Media reports this week that speed cameras have failed to reduce accidents and have even led to a rise in some places fail to take random variation into account
"Speed cameras failed to cut accidents on many roads and actually led to a rise in casualties on some routes," claims the The Telegraph.
Can this be true? Have speed cameras "failed" to cut accidents? Have they "led" to a rise in accidents? A little careful consideration reveals that both of these bold claims may be false.
When considering a large number of speed cameras and councils, we would expect, by random chance alone, that there will be some sites where the number of accidents will have increased after the installation of a speed camera. This will happen even if the probabilities of an accident happening have actually gone down.
Any analysis must take randomness into consideration. Figures provided by the councils must also enable people to get the whole picture, which means giving the public information about how variable the data are. For example, the Humberside council data only gives three-year averages for casualties and collisions and no indication of the variability around this value.
Whether or not a particular accident will occur is a mathematically random event – it can't be predicted. The number of accidents that happen each hour or each day or each year is a random number that will depend on all sorts of factors, such as the speed limit on the stretch of road on which a particular camera is placed, how busy the road is and the weather. We cannot predict exactly how many accidents there will be in any given time, but we can estimate it with some degree of certainty.
The key to understanding the speed camera data is an appreciation of this measure of certainty. Statisticians call this number the variance – how spread-out the numbers are. In the case of speed cameras, this is a measure of the variability in the number of accidents. For example, if the number of accidents was about the same every year then this is a small variance and if the number of accidents changes drastically each year then this is a large variance.
Imagine that whether or not there is an accident is like tossing a coin. If this is a fair coin then the probability of a car having an accident would be one half or 50:50. This is the same as saying that it is equally likely that a car will have an accident or will not have an accident.
With this in mind, we can look at the speed camera claims in the Telegraph. These claims are an example of the prosecutor's fallacy, which can arise when making multiple comparisons – such as over years or between councils.
If we were to test 100 fair coins, by tossing each one 10 times, the chance of getting heads will of course be one half. So one would expect that when flipping 100 fair coins 10 times each, to see a particular coin chosen beforehand come up heads 9 or 10 times would be very unlikely.
However, if we are not concerned about which particular coin might behave this way, then seeing some coin come up heads 9 or 10 times would be more likely than not, just by chance alone.
As a result, tossing 100 coins would more likely than not lead us to the mistaken conclusion that at least one coin is not fair. We would come to the entirely false conclusion that some of the coins have a higher probability of landing heads than tails.
If a car accident is equivalent to getting a head with a coin, simply by random chance alone, we would expect to see a rise in the number of accidents at some locations, despite the installation of a speed camera.
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Source: http://www.guardian.co.uk/science/blog/2011/aug/25/speed-cameras-accidents-maths
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