Percentages Level the Playing Field

In a recent article in the NY Times on how mistakes made by doctors lead to harm for patients, this sentence conveyed the statistics: “Of the 651 hospital patients studied, 309 had errors on their medication lists, and over half those errors had the potential to cause harm”
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By itself, the sentence seems to convey the gravity of the situation. Yet the average reader is unable to gain a foothold with the numbers. What if more (or less) patients were studied, what would the number of mistakes be then? And how would you compare different studies over time to gauge whether the situation was improving or not?

What comes to the rescue is converting the numbers to a proportion or a percentage. If x out of n meets a certain criterion, then x/n is the proportion of those who met the criterion. If you multiply this proportion by 100, you convert it to a percentage. The power of percentage derives from the fact that it is out of 100. “Per Cent” simply means “Out of 100.” Once we reduce two numbers (one out of the other) to their equivalent percentage, we level the playing field because each number is calculated out of the same number, that is, 100.

The more important question is, can we calculate the percent to an approximation in our head?

Yes, we can.

Consider the two numbers above. Out of 651 patients, 309 were given the wrong medication in hospitals. If we approximate 651 to 600 and 309 to 300, we see that 300 is half of 600, which is 50%. In other words, we can say right away that approximately 50% were given the wrong medication, and half of them, that is, 25%, ran the risk of serious harm done to them.

When the numbers are reduced to percentages, it is easy to get a sense of what is going on, compared to absolute numbers. From there, it is a relatively straightforward matter to approximate whether the percentages have been under or overestimated. Because 651 was reduced to 600, and because it is doing the dividing, (remember, the larger the denominator, the smaller the proportion and the smaller the denominator, the larger the proportion) we overestimated the error. So instead of 50%, we may guess that the number is something like 45%. (Actual percentage of 309/600 comes to 47%).

When numbers are calculated out of 100, we have a way of comparing different numbers out of different totals, in addition to gaining a better understanding with the two numbers themselves. The mathematical playing field is leveled, because all comparisons are out of 100. What’s exciting is that you can calculate that percentage in your head to a good approximation.
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When are proportions or percentages not a good idea? When the sample size (the number in the denominator n) is small, the variation in what you are looking to match (the numerator x) is larger, which distorts the comparison. But that should not keep us from appreciating the power of percentages to level the playing field in many instances.