## Grayton support down by 19%

On your feet. No hesitating. This will count toward your overall Language Log grade: Take a glance at the latest xkcd cartoon and tell me Senator Grayton's present approval rating.

The sort of person Language Log doesn't want you to be will see neither the mathematical issue here nor the linguistic one.

If you are like about 78.3% of the population (that's a guesstimate; it could be off either way by, oh, 19%) you will say "One percent".

What the xkcd cartoonist is worried about is that those people could be regarded as correct if "plunge by 19%" is taken to mean "decrease by 19 percentage points from N/100 to (N–19)/100"; but you can't be sure what the voice on the TV meant, because the level of carefulness in media talk about numbers is scandalously below where it should be.

The mathematically informed but linguistically unobservant psephologist (say, an election specialist with a good head for figures who never took a linguistic course and doesn't read Language Log) will know instinctively that a drop from 20% to 1% overnight is implausible (that would mean that 95% of the people who liked him have changed their minds, which would be an implausibly cataclysmic plunge), and, using common sense and fast mental arithmetic, will say, "Obvious: Grayton is at 16.2% now; what's the problem?"

You get the latter figure by assuming that only 19 out of each 100 people who supported Grayton have changed their minds, so we need to subtract 19% of the 20% figure that we start with. So: 19% of 20 is 3.8, and 20 minus 3.8 is 16.2, giving us the present percentage approving of Grayton, provided the TV voice did not mean "his support has since plunged by 19 percentage points from 20% all the way down to 1%", but instead meant "his support has since plunged from 20% to a figure that is only 81% of that, i.e., from 20/100 to the new value of 20/100 – ((19/100)×(20/100))." It probably did mean the latter, but didn't know how to say it in an intelligible way.

It's important that there are intelligible ways. Why make the reader do multiplication of fractions to learn where Grayton stands in the polls? Why not just say "his support has since plunged from 20% to 16.2%"? The mistake is in trying to use language about the fraction by which the support has fallen when language about the current level of support would be far more intelligible. You really can visualize the difference between (1) a crowd of a thousand people in which 200 are Grayton supporters and (2) the same crowd but with only 162 Graytonites (though you might not describe the difference as a "plunge"). You can sort of see the difference in your mind's eye.

But now try visualizing a crowd in which 19 out of each 100 of a Grayton faction that consists of one-fifth of the people present raise their hands. Harder, isn't it? And now try visualizing the difference between the size of the remaining subpart of that 20 percent (the ones in the Grayton faction who aren't raising their hands), and the size of the entire original faction. Virtually impossible. The TV news has gone for exactly the wrong way of stating the numbers: in terms of the difference between the two support levels expressed as a fraction of the old one.

I hope you got all this, because it is just the sort of thing that will come up on the Christmas Quiz. I hope you didn't think every time you saw "Linguistics in the comics" under a Language Log post you would get some silly word joke or some tantrum from Alice in the Dilbert strip or something, and your brain would have a minute of free time to coast. Think again. Think all the time. You are a Language Log reader.

[Comments are off; use the time saved for thinking. One of the things I'm thinking about is why I managed to spell "Grayton" as "Grayson" in this post yesterday. Presumably frequency: I knew the name Grayson from other sources, but not the name Grayton. And — technical detail — in the WordPress software that we use, while you're typing the text you cannot see the graphics.]