## Percentage vs Percentage Point: A Primer

If your (ordinary least squares) regression coefficient is .047, that is an increase of .047 points, or an increase of 4.7 percentage points.  When X goes up by 1, Y goes up by 4.7 percentage points (or 4.7 ppt for short).

It is not an increase of 4.7%.

To determine what percent change it is, you need to start with a base or an average. If, for example, the mean of the Y variable is .47, then an increase of .047 would be: .047/.47*100 = an increase of 10% off the mean.

Note that 10% is not the same as 4.7%.

Percentage vs. percentage point is a way that people lie with statistics.  A small percentage point change can look large in percentage terms and a large percent change can look small in percentage point terms.  Most people don’t know the difference, and think both mean percent.

*disclaimer:  if both your X and Y variables are in natural logs then, because of the beauty of Taylor approximations, the regression coefficient can be read as a percent with certain assumptions about the size of the change etc.

### 16 Responses to “Percentage vs Percentage Point: A Primer”

1. bogart Says:

Thank you. It’s OK to print this and post on my office door … right?

While we’re at it, just because doing (or experiencing, or having) X doubles the risk that you will experience problem Y, that doesn’t necessarily mean that you should be alarmed that you’ve been doing (etc.) X every single day since you were 13. A good starting point to assess the appropriate level of concern / need for change (if feasible) would be to find out what the average risk of having problem Y is. If it is 1/10,000, then unless Y is really, truly awful (and maybe even then, if X is, you know, fun, or has benefits besides its impact on Y), you may decide to continue doing X.

• nicoleandmaggie Says:

YES.

Doubling your risk of death from .000000000005 to .000000000010 isn’t actually changing your risk of death all that much. It’s one of those “and yet we get out of bed in the morning most days anyway.”

Generally what we want to do in those kinds of situations is compare to other risks that we take daily. Is this more or less risky than crossing the street, getting in a car, getting in a plane etc.

Additionally there’s the “what’s the cost to avoid X”… there’s not much risk to eating unheated deli meat as a pregnant lady, but when the risk hits it’s terrible, and it’s pretty easy to microwave a sandwich. So one avoids eating unheated deli meat while pregnant.

• bogart Says:

Right. Although OTOH the comparison is (strictly) legit only to the extent that the risks are substitutable rather than additive. I.e. claiming that “riding a horse” is no more dangerous than “driving to the barn” is clearly faulty, since (presumably) doing the former requires also doing the latter. Obviously that’s an example unusual in the linking of the two risks, but more generally, while useful as a way of wrapping one’s head around “how much risk,” I do then try to consider whether I do, in fact, want to add (any) risk in a particular realm or, you know, not. As you note, the cost/benefit calculation can be an important (decisive) one.

• nicoleandmaggie Says:

If you stay in bed, you may get hit by an asteroid. It happens.

• bogart Says:

Ooh! Tangential conviction: We (modern, “Western” humans) greatly underestimate and underappreciate irony’s role as a causal force.

• bardiac Says:

And if you stay in bed, you’ll get bedsores and sick and die anyway. Going out sounds WAY more fun!

Augh, thank you! Just the other day I was plodding through a client’s grant application in which one of the contributors blithely used “percentage” and “percentage point” interchangeably. {sigh} The authors are higher ed administrators…could this explain something about the environment in which we work?

3. But alarming people with meaningless statistics is so much fun! People hear “an 80% increase risk in getting a certain kind of cancer” as the same thing as “an 80% risk of getting cancer.” Even though the former may just mean raising your chances of 1 in 100,000 to 1.8 in 100,000. But that makes a much less cool headline.
Perhaps a post on Type II errors someday?

• nicoleandmaggie Says:

We actually have one in the context of Moral Hazard (e.g. why even though your relative doesn’t deserve to be on disability doesn’t mean that we should get rid of the entire program). It goes into a bit more depth about targeting, elasticity of demand, and how kids don’t get to choose.

http://nicoleandmaggie.wordpress.com/2012/01/30/moral-hazard-or-why-ron-paul-says-its-ok-to-feed-kids/

• Oh, I think it deserves its own post. As for the moral hazard one, I see you’re hinting at the idea that anecdotes aren’t data too… In a world of 7 billion people, you can find anecdotes of anything. Another post!

• nicoleandmaggie Says:

Perhaps we should have separated that one into individual components. But you still get idiots who Just Don’t Get It, and who don’t want to get it, who also apparently hate kids (well, hate poor kids) in the comments and they’re really tiresome. Still, easier to have a reasoned argument with them during the summer than the school year. During the school year I always feel like, “Man, I get students willing to pay large amounts of money to learn from me, I don’t have to give it away for free from someone who doesn’t actually want to learn anything.”

4. I greatly enjoy thinking about how poorly people do with assessing and mitigating relative risk. One very interesting thing I found–although I haven’t found an original supported source–was an estimate that more extra people died in car crashes in the aftermath of 9/11 because they fearfully decided to drive instead of fly than died in the 9/11 attacks themselves.

5. James P. Scanlan Says:

I maintain a web page discussing the way the even in scientific journals researchers use percent (or percentage) interchangeably with percentage points:

http://www.jpscanlan.com/vignettes/percentgepoints.html

Other pages addressing similar or related issues include:
The following page discusses the predominance in scientific journals of describing a rate, for example, of 3% as “three times higher” than a rate of 1% rather than as “three times as high” even though the former usage technically means “four times as high.” The pages also discusses that virtually all definitions of “multiplication” are incorrect and, if followed, would cause one to find that 3 times 3 is 12: http://www.jpscanlan.com/vignettes/timeshigherissues.html
The following page discusses that way that researchers, especially in the discussion of racial disparities in cancer outcomes, refer to relative differences in survival and relative differences in mortality interchangeably, often stating that they are examining one while examining the other. They do so without recognizing that the two relative differences tend to change in opposite direction as cancer survival generally changes:

http://jpscanlan.com/mortalityandsurvival2.html

The following page discusses a situation where national magazines confused the proportions blacks and white comprised of persons in college with the proportions of each group attending college, causing the size of the racial difference in college attendance to seem vastly larger than it was:

http://jpscanlan.com/vignettes/journalistsstatistics.html

The following page discusses that, even when researcher describe measures of differences between rates accurately, much of what they say is unsound because of the failure to recognize the ways that the measure employed tends to be systematically affected by the prevalence of an outcome:

http://jpscanlan.com/scanlansrule.html