The Art of Statistics: How to Learn from Data

by David Spiegelhalter

Suppose I have a coin, and I ask you for your probability that it will come up heads. You happily answer ‘50:50’, or similar. Then I flip it, cover up the result before either of us sees it, and again ask for your probability that it is heads. If you are typical of my experience, you may, after a pause, rather grudgingly say ‘50:50’. Then I take a quick look at the coin, without showing you, and repeat the question. Again, if you are like most people, you eventually mumble ‘50:50’. This simple exercise reveals a major distinction between two types of uncertainty: what is known as aleatory uncertainty before I flip the coin – the ‘chance’ of an unpredictable event – and epistemic uncertainty after I flip the coin – an expression of our personal ignorance about an event that is fixed but unknown.

Fortunately, we saw in the last chapter that the number of homicides each day act as if they are random observations drawn with a Poisson distribution from a metaphorical population of alternative possible histories. This in turn means the total over the whole year can be considered as a single observation from a Poisson distribution with mean m equal to the (rather hypothetical) ‘true’ underlying annual rate. Our interest is whether m changes from year to year. The standard deviation of this Poisson distribution is the square root of m,

written , which is also the standard error of our estimate. This would allow us to create a confidence interval, if only we knew m. But we don’t (that’s the whole point of the exercise). Consider the 2014–2015 period, when there were 497 homicides, which is our estimate for the underlying rate m that year. We can use this estimate for m to estimate the standard error as = 22.3. This gives a margin of error of ± 1.96 × 22.3 = ± 43.7. So we can finally get to our approximate 95% interval for m as 497 ± 43.7 = 453.3 to 540.7. Since 95% confidence intervals are often assumed to be plus or minus 1.96 standard errors, this means that we can be 95% ‘confident’ that during this period the true underlying homicide rate lies between 453 and 541 per year.