The bell-hop’s tale: Avoiding getting tricked by numbers
There is an old mathematical puzzle about three mathematicians and a bell-hop which is a good lesson in how numbers can be used to deceive as well as inform.
Three mathematicians travelling to a conference out of town decide to save money by sharing a room (clearly these mathematicians are academics rather than mathematicians working on exotic products for investment banks).
At the front desk, they pay $300 for the night for their room.
The mathematicians have already headed off to their room when the hotel manager realises that there is a special on that day and he should have only charged $250 for the room.
So he sends the bell-hop after them with five $10 dollar bills in change (they are in the US, the land of bell-hops). On the way up, the bell-hop decides that it will be too hard to split the $50 three ways, so he pockets two of the bills and then gives one to each of the mathematicians.
This means that the mathematicians ended up paying $90 each, for a total of $270 and the bell-hop kept $20, bringing to the total to $290.
So where is the missing $10?
The answer’s at the bottom of the post, but this puzzle is the mathematical equivalent of a magician’s mis-direction, drawing your attention away from where he is palming the card, coin or dove. It isn’t too hard to see through, but it can often catch people out at first. What it shows is that if a simple puzzle like this can use numbers in a misleading way, how much easier is it for people to be misled by the barrage of numbers and statistics that appear in the media every day?
Misleading numbers may simply be the result of arithmetic blunders, as in the case picked up by Media Watch where Channel Nine’s “Money for Jam” concluded that saving $25 dollars each week adds up to $5000 in a year (more than $10 missing there).
More often, statistics are used in a selective, distorted or exaggerated way to help prove a particular point. A good example is in a viral video about social media that many of you may have seen.
The social media blog Socialnomics has a detailed point-by-point exposition of the abuse of statistics in that video. Another example is the report by the UK Department of Energy and Climate Change that UK CO2 emissions have fallen by 12.8% since 1990.
However, as the House of Commons Environmental Audit Committee pointed out, this figure includes carbon credits purchased through the EU emissions trading scheme and the real reduction of emissions is in fact closer to 8.5%.
So, the lesson from the bell-hop is to be aware. Do not take numerical explanations at face value without thinking them through yourself to see if they make sense. Sometimes they will, but often they will not.
Here are a few more places to dig into the use and abuse of numbers:
1. Mathematician and author John Allen Paulos literally wrote the book on Innumeracy. A few years ago, he wrote about misleading numbers in the news, zeroing in on social security, illegal immigration and civilian deaths in Iraq. It’s safe to say that the media has not learned much since then.
2. Misleading statistics seem to crop up in health and medicine more than anywhere else. Unfortunately, statistical illiteracy seems to be rife among medical professionals, not just the general public. For some good ideas on how to avoid mistakes when dealing with medical statistics, the paper Simple tools for understanding risks: from innumeracy to insight is free to download if you complete a registration form with the online BMJ journal.
3. Ben Goldacre’s Bad Science blog is rich source of dodgy science of all varieties and, needless to say, the abuse of statistics is a common theme. A good example is this post, which tears apart some crime statistics from the UK’s Home Office.
4. Graphs can be a powerful way to communicate data, but they can also be used to distort data. More that 50 years ago, Darrell Huff’s book “How to Lie With Statistics” examined many of tricks of the charting trade, but for a more contemporary look at the good, the bad and the ugly of graphs in the media and online, a good place to visit is the blog Junk Charts.
By the way, if you haven’t worked out where the missing $10 went yet, the trick is that you should subtract the bell-hop’s $20 from the $270 (not add it) to get to the $250 cost of the room that the hotel was paid.
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