27 January 2011

## Do the math

*Posted by Callan Bentley*

A video, “The Dreaded Stairs,” has been getting some circulation lately on Facebook. It shows what happens when a staircase (beside an escalator) gets a makeover which features piano-style keys which make sounds when they are trod upon. Watch the video if you would like; I’m going to focus on the accompanying blurb, which reads:

There is a set of stairs, with a moving escalator next to it …. both of which lead to the same spot on the floor of the upper level. At first no one took the stairs, almost 97% of the people took the escalator. Okay. I think that could be a normal expected result.

Then a group of engineers got together, and decided they wanted to change the percentage around.

Notice what these scientists did. Clever huh. And now they have reversed the percentages, as a whopping 66% more people take the stairs, than ride the escalator.

Now that sounds pretty impressive at first glance — this clever (and I’m sure expensive) treatment resulted in the before/after numbers going from ‘97% escalator’ to ‘66% stairs.’ But wait a second… are those two numbers actually comparable? Is that really what’s being presented? Or only what’s being implied?

If 97% of pre-piano-treatment people took the escalator, that means:

100% – 97% = 3%

3% didn’t take the escalator (and presumably took the stairs instead).

The second number is not ‘66% of people took the stairs’ post-treatment. That would be direct and unambiguous, and wouldn’t have inspired me to chew into this with a blog post. Instead, it says that “66% more.” The key word here is “more,” and how we choose to interpret it. One possibility is that “more” refers to the pre-treatment state of affairs. This was the first meaning of “more” to jump to my mind — a comparison of post-piano-treatment to pre-treatment. If this interpretation is the correct one, it means ‘66% more than the original 3%.’

3% x 1.66 = 4.98%

If this interpretation of the “more” in “a whopping 66% more” is correct, then the video blurb is intentionally misleading. Going from 3% to 4.98% is not particularly noteworthy, especially if no accompanying statistical treatment shows that it is in fact a significant increase from the pre-treatment state. A variation of less than 2% may well be statistically significant, but no evidence is provided for that assertion.

A second, more charitable interpretation of “66% more” might be that the post-treatment situation has a new *status quo* wherein x% of people take the escalator, and y% of people take the stairs. In other words, “more” might refer to the present ‘stairs’ *versus *‘escalator,’ and not to ‘before’ *versus* ‘after.’

So let’s try that interpretation out instead. According to the blurb, if x is the percentage of escalator people and y is the percentage of stairs people,

y = x + 0.66x

y = 1.66x

Because the stairs and the escalator are the only two options,

x + y = 100%

Therefore,

x + 1.66x = 100%

2.66x = 100%

(2.66x)/2.66 = (100%)/2.66

x = 37.59%

And, since y = 1.66x, then y = 62.39%

So, if this second interpretation of “more” is the one intended by the authors of the video blurb, then the actual increase in stairs usage is from 3% before the piano treatment to 62.39% after the piano treatment.

62.39% / 3% = 20.8

If the second interpretation of “more” is correct, the treatment resulted in almost** 21 times** the pre-treatment number of people who take the stairs! It represents **2100%** of the original number! Another way of saying that is an increase of 2000% beyond the original. That’s a great number regardless of the particulars of how it is expressed, and worth crowing about. One wonders why the written blurb accompanying the video doesn’t tout this number instead of the relatively weak phrase “a whopping 60% more.”

The fact the the blurb authors opted for “60% more” instead of “21 times more” leaves me inclined to think that my first interpretation of the word “more” was correct. Which means of course, that compared to the alternative, “more” means less (than it is implied to mean).

A bit of selective video editing, and a mathematically unreflective population of video viewers, and you’re left with the public perception of an apparent masterwork of engineering achievement. Soon all our stairs will be coated with piano keys …and perhaps 2% more of us will take them.

Funny post. I agree that people seem to get in a muddle with percentages, and I think English was the second language of these guys (are they Swedish?).

Anyway, I think there’s another interpretation. People often say x percent more, when they mean x percentage points more. In this case that would be a final proportion of 3% + 66% = 69%. People often do this with, say, rock porosity. ‘We start with 15% porosity then increase that by 10%.’ Better to say ‘percentage points’ if that’s what’s meant. (Or ‘porosity units’ in this case).

By the way, the video’s not really new… I think it’s been doing the rounds since October 2009.

Thanks Matt — that’s an excellent point (about percentage points) that had not occurred to me.

I’ll nix the “new” in the introductory sentence too. Thanks for that.

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