# Proper and improper use of percent changes or differences

## 2016/04/01

I’m often quite confused by the meanings of percentages when used to describe changes or differences. Maybe everyone thinks it’s obvious what is meant, but I think that means many of us are saying different things.

I assert: percent changes are unambiguous when

1. they** are between 1% and 99% **and

2. they refer to changes (or differences) in quantities that are themselves not percents.

This implies that there are two problems that can arise when reporting percent changes or differences. First, there are two ways to interpret a percent change outside the 1% to 99% range. A percent inside this range is easy to understand: if I say the old price was $10 and it increased by 50%, that means that new cost is$15 (i.e., 1.5 times the old price). I think we all agree on that.

How about percents between 100% and 200%? If I say the price increased 150%, what does that mean? I think many people would take that to mean that the price is what is used to be plus another 150% of the original, making a total of 2.5 times the original value: a 150% increase from $10 would be$25. I worry, though, that some people say, “increased by 150%” to mean “increased to 150% of the original”. In the other way of talking, this is an increase by (or “of”) 50% increase from $10 to$15.

Percentages over 200% are especially odd, but they appear to be popular. I suppose people like them because they sound shocking and enormous, since we’re used to hearing about percent changes between 1% and 99%. The fact that percents like this sound shocking should be a sign that they are confusing and unexpected!

For example, if the price of something “increased by 200%”, I’m not sure if the price went up to 200% of the original or it increased by 200% of the original: is the new price twice the original (“to to 200%”) or is it three times the original (“up by 200%”)? Amusingly, this is a difference of 100%. If a price went from $10 to$20, you can say it increased by 100% or it increased to 200% of its original price. This is a mess: you should just say the “times” difference. The new price is two times the old price. That’s easy for everyone to understand.

For particularly large percentages, this distinction between by and to doesn’t really make a difference, but it doesn’t make the huge percentages any less obfuscatory. For example, news sources said Martin Shrekli raised the price of a drug by 5,000% or 5,500%. The price of a pill had increased from $13.50 to$750, so the fold increase was $= 55.56$. Multiply that by 100, add a percent sign, and you get 5,556%. I suppose that’s an awkward number, so writers rounded it down to 5,000% or 5,500%. At this point, it’s irrelevant whether the drug increased by 5,500% of its original price or increased to 5,500% of its original price. The difference between those two ideas is 100%, and the reported percentages are rounded by 500% or 1,000% anyway. This is still weird and confusing, and it’s better to say the drug’s price increased to 55 times its original. Everyone knows what they means.

The second pitfall of communicating percent changes is that there are two ways to interpret percent changes in quantities that are themselves percents (or “rates”). For example, you might hear that the national unemployment rate decreased 0.8% between January 2015 and January 2016. If I told you the 2015 rate was 5.7%, you would probably compute the 2016 rate by subtraction: $5.7% - 0.8% = 4.9%$. But what if I told you the unemployment rate decreased 14%? Then you would probably compute the change by multiplication: $5.7% (1.0 - 0.14) = 0.8%$. So did the unemployment rate decrease by 0.8% or by 14%? It seems crazy to me that I can say two different numbers using the same language and you decide to do two different mathematical operations!

In the case of unemployment rates, we were able to choose which operation to do because we had good contextual information. If you knew the unemployment rate was less than 14%, and I told you the change was 14%, then you know it’s crazy to subtract. You would have to multiply. You might also have some sense that the unemployment rate is going down more than by a teeny little bit, so if I told you the change was 0.8%, you would know that I mean subtraction, since it’s not that the rate changed by only eight parts in a hundred.

What if you don’t have good contextual knowledge about the percents? What if I told you that the percent of Asian-Americans increased 1.1% between 2000 and 2010? In 2000, 3.8% of Americans were Asian-American. Does that mean you should add percentages ($3.8% + 1.1% = 4.9%$; i.e., the fraction of Americans that are Asian grew by about a third) or multiply (i.e., the fraction of Americans that are Asian basically didn’t change; $3.8% 1.011 3.8%$)?

Because it requires contextual information to know whether a percent change in a percent means adding or multiplying, you should say the percent changed from X% to Y%. If the percent increased a lot, say from 3% to 15%, you could say it “increased five-fold”. That being said, it’s about the same number of words and more informative to say “from 3% to 15%”.

In summary:

• For big increases or decreases in countable things like dollars, write the change using “folds”, “times”, or fractions. (e.g., The price of gas increased three-fold this quarter or The number of arrests in 1990 was five times the number it is now)

• For small increases (between 1% and 99%) in countable things, use a percent with the word “by”. (e.g., The price of corn increased by 25% or The number of convictions decreased by 20%)

• For changes in percents, fractions, or rates, say the old and new value. (e.g., The fraction of Americans that are under age 25 decreased from 25% to 22% or The unemployment rate decreased from 10% to 8%)

Happy numbering!