Percentages can sometimes be misleading, even for those who feel comfortable with numbers. The word comes from the Latin: per cent means “for each hundred.” This is why 95 percent means 95 for each hundred. It is also why the percentage is the ratio in terms of hundreds. Suppose a poll finds that 620 people out of 1250 polled prefer chocolate to vanilla ice cream. The ratio of chocolate lovers to the whole population is 620/1250 = .496. The percentage of those who prefer chocolate is the ratio times 100; in this case .496 x 100 = 49.6% of those questioned prefer chocolate over vanilla.
At times we do percentages in our heads rather quickly, which can lead to erroneous conclusions. For example, suppose a shopkeeper raised the price of a jacket by 50 percent, and then discounted it by 50 percent. The price would not be the same price as it started! Suppose the jacket cost $100. After the 50 percent increase, it would cost $150. Take 50 percent off of $150, for a price of $75, rather than the original $100.
Another common trap is the following: Suppose the homicide rate in a city went down 50 percent in one year, and another 30 percent the following year. Did the rate go down by 80 percent over the two-year period? No! Suppose the initial rate was 100 homicides for every 100,000 residents. After a 50 percent reduction, the rate is 50 homicides for every 100,000 residents. Another thirty percent reduction means an additional .3 x 50 = 15 percent reduction. The final rate is then 50 - 15=35 homicides for every 100,000 residents. On the other hand, an 80 percent reduction of the original rate is .8 x 100 = 80 fewer deaths for each 100,000 residents, for a final rate of 20 homicides for every 100,000 residents. So be wary of “additional amounts” — you can’t just add the percentages.
Percentages can also mislead us if we use them when comparing diverse groups of people. The press often comes out with headlines such as “ Mississippi has the worst maternal death rate in the country” (note: MI is chosen for illustrative purposes only) or other state-by-state comparisons. The implication is that Mississippi doesn’t have as good a health care system.
Now it also turns out that blacks as a population have a worse maternal death rate than whites in the U.S. And the poor have a worse death rate than the rich. It turns out that Mississippi has a higher rate of blacks, and a higher rate of poor people than other states.
In fact, one can cook up numbers in which Mississippi has a lower maternal death rate for both blacks and for whites than another state, and yet has a higher overall death rate! Suppose that in Mississippi 1 in 10,000 black women die in childbirth, and 1 in 20,000 white women die in childbirth. Suppose in Illinois, 1 in 9,000 black women die in childbirth, and 1 in 18,000 white women. Mississippi has a lower rate for both groups. But if Mississippi has a population which is 50% black, and 50% white, then about 1 in 15,000 women die in childbirth. In contrast, if Illinois is 10 percent black, and 90 percent white, then its death rate will only be .9 in 15,000 deaths in childbirth, a slightly lower rate than in Mississippi.
Lastly, we expose the classic error involved with percentages. Suppose that a murder occurs in a small town of 10,000 people, and DNA evidence is collected. Suppose also that DNA on everyone in the town is on record. Lastly, suppose that DNA identification is 99.9 percent specific. If we found someone in town whose DNA matches that of the perpetrator of the murder, how likely are you as a juror to believe that he was guilty of murder?
If you were clever with your numbers, you would reason that the DNA would identify 1 in 1,000 people, and that means that there are 10 people in a town of 10,000 who have DNA matching that collected at the scene of the crime. A random person accused of the crime whose DNA matches that found on the scene has a one in ten chance of being the actual criminal. Not exactly proof of guilt. This reasoning gives a different perspective on DNA screening of criminals, especially if it is the only evidence used to convict.
It’s worth the time to reason about a statement involving percentages with examples and real numbers. Our intuition is not always our best friend.