How Numbers Can Easily Mislead

I see a lot of headline-grabbing warnings in the media about how activity x, y or z is a huge danger to a, b or c. They bandy figures around like if you do x you'll be 80% more likely to get this type of cancer, or y increases the risk of a heart attack in men by 70%, or z makes a more than 5 degree temperature more likely by 75%.

When stated like that, activities x, y and z can quite easily cause alarm - and they often do, whether it's how many cups of tea you drink a day, how often you have a fried breakfast, how many cars are on the road or whether you smoke in a house with children - someone has got something to say about it and some stats to throw at it.

But those figures mean very little unless you know the base probabilities to begin with. For example, suppose drinking an extra two cups of tea a day (from, say, 4 to 6 cups) increases the probability of your getting prostate cancer by 40%. On first inspection it may sound advisable to stick to the 4 cups of tea a day.

But suppose drinking 4 cups of tea a day only makes your chance of getting prostate cancer 15% - a 40% increase in probability from 4 cups a day to 6 cups a day only adds another 6% to your chance, which is still only a 21% chance. You may well feel that all those extra cups of tea over your lifetime is worth a 6% increase in probability. However, if you just saw the headline "2 extra cups of tea increases the chance of prostate cancer by a whopping 40%" then taken at face value it may put you off tea for life.

We routinely hear claims of the kind that eating two rashers of bacon a day raises the risk of bowel cancer by 18%. But without a base rate (how common is bowel cancer?) this information is not very useful. As it happens, in the UK, bowel cancer affects six out of 100 people; so a bacon-rich diet would cause one additional case of bowel cancer per 100 people.

Here's another example. Suppose 40 million of the UK population ticks a yes or no box to say whether they trust the media, and then the media tries to clean up its act. Next year there is a repeat poll and the results show an 18% increase in people who now trust the media. At first glance that sounds like it could be quite a lot of people changing their mind - after all, 18% of 40 million is 7.2 million people.

But, of course, that's totally the wrong way to think about it, because we need to know the base rate - that is, the number of people who trusted the media in the first year. Apparently the actual figure is that only 6 out of every 100 people say they trust the media, so an 18% increase the year after is only an extra 1 person in every 100 now trusting the media.

Numbers are misleading - they can shock in large quantities, but that often skews the real picture. When a few years ago Vince Cable projected that our joining the Euro would increase our GDP by a few billion pounds, lots of Liberal Democrats got excited, and many pressed for us to join.

What should have been obvious is that that GDP figure is spread over the entire population of Britain, and doesn't add up to much at an individual level (for example, 4 billion divided by 63 million works out at just over £60 each). Would you want to lose the pound sterling for an extra sixty quid in your pocket? (that was then, of course - now almost every Brit is glad we didn't touch the Euro with a barge pole).

Yet another example. Imagine that there is a new illness discovered, colloquially called 'MXDA', the symptoms of which are swollen hands and occasionally swollen feet (with neither causing the other, and either can occur independently of the other). With MXDA swollen hands occur in 99/100 people diagnosed with it. Swollen feet occur in only 1/100 people diagnosed with it. Consider this question: which of the two following statements is most probable:

A) George contracted MXDA and had swollen feet

B) George contracted MXDA and had swollen feet and swollen hands

The vast majority of people answer B, even though the answer is obviously A. It's fairly evident that even though George has swollen feet, there is still a 1 in 100 chance that he does not have swollen hands, and given that the two probabilities are independent, it is impossible that B is more probable than A. Belief to the contrary is known in philosophy as the 'conjunction fallacy'.

Understanding base rates, probabilities and logical thinking can help us in everyday life decisions too. Take insuring your household products (a subject I once blogged about here) Using similar logic to the above thinking, then in terms of probability when it comes to insurance of household products the cards are stacked in the favour of the insurer not the insured. It should be fairly evident why: insurers must cover the cost of pay outs and the administration costs to sell insurance, so insurance must be a winning hand for the insurer overall (ditto casino owners, bookmakers, amusement arcade owners, and so on - the fact that they are in business at their customers' expense tells us all we need to know).

Should someone who has spent £20,000 on a conservatory spend an additional £30 to insure against it being damaged by the weather? The obvious answer seems like yes, but it depends on the 'base rate' - the odds of the conservatory being damaged by the weather. If the odds of it being damaged are 1000/1 he'll be £10 down on the deal. Maybe it's still worth it, but what about £3,000 insurance against a 1000/1 chance that a £2 million conservatory will be damaged? It's the same as before in terms of odds and ratio, but our man may think an extra £3000 could be better spent elsewhere.

One final point, when it comes to perceived rationality things aren't always as they seem - sometimes it's important to think a bit further outside the box. For example, generally it is thought that people who understand probability won't buy a lottery ticket* because they know the vanishingly small chance of winning the jackpot pretty much makes any lottery ticket purchase tantamount to a waste of money.

That might be true if those gambling odds were all there is to the purchase, but there are other factors that might make a ticket purchase worthwhile, just as there are other reasons why going for a night out at the casino is not necessarily irrational despite the probability being in the casino owner's favour. In buying a lottery ticket you might also be buying the dream and the excitement, which may well be worth the value, particularly if you enjoy the whole TV show that goes with the lottery experience. And in case you're wondering, I don't buy a lottery ticket, I'm not an idiot! Haha! Just kidding!  

* I remember reading about the high number of lottery winners who squander their fortune, some of whom even go bankrupt within a few years. It could be that the paradox of lottery players is that if they are willing to spend a few pounds each week on lottery tickets in the first place they are not likely to be the kind of person who optimises their spending commensurate with their budget. Therefore, one would expect that a high number of lottery winners had proclivities for profligacy. "Proclivities for profligacy" - That's one hell of a statement.