## Women on S&P500 boards of directors

This is a co-post with FogOfWar.

Here’s an interesting article about how many board of directors for S&P500 companies consist entirely of men. Turns out it’s 47. Well, but we’d expect there to be *some* number of boards (out of 500) which consist entirely of men even if half of the overall set of board members are women. So the natural question arises, what is the most likely actual proportion of women given this number 47 out of 500?

In fact we know that many people are on multiple boards but for the sake of this discussion let’s assume that there’s a line of board seekers standing outside waiting to get in, and that we will randomly distribute them to boards as they walk inside, and we are wondering how many of them are women given that we end up with 47 all-men boards out of 500. Also, let’s assume there are 8 slots per board, which is of course a guess but we can see how robust that guess is by changing it at the end.

By the way, I can think of two arguments as to why the simplification that nobody is on multiple boards argument might skew the results. On the one hand, we all know it’s an old boys network so there are a bunch of connections that a few select men enjoy which puts them on a bunch of boards, which probably means the average number of boards that a man is on, who is on at least one board, is pretty large. On the other hand, it’s also well-known that, in order to seem like you’re diverse and modern, companies are trying to get at least a token woman on their board, and for some reason consider the task of finding a qualified woman really difficult. Thus I imagine it’s quite likely that once a woman has been invited to be on a board, and she’s magically dubbed “qualified,” then approximately 200 other boards will immediately invite that same woman to be on their board (“Oh my god, they’ve actually found a qualified woman!”). In other words I imagine that the average number of boards a given *woman* is on, assuming she’s on one, is probably even higher than for men, so our simplifying assumptions will in the end be *overestimating the number of women on boards.* But this is just a guess.

Now that I’ve written that argument down, I realize another reason our calculation below will be overestimating women is this concept of tokenism- once a board has one woman they may think their job is done, so to speak, in the diversity department. I’m wishing I could really get my hands on the sizes and composition of each board and see how many of them have exactly one woman (and compare that to what you’d expect with random placement). This could potentially prove (in the sense of providing statistically significant evidence for) a culture of tokenism. If anyone reading this knows how to get their hands on that data, please write!

Now to the calculation. Assuming, once more, that each board member is on exactly one board and that there are 8 people (randomly distributed) per board, what is the most likely percentage of overall women given that we are seeing 47 all-male boards out of 500? This boils down to a biased coin problem (with the two sides labeld “F” and “M” for female and male) where we are looking for the bias. For each board we flip the coin 8 times and see how many “F”s we get and how many “M”s we get and that gives us our board.

First, what would the expected number of all-male boards be if the coin is unbiased? Since expectation is additive and we are modeling the boards as independent, we just need to figure out the probability that *one* board is all-male and multiply by 500. But for an unbiased coin that boils down to (1/2)^8 = 0.39%, so after multiplying by 500 we get 1.95, in other words we’d expect 2 all-male boards. So the numbers are definitely telling us that we should not be expecting 50% women. What is the most likely number of women then? In this case we work backwards: we know the answer is 47, so divide that by 500 to get 0.094, and now find the probability p of the biased coin landing on F so that all-maleness has probability 0.094. This is another way of saying that (1-p)^8 = 0.094, or that 1-p is 0.744, the eighth root of 0.094. So our best guess is p = 25.6%. Here’s a table with other numbers depending on the assumed size of the boards:

If anyone reading this has a good sense of the distribution of the size of boards for the S&P500, please write or comment, so I can improve our estimates.

For public companies, this information is public and easy to get (if you are patient or hire an undergrad to do it). E.g. IBM has 2 women out of 12, Google has 2 out of 9, Apple has 1 out of 7, and those 5 women are distinct. Just google “IBM board of directors”, etc.

This link claims the average board size for S&P 500 companies in 2010 was 10.7. (See the last citation.)

http://blogs.law.harvard.edu/corpgov/2011/04/10/the-shifting-landscape-of-corporate-governance/

But didn’t the article you link already answer your basic question?

“. . . the number of women on corporate boards of companies on the S&P 500 dropped to 16 percent from 16.6 percent, according to Bloomberg.”

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Awesome that you gave me those numbers and EVEN read the article all the way through!! I’d still love to get all those numbers because it would be interesting to understand these numbers more thoroughly… and since the answer is 16, I’m wondering if it’s because of the size of the boards we have wrong or because of the undercounting of women because of being on multiple boards or because of the tokenism issue.

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Here is a possible argument for tokenism. Given that 16% of board members are women (according to that article), the chance that a 10.7 person board would be all male is (0.84)^10.7 = 15.5%. This leads to an expectation of 77 out of 500 boards being all male.

The assumption in this calculation is just that each person on each board is chosen independently, with a 16% chance of choosing a female each time. If you look at the statistic “number of females per board” you should get a binomial distribution, with mean around (0.16)(10.7)=1.7 and with (once again) the value of 0 occurring about 15.5% of the time.

Since the value 0 actually occurs much less than that (only 47 out of 500) this suggests that the assumption is wrong: there is a bias against having 0 female board members. It also follows that there will be fewer boards with lots of women than you would expect from the Bernoulli trials.

(In other words, of the (500)(10.7)=5350 board members counted with multiplicity, 16% or 856 are female. The way they are distributed appears to be more spread out than if it were random, perhaps due to tokenism. However, accounting for multiplicities would skew the results in this same direction, so one should really factor that out first.)

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I guess it’s also worth pointing out that, although of course there’s no such thing as a board with 10.7 people on it, the inequality that you get from averaging works in favor your argument. For instance, if half of all boards had 8.7 people, and the other half had 12.7, then under the independence assumption the rate of all-male boards would go up to 16.4%. (I suppose the right thing to say is that 0.84^x is concave-up.)

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Incidentally, is it obvious (or even suggested) that tokenism is “bad”?

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First, Cathy gives too much credit with “co-post”–I deserve at most 25.6% for seeing the article, thinking “Hm, that’s an interesting data point, but what’s the implied probability?”, spending 5 minutes fucking around on a spreadsheet and then shooting it over to Cathy who actually wrote it up in a coherent blog post!

Second, we could debate whether tokenism is bad or good, but a more mathy question is how to evaluate and quantify if tokenism is occurring. I like the above analysis–better than my original thought on the subject which was to determine the conditional probability of a F board member in the next 3 selections after a F is selected for a board. If it reverts to below the overall F implied probability, that might be evidence for tokenism occurring.

Third, I can think of another way networked boards (once you’re on one board, you’re invited to be on many) would skew the results: it decreases the overall number of coin flips, and thus the expected results. For an extreme example: what if there were only 15 people in the “board universe”, and each board picked 10 of those 15 people. How many boards would one expect to be all F in this case (depending upon the composition of the initial 15)? My intuition is the smaller the universe of “networked” board candidates, the higher the probability of unusual results (law of small numbers). I suspect anecdotally that if you tracked individuals on multiple boards, you’d find that women who “break in” serve on as many boards as men (possibly more due to tokenism effects).

Lastly, we haven’t looked at all at what the expected P(F) should be–i.e., the percentage of candidates who are qualified and interested in the role who are female (lots of people are smart, but not a lot of people are willing to work endless hours for decades in corporate america striving to reach upper management and often ignoring their children in the process). Self-selection is a very significant factor for a whole host of reasons, and is also a very tricky variable to put to data.

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