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A statistical look at a 50% quota

Even if you support the ideology, you would have to believe in some crazy statistical assumptions to back a 50% gender quota (or target).

This is not a post about the merits of affirmative action in principle. People argue that plenty; I’ve little to add. It’s in basic statistics that Labour’s proposal to target “at least 50%” female representation loses any sense very quickly.

To see why, let’s start with a few basic assumptions. Some over-simplify things a bit, but they’re all charitable to quotas in principle.

  1. Someone’s competence and their gender are independent events.
    By this I mean that if you’re told someone’s gender, you learn nothing about their competence, even in statistical generalities.* Conversely, if I told you that someone was “great”, “useless” or “above average”, you’d have no better chance of guessing their gender than a coin.
  2. People are not all equally competent.
    This should be pretty uncontroversial. Note that this is a comment on individuals, not groups.
  3. The statistical distributions of competency are the same for men and women.
    This follows directly from (1). Note that this doesn’t just say on average. It’s the entire distribution, so it applies among the highest and lowest echelons of a(n infinitely large) population too. To be more precise, we say that the competences of men and women are independent and identically distributed.
  4. Exactly 50% of the population is female, and exactly 50% is male.
    This isn’t strictly true (females slightly outnumber males and we ignore transsexuals) but normally in these discussions we assume that it’s close enough.

Assume also (for the sake of demonstration) that our objective is always to choose the most competent people. Our motivation for instituting a quota, of course, is that we think there are other barriers to our achieving our competence-based objective.

From my list of four assumptions above, the following statement does logically follow:

  • If every year (or election), you choose the best N people for something, the long-run average of the proportion of everyone chosen that are male and female will both tend toward 50%.

But in order to support a 50% quota, or even just a target, for every single election cycle, you would also have to believe this statement:

  • If every year (or election), you choose the best N people for something, then the proportion of men and women in each group will be 50%, allowing for rounding.

Picking it apart
How true would this be? Remember our first assumption: gender and competence are independent. This means that if you choose the best N people, as far as gender is concerned that is as good as picking N people at random.

So let’s start with a simple case, N = 2. You might remember doing this at school, except with parents having two children. There are four combinations: two men, two women, a man then a woman, and a woman then a man (the order matters). That gives a 2/4, or 50% chance, of there being exactly 50%.

As you increase N, the likelihood of the split being exactly 50% decreases. For four people it’s 6/16 or 37.5%; for ten it’s 24.6%; for 34 (the size of Labour’s caucus), it’s 13.6%.

This might seem counter-intuitive, given that I also just said that the long-run average would tend towards 50%. Surely this means that as N gets bigger and bigger, the probability of it hitting 50% must be higher? Well, yes and no. The probability of it being exactly 500,000 out of a million, not 499,999 or 500,001, is tiny (0.08%). But the probability distribution of gender split also narrows, so it’s more likely to be close to 50%. So while the odds of being between 4/10 and 6/10 are just 65.6%, the odds of being between 4,000 and 6,000 out of 10,000 are 100%, rounded to 90 significant figures.

binodist

A flawed policy
What does this mean in the context of quotas? Well, it means that if you think that in the absence of barriers we would see 50% of caucus being female every year, you’re statistically incorrect. We would actually expect it to vary from year to year. With a group of 30 to 40, it would vary quite a bit, both over and under, but would average out to 50% in the long run.

Obviously this hasn’t been the case to date. Men are over-represented in every legislature, including ours, and have been as long as anyone can remember. So on the statistics, if we accept all of the above assumptions, then it’s fair to say there are undue barriers preventing women from being nominated.

But to yearn for an even split every election year, you would have to reject at least one assumption. You might believe that all individuals are identically competent. You might believe that if I told you the best person was female, then it’s a good guess that the second-best is male, i.e. that competence and gender are not independent. I would be surprised if anyone did believe either of those, but I’m not objecting to it here. Proponents may also have values other than strict competence at heart (e.g. diversity), which is fine too.

It’s just that those aren’t typically the arguments that get raised when we talk about 50% gender quotas. The cause of gender equality is worthy: no-one, left or right, seriously contests it; they just differ over how to achieve it. But a single caucus—a single sample of 30 to 50 people—is not a reliable metric of whether we’re there.


* The reason I say this expressly is that I don’t just mean it in the “statistics don’t apply to the individual” sense. I mean it statistically, as in it doesn’t even tell you about the odds.
† I originally tried to get a figure for one million, but the probability of it not being between 400,000 and 600,000 is too small for a computer to express with a 64-bit double-precision floating point number.

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