More Pie Than Plate

intermediate3 min readeconomics · statistics · mathematics · business

A component’s share of net change can exceed the whole when other components are negative. Ellenberg’s job-growth example shows why such percentages need the underlying gains and losses to make sense.

THE IDEA

Proportions can be misleading even in simpler, seemingly less ambiguous cases.

A working paper discussed by Ellenberg, 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%, 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 widely repeated, from The Economist to a book by Bill Clinton. But you have to be careful about what it means. Ninety-eight percent is really, really close to 100%. So does the study say that 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. Jobs in the tradable sector grew by a mere 620,000 between 1990 and 2008, that’s true. 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%!

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 that negative numbers were numbers at all—they do not, after all, represent quantities in exactly the same way positive numbers do. I can have seven apples in my hand, but not negative seven. The great sixteenth-century algebraists, like Cardano and François Viète, argued furiously about whether a negative times a negative equaled a positive; or rather, they understood that consistency seemed to demand that this be so, but there was real division about whether this had been proved factual or was only a notational expedient.

When to use it

Analyzing economic growth or job data where sectors can lose jobs.
Evaluating financial performance when profits or revenues might decline.
Interpreting survey results where net changes are small and percentages can be inflated.
Presenting market share changes when a competitor exits the market.

FROM THEORY TO EVERYDAY LIFE

In practice

U.S. Job Growth 1990–2008

Between 1990 and 2008, the U.S. economy gained 27.3 million net jobs, of which 26.7 million (98%) came from the nontradable sector. While that sounds overwhelmingly concentrated, the tradable sector actually grew by 620,000 jobs—had it declined, the percentage could have exceeded the whole, as shown when the tradable sector lost 3 million jobs from 2000 to 2008, causing the nontradable sector's share to become 175%.

Negative Numbers in Percentages

When total net change is small or negative, a component’s percentage can exceed 100% or become negative. The arithmetic can be valid while the interpretation is confusing. For instance, if a company's profits drop from $10M to -$2M, reporting a '-120%' change is clearer when accompanied by the profit and loss amounts.

✳

A MOMENT TO REFLECT

Put it to work.

When might you be tempted to use percentages on data that includes negative numbers? How could that mislead your audience?
Can you think of a real-world report or statistic that used percentages in a misleading way because the denominator was small or negative?
How would you reframe a statistic that involves negative numbers to avoid the 'more pie than plate' fallacy?