for the Romney campaign—if they’d started their reckoning in February, the first full month of the Obama presidency, they could have pointed out that women accounted for over 3,000% of all jobs lost on Obama’s watch!
But that would have signaled to any but the thickest voters that this percentage was somehow not the right measure.
What actually happened to men and women in the workforce between Obama’s inauguration and March 2012? Two things. Between January 2009 and February 2010, employment plunged for both men and women as the recession and its aftermath took their toll.
January 2009−February 2010:
Net job loss for men: 2,971,000
Net job loss for women: 1,546,000
And then, post-recession, the employment picture started slowly improving:
February 2010−March 2012:
Net job gain for men: 2,714,000
Net job gain for women: 863,000
During the steep decline, men took it on the chin, suffering almost twice as many job losses as women. And in the recovery, men account for 75% of the jobs gained. When you add both periods together, the men’s figures happen to cancel out almost exactly, leaving them with about as many jobs at the end as the beginning. But the idea that the current economic period has been almost exclusively bad for women is badly misguided.
The Washington Post graded the Romney campaign’s 92.3% figure as “true but false.” That classification drew mockery by Romney supporters, but I think it’s just right, and has something deep to say about the use of numbers in politics. There’s no question about the accuracy of the number. You divide the net jobs lost by women by the net jobs lost, and you get 92.3%.
But that makes the claim “true” only in a very weak sense. It’s as if the Obama campaign had released a statement saying, “Mitt Romney has never denied allegations that for years he’s operated a bicontinental cocaine-trafficking ring in Colombia and Salt Lake City.”
That statement is also 100% true! But it’s designed to create a false impression. So “true but false” is a pretty fair assessment. It’s the right answer to the wrong question. Which makes it worse, in a way, than a plain miscalculation. It’s easy to think of the quantitative analysis of policy as something you do with a calculator. But the calculator only enters once you’ve figured out what calculation you want to do.
I blame word problems. They give a badly wrong impression of the relation between mathematics and reality. “Bobby has three hundred marbles and gives 30% of them to Jenny. He gives half as many to Jimmy as he gave to Jenny. How many does he have left?” That looks like it’s about the real world, but it’s just an arithmetic problem in a not very convincing disguise. The word problem has nothing to do with marbles. It might as well just say: type “300 − (0.30 × 300) − (0.30 × 300)/2 =” into your calculator and copy down the answer!
But real-world questions aren’t like word problems. A real-world problem is something like “Has the recession and its aftermath been especially bad for women in the workforce, and if so, to what extent is this the result of Obama administration policies?” Your calculator doesn’t have a button for this. Because in order to give a sensible answer, you need to know more than just numbers. What shape do the job-loss curves for men and women have in a typical recession? Was this recession notably different in that respect? What kind of jobs are disproportionately held by women, and what decisions has Obama made that affect that sector of the economy? It’s only after you’ve started to formulate these questions that you take out the calculator. But at that point the real mental work is already finished. Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.
Includes: hidden messages in the Torah, the dangers of wiggle room, null hypothesis significance testing, B. F. Skinner vs. William Shakespeare, “Turbo Sexophonic Delight,” the clumpiness of prime numbers, torturing the data until it confesses, the right way to teach creationism in public schools
SIX
THE BALTIMORE STOCKBROKER AND THE BIBLE CODE
People use mathematics to get a handle on problems ranging from the everyday (“How long should I expect to wait for the next bus?”) to the cosmic (“What did the universe look like three trillionths of a second after the Big Bang?”).
But there’s a realm of questions out beyond cosmic, questions about The Meaning and Origin of It All, questions you might think mathematics could have no purchase on.
Never underestimate the territorial ambitions of mathematics! You want to know about God? There are mathematicians on the case.
The idea that earthly humans can learn about the divine world by rational observation is a very old one, as old, according to the twelfth-century Jewish scholar Maimonides, as monotheism itself. Maimonides’s central work, the Mishneh Torah, gives this account of Abraham’s revelation:
After Abraham was weaned, while still an infant, his mind began to reflect. By day and by night he was thinking and wondering: “How is it possible that this [celestial] sphere should continuously be guiding the world and have no one to guide it and cause it to turn round; for it cannot be that it turns round of itself?” . . . His mind was busily working and reflecting until he had attained the way of truth, apprehended the correct line of thought, and knew that there is one God, that He guides the celestial sphere and created everything, and that among all that exist, there is no god besides Him. . . . He then began to proclaim to the whole world with great power and to instruct the people that the entire universe had but one Creator and that Him it was right to worship. . . . When the people flocked to him and questioned him regarding his assertions, he would instruct each one according to his capacity till he had brought him to the way of truth, and thus thousands and tens of thousands joined him.
This vision of religious belief is extremely congenial to the mathematical mind. You believe in God not because you were touched by an angel, not because your heart opened up one day and let the sunshine in, and certainly not because of something your parents told you, but because God is a thing that must be, as surely as 8 times 6 must be the same as 6 times 8.
Nowadays, the Abrahamic argument—just look at everything, how could it all be so awesome if there weren’t a designer behind it?—has been judged wanting, at least in most scientific circles. But then again, now we have microscopes and telescopes and computers. We are not restricted to gaping at the moon from our cribs. We have data, lots of data, and we have the tools to mess with it.
The favorite data set of the rabbinical scholar is the Torah, which is, after all, a sequentially arranged string of characters drawn from a finite alphabet, which we attempt faithfully to transmit without error from synagogue to synagogue. Despite being written on parchment, it’s the original digital signal.
And when a group of researchers at the Hebrew University in Jerusalem started analyzing that signal, in the mid-1990s, they found something very strange; or, depending on your theological perspective, not strange at all. The researchers came from different disciplines: Eliyahu Rips was a senior professor of mathematics, a well-known group theorist; Yoav Rosenberg a graduate student in computer science; and Doron Witztum a former student with a master’s degree in physics. But all shared a taste for the strand of Torah study that searches for esoteric texts hidden beneath the stories, genealogies, and admonitions that make up the Torah’s surface. Their tool of choice was the “equidistant letter sequence,” henceforth ELS, a string of text obtained by plucking characters from the Torah at regular intervals. For example, in the phrase
DON YOUR BRACES ASKEW
you can read every fifth letter, starting from the first, to get
DON YOUR BRACES ASKEW
so the ELS would be DUCK, whether as warning or waterfowl identification to be determined from context.
Most ELSs don’t spell words; if I make an ELS out of every third letter in the sentence you’re reading, I get gibberish like MTSOSLO . . . , which is more typical. Still, the Torah is a long document, and if you look for patterns, you’ll find them.
As a mode of religious inquiry, this seems strange at first. Is the God of the Old Testament really the kind of deity who signals his presence by showing up in a word search? In the Torah, when God wants you to know he’s there, you know—ninety-year-old women get pregnant, bushes catch fire and talk, dinner falls from the sky.
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 8