Professor Edward Hinds FRS is chair of the Equality and Diversity Network.

As Chair of the Royal Society’s Equality and Diversity Network (EDAN), I’m interested in measuring whether the Royal Society treats people fairly. If there is a bias, we should know about it so that we can fix it.  A good place to start is the election of Fellows of the Royal Society. Each year 44 new Fellows are elected, typically including four women. This year, only two women were elected, which has understandably caused some alarm. This is a point worth thinking about.

Let me propose the following ideal model for the purpose of this short discussion. The Sectional Committees that elect Fellows have a pool of candidates to consider, of which roughly 9% are women.  They compare the candidates on the basis of scientific merit alone, disregarding whether they are male or female. This is what they are supposed to be doing so the model is a reasonable starting point. In this model, four out of the 44 Fellows elected each year are women, that being 9%. Of course, we do not expect exactly four women every year because the sex of the fellows is not taken into account when they are elected, so it is random. In fact there is only a 1-in-5 chance of choosing four women, so on average it will happen 2.4 times over 12 years – that’s what the highest blue dot shows in the figure. The blue dots on either side of this show that three and five are almost as likely as four. Even two and six are likely to happen about one and a half times each out of twelve years.

You can see this for yourself if you take a pack of 44 cards including the four aces, so that the chance of picking an ace is 9%. Shuffle the cards and look at the top one – that’s one Fellow elected. Put it back (to maintain the 9% of aces) and repeat – that’s the next Fellow.  After doing this 44 times, the number of aces you chose represents the number of women you elected.  Most probably it was not 4. Please don’t think that the election of Fellows is as random as shuffling the cards! It is a very considered process, taking many relevant things into account. But (in my model at least) the sex of the candidate is as random as the card selection and the number of women elected has the same distribution as that of the aces chosen from the pack of cards. If you are statistically inclined you can calculate this using the binomial distribution, which gives the blue dots in the figure, but the cards are more fun.

WomenElected

Number of times (frequency) a given number of women are elected over twelve elections. Blue dots: Average frequency. Grey area: region where 90% of actual frequencies should lie. Red dots: actual frequency of numbers elected over the last 12 years.

Correct operation of the Sectional Committees, as proposed in my model, naturally produces year-to-year variations in the number of female Fellows elected. The interesting point is that those variations are entirely consistent with what has actually happened over the last twelve years, shown by the red dots in the figure. The highest red dot represents the election of five female fellows, which has happened four times. The next highest are at two and four, both of which have been elected three times. Three and nine have each occurred once. There is noise in the red data, which is inevitable. The shaded area indicates the range of variation that we should expect 90% of the red points to occupy. And that is exactly what we see: ten of the eleven points lie within this area while one lies just outside. There is nothing in the data to suggest the model is wrong (for the experts the reduced χ2 is 1.1). So I think the more pertinent question is not ‘why were only two women elected this year?’ but rather ‘why is the average fraction of women only 9%?’. The evidence suggests that the elections are fair but we need to think further about the nomination. I will write about this new question soon.

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