Busted!

Once again, the Mythbusters have shown that they are careless with basic mathematics. They have shown that they can’t reliably calculate percentages, which makes me worry about all of other math that they do. They described dimpled golf balls as going 37% farther, when they actually went 61% farther!

In Episode 127 – Dirty vs. Clean Car they test whether a dirty car will get better mileage than a fast car. As part of this test they measure how much farther a dimpled golf ball will go compared to a smooth golf ball. They describe the results in three different ways:

- The dimpled golf balls appear to go about twice as far
- The dimpled golf balls go about 893 feet compared to about 556 feet for the smooth golf balls (those are either exactly what they said or very close)
- The dimpled golf balls go 37% farther than the smooth golf balls

This series of statements seem suspicious. Twice as far would be 100% farther, which is a long way from the 37% farther that they calculated. The ‘twice as far’ was just an eyeballed estimate, but the calculation of 37% farther seems improbably inconsistent with this estimate.

Looking at the numbers and applying standard percentage calculations we see that the percentage change from non-dimpled (556 feet) to dimpled (893 feet) balls was:

(893 / 556) * 100% – 100% = 60.6%

The multiplication by 100% is needed to change the result from a simple ratio to a percentage, and then we subtract 100% because we want to calculate the percentage increase.

The result of 60.6% shows that their eyeballed estimate was actually reasonably good, but their math was terrible. So, where did the 37% number come from? Let’s try to find out.

Given those same numbers we could reverse the numbers to see the percentage change from dimpled (893 feet) to non-dimpled (556) feet balls:

(556 / 893) * 100% – 100% = –37.7%

It’s not perfect (37.7% should be rounded to 38%) but it looks like this is the calculation that they did. They calculated one thing (the non-dimpled balls travelled about a 37% shorter distance) but said something else (the dimpled balls travelled about 37% farther). The two statements are quite different.

The simplest example of this non-transitivity is if we have two people, one who is three feet tall and the other who is six feet tall. We can say that the tall person is 100% taller, or we can say that the short person is 50% shorter. Different numbers for different statements.

Adam and Jamie need to think more about what they’re saying, or get better fact checkers.

One has to remember that MythBusters is TV show, which means it exists primarily to sell advertising. Adam and Jamie seem like great guys, and I’m sure they want to put out a quality show, but the realities of production and editing turn it into a trainwreck. I’ve rarely found their experiments to be sufficiently rigourous to satisfy my own curiosity. The first few years, they were decent, but I’d say after the 3rd or 4th year things really went downhill.

Perhaps if they cut out the 3 lackeys and their thinly veiled infomercial segments, along with all the repetition and that blasted announcer dumbing things down to caveman language, maybe they’d have enough time to do things properly. I’d rather watch a single myth beaten to death, than the current brain-dead format.

It only keeps getting worse too.

“The multiplication by a hundred is needed to change the result from a simple ratio to a percentage” – No. 1 = 100%. You could multiply by 100%, which is the same as multiplying by 1, to make it clear, but multiplying by 100 is wrong. Same for -100 – it should be -100%, or -1.

Good point — I should be multiplying by 100% (which is equal to one) not 100. This demonstrates correct use of units. I’ll fix the post.

Finding the correct percentage basis gets particularly tricky with time and velocity. For example, if Alice and Bob race and I say that Alice was 50% faster than Bob, that could mean two things:

1. Alice’s speed was 50% higher than Bob’s speed

2. Alice’s race time was 50% shorter than Bob’s race time

I was trying to see whether there is a similar generous way of interpreting the Mythbusters’ numbers, but I don’t see one. So, I do agree that they seem to have dropped the ball on that one.

I disagree that there are two interpretations to your sample statement. If you say that Alice is 50% faster than Bob then that means that Alice’s average speed was 1.5x Bob’s average speed.

If you want to say that Alice’s time was 50% shorter then you need to say “Alice’s race time was 50% shorter”. I see no ambiguity. I agree that some people are careless with this distinction, but the distinction is real.

Non-transitive? Don’t you mean non-symmetric? Or possibly non-commutative?

Either you have fallen victim to Muphry’s law (http://en.wikipedia.org/wiki/Muphry's_law), or I have. 😉

I’m actually not sure what I meant. I think you are correct that non-transitive is not correct. Commutative is probably the best replacement.

I thought you’d made a typo when you wrote Muphry’s law, but luckily I followed the link and read about it before suggesting that. I had not heard of it before but I like it.

I had a personal story about this, a salesman was asked to add 30% to a price… Here’s the strange calculation he did. He figured that 100%-30%=70%, so he uses the original price as 70% and finds what the 100% is:

$retail = ($price/70%)*100%

He was asked to add 30% of the original price, but instead he unknowingly added 30% of the higher retail price…

The difference, 100$ + 100$*0.30 = 130$, versus (100$/70%)*100% = 142$. Not much, but still funny. So technically he added 30% of 142$ instead of 100$.

I think the mistake is so common because people forget that value of a single percent depends on what is considered 100%. It can especially get confusing in speech. In statements like A is faster/slower than B by n%, B is the 100% and A is being compared to it. This is a pretty easy rule to remember: whatever you are comparing to is the 100%.

Here’s a neat and related bit of math, if some A is bigger than B by n%, then by what m% is B smaller than A? I came up with a little formula that inter-converts between those two percentages, it’s actually possible to convert from m to n, and back, without ever knowing what original values of A or B were.

A = (B/100%) * (100%+n%) – find A from B

[B%A]=(B/A)*100% – percentage B of A

m% = 100% – [B%A]

It simplifies to

m = 100n/100+n

For example you could convert 61% to 38%, and -38% to -61%. The minus signs are needed because the statement was originally written “bigger to smaller”. This can be a convenient shortcut for making correct comparative statements with percentages. If my car is faster than some other car by 25%, then the other car is slower by 100*25/100+25, or 20%. If my car is slower than the other car by 25%, then the other car is faster by 100*25/100-25, or 33%. (minus in 100*25 is optional)

That is an obscenely long-winded way to say “reciprocal”. Most people just write it as “1/n” 😉

Uhh, yes i suppose that’s a bit more straightforward, but still it wouldn’t automatically tell you the difference 🙂

Sorry to annoy you, but you won’t believe this, this is hilarious, your way actually simplifies to the same expression. 1/n won’t work, because n is by how much it’s bigger, so it’s (100% + n%), also since it’s %, then 1 becomes 100%: 100%/(100%+n%). Then to convert it back to % from ratio, multiplying by 100 and subtracting everything from 100 to get the difference: 100 – (100/(100+n))*100 = 100n/(100+n)… Tada! I win 🙂

Ha, that’s a neat way of calculating the percentage increase. I would’ve done the following:

893 – 556 = 337

337/556 * 100 = 60,6 %

But I like your way better! 😀

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