Percentages are everywhere in the news, and they are easy to misread. A few simple rules clear up most confusion.
Percent change
“Percent” means “per hundred.” To find a percent change, divide the change by the starting value and multiply by 100. If a bus fare rises from $2.00 to $2.50, the change is $0.50, and $0.50 divided by $2.00 is 0.25, so the fare rose 25 percent.
Percent versus percentage points
When the thing being measured is itself a percentage, two different descriptions are possible. Suppose unemployment rises from 4 percent to 6 percent. The rate rose by 2 percentage points, the simple difference between 6 and 4. But it also rose by 50 percent, because 2 is half of 4. A rise from 4 percent to 6 percent is a rise of 2 percentage points, or 50 percent. Both are correct; mixing them up is not. A headline saying “unemployment up 2 percent” is ambiguous at best.
Base effects
Percent changes depend heavily on the starting point, or base. If a small town had 2 robberies last year and 4 this year, robberies rose 100 percent, which sounds like a crime wave but is two extra events. Small changes in small numbers create huge percent changes, so always ask for the actual counts.
Percent changes are also not symmetrical. If a $100 investment falls 50 percent, it is worth $50. If it then rises 50 percent, it gains half of $50, which is $25, ending at $75. A fall of 50 percent followed by a rise of 50 percent leaves you at 75 percent of where you started. To recover from a 50 percent loss you need a 100 percent gain.
Rates and per capita figures
Bigger places have more of almost everything: more crimes, more hospitals, more births. To compare places of different size, convert counts into rates, such as a number per 1,000 or per 100,000 people. A figure divided by the population is called a per capita figure, Latin for “per head.”
Suppose Town A has 50 bicycle thefts among 10,000 residents, and City B has 500 among 200,000. City B has ten times as many thefts. But Town A has 50 divided by 10,000, or 5 thefts per 1,000 residents, while City B has 500 divided by 200,000, or 2.5 per 1,000. Per person, Town A has twice the rate.
Absolute and relative risk
Health news often reports changes in risk. Suppose a treatment cuts the chance of a disease from 2 in 100 to 1 in 100. The relative risk reduction is 50 percent, because the risk was halved. The absolute risk reduction is 1 percentage point, from 2 percent to 1 percent. So the treatment gives a 50 percent relative reduction and a 1 percentage point absolute reduction. Both are true, but “halves your risk” sounds far more dramatic.
A useful related measure is the number needed to treat: how many people must receive a treatment for one of them to benefit. It equals 1 divided by the absolute risk reduction. Here, 1 divided by 0.01 is 100, so 100 people must be treated to prevent one case.
A historical warning
In 1995 British authorities warned that certain newer contraceptive pills roughly doubled the risk of a dangerous blood clot compared with older pills. In relative terms the warning was accurate. In absolute terms, the risk rose from about 1 in 7,000 women to about 2 in 7,000. The doubling was large in relative terms but small in absolute terms. Many women stopped taking the pill abruptly, and researchers later linked the scare to a rise in unplanned pregnancies and abortions. Reporting both the relative and the absolute numbers would have given a truer picture.
Worked example: reading a health headline
A headline announces: “Eating Snack X every day linked to 20 percent higher risk of Disease Y.” Here is how to make sense of it, using invented figures.
Find the baseline. The study reports that among people who rarely ate Snack X, about 5 in 1,000 developed Disease Y over ten years.
Apply the relative increase. A 20 percent higher risk means 5 multiplied by 1.2, which is 6 in 1,000 among daily eaters.
Find the absolute difference. The difference is 6 minus 5, or 1 extra case per 1,000 people over ten years. As a percentage, the risk went from 0.5 percent to 0.6 percent, a rise of 0.1 percentage points.
Turn it around. If 1,000 people ate Snack X daily for ten years, we would expect about one extra case of Disease Y. The other 999 would have the same outcome either way.
Check the kind of study. The word “linked” usually signals an observational study. People who eat Snack X every day might differ in other ways, such as diet, exercise or income. Level 5 explains why that matters.
So a fair summary might be: “In one observational study, daily Snack X eaters had about one extra case of Disease Y per 1,000 people over ten years; the cause is unclear.” That is far less alarming than the headline, though not nothing. Whether it matters for you depends on your own baseline risk and how much you enjoy Snack X.
The habit to build is simple: whenever you see a relative change, ask “out of how many?” and work out the absolute numbers.