The Art of Statistics: Learning from Data

by David Spiegelhalter

This tail-area is known as a P-value, one of the most prominent concepts in statistics as practised today, and which therefore deserves a formal definition in the text: A P-value is the probability of getting a result at least as extreme as we did, if the null hypothesis (and all other modelling assumptions) were really true. The issue, of course, is what do we mean by ‘extreme’? Our current P-value of 0.45 is one-tailed, since it only measures how likely it is that we would have observed such an extreme value in favour of females, were the null hypothesis really true. This P-value corresponds to what is known as a one-sided test. But an observed proportion in favour of males would also have led us to suspect the null hypothesis did not hold. We should therefore also calculate the chance of getting an observed difference of at least 7%, in either direction. This is known as a two-tailed P-value, corresponding to a two-sided test. This total tail area turns out to be 0.89, and since this value is near one it indicates that the observed value is near the centre of the null distribution. Of course this could be seen immediately from Figure 10.2, but such graphs will not always be available and we need a number to formally summarize the extremeness of our data. Arbuthnot provided the first recorded example of this process: under the null hypothesis that boys and girls were equally likely to be born, the probability of boys exceeding girls in all 82 years was 1/282. This only d...

P value, single tail and double tail