Four New Jersey women in March accused the Campbell Soup Company of misleading customers with claims of lower sodium levels in its "25% Less Sodium Tomato Soup." Whether the soup has more or less sodium than regular versions is not for me to investigate. I want to focus on the "25% less" phrase — a type of claim we see regularly in ads and new product labels — and in the process provide some numerical literacy skills to our arsenal of skeptical thinking tools.
In an age when quantitative thinking is at a premium and "innumeracy," as cognitive scientist Douglas R. Hofstadter termed it, is a problem, many people easily misinterpret numbers and become wary about statistics. Sometimes this skepticism is for good reason - remember that oft-cited phrase "there are three kinds of lies: lies, damned lies and statistics."
But turning our backs on numbers is a mistake. We require critical thinking skills to make sense of data that appear in commercials, politician-mediated public opinion polls, official documents and research studies.
Consider this paragraph from a New York Times article about the increase in multiracial people in the latest 2010 U.S. Census: "In North Carolina, the mixed-race population doubled. In Georgia, it expanded by more than 80 percent, and by nearly as much in Kentucky and Tennessee. In Indiana, Iowa and South Dakota, the multiracial population increased by about 70 percent." A few paragraphs later the article reports a possible national multiracial growth rate of 35 percent, maybe even a 50 percent increase from the last census in 2000 when 2.4 percent of Americans selected more than one race.
With these numbers coming at you fast and furious, it takes a moment to reflect on what is actually being said and what information is missing.
First, it's essential to understand how percentages are calculated. At the simplest level, percentages are meant to standardize comparisons among two or more groups. For example, it would be accurate to report that Californians apply for the most passports in the U.S., averaging more than 2 million a year. As the most populous state, this isn't surprising. But it turns out that California is behind five other states, including New Jersey and Alaska, when taking into account the relative population of the states — in other words, the percent of citizens with passports. (I leave it up to you to create your own inventive explanations why these two states have more citizens per capita with passports!)
What can get confusing is how the numbers are reported.
A percentage point is the simple difference between two percentages, so that a change from 20 percent in the population supporting Candidate A in the election to 30 percent a few months later indicates an increase in 10 percentage points for the struggling politician. Yet, this 30 percent is an increase of 50 percent over the original 20 percent. Got that?!
To calculate this, you take the percentage point difference and then divide it by the original percentage (10/20=.50). You can now claim in your ads that your candidate has improved by 50 percent in the favorability rating (rather than the way the candidate's opponent would present the same story saying there was merely a 10 percentage point increase)! Perhaps some people will erroneously conclude your candidate went up 50 points from 20 to 70 percent in the polls — and maybe even became less salty in the process — would be a happy, if not entirely unexpected, accident.
Going back to the Census figures quoted in The New York Times, it's one thing to claim that the multiracial population may increase 50 percent, but when the original figure is only 2.4 percent of Americans, a 50 percent increase simply means that the 2010 multiracial population could end up around 3.6 percent of the population. The number 50 surely sounds more impressive than the smaller 3.6 figure. Manipulating these numbers can create misleading impressions, sometimes done with intention.
Imagine you were creating an ad for a new baldness cure. You can easily fool a non-skeptical audience by exclaiming that after trying your fabulous wonder product for several months, there's been a 50 percent increase in satisfaction. Of course, you don't reveal that only 3 percent love the product after six months compared to a measly 2 percent raving about it as the sure-fire cure after month one.
Being able to interpret numbers correctly often comes down to reporting all the relevant data. In The New York Times article, we really do not have enough information to know exactly what the multiracial population is in the states listed. That they doubled in North Carolina and increased 80 percent in Georgia tells us little about the actual 2000 or 2010 census figures in those states.
The Chicago Guide to Writing about Numbers reminds us always to identify clearly the items being compared and to express differences as percentages of the initial reference values. An increase of 50 cents when purchasing a cappuccino is a little more annoying that the same amount tacked onto the price of a new car.
When it comes to statistics and developing critical quantitative thinking skills, also remember what some wag once opined: "98 percent of all statistics are made up."