Field of Science

On average

The average, when it means the mean, of something is mathematically very easy to calculate. I doubt that it's necessary to refresh anyone's mind, but just on case: the mean of a set of numbers is the sum of those numbers divided by the number of numbers. And recall that two sets of numbers can have the same mean, and yet have different distributions. The shape can be different (e.g. a uniform vs. a Poisson distribution), and if the type of distribution is the same for two sets, then the variance can be different, while the mean is still the same.


The set {4,5,6} and the set {1,5,9} both have mean, µ=5, but their variance is different (greater for the second set).

If two sets of numbers have different means, then a random drawing from the set with the higher mean is expected to yield a higher value. In other words, drawing one number at random from each set will more often than not result in a greater number from the set with the higher mean, compared to the set with the lower mean, if the two sets are drawn from the same type of distribution (e.g. Gaussian).

Take men and women. Men are on average taller than women. If you need something from the top shelf, then it's a safer bet to ask a random man than a random woman. Clear as crystal. That some women are very tall and many women are taller than many men doesn't change this. And yet, I have many times heard an objection along the lines of "but my dog's best friend's owner's sister is 6 foot 5." Aha. Fascinating. Where are you going with this? Most often people are just trying to carry a conversation. Real life examples are often more interesting than statistics.

But sometimes people revert to this irrelevant case-by-case argument even when talking about statistics. If we talk about the requirements for becoming a firefighter, for example. The physical requirements are quite tough, so we can ask what that would mean for the gender distribution of firefighters. If we posit that men are - on average - stronger than women, then the expectation would be that more men than women become firefighters (ignoring all other qualifications and personal preferences, which I here assume to be similar for the two sexes).

For example, if someone had said the following, then that person would be completely ignorant of this significance of averages.
What makes you define women as "the weaker sex"? There's significant overlap in the distribution of body mass and strength between men and women. Some people are stronger than I am. Some people are weaker than I am. That includes both men and women. And I, as a woman, often hold the door for others of either gender.
Yes, this woman may be stronger than some men, but that doesn't change the fact that women on average are not as strong as men, does it?

Additionally, if men and women have different averages in many physical traits (e.g. strength, height, weight, speed, lung capacity, sleep requirement, vision acuity, hearing, etc.), and if having an excellent score in all of these is imperative for fighting fires, then there will be a difference in the physical qualifications between men and women - on average. That is, averaging or scoring these eight averages in some way relevant to the physical requirements of firefighters would not be expected to cancel out exactly, even when men score better on some on average and women better on others on average. That job performance depends on just one trait thus doesn't mean that we should suddenly expect men and women to be equally likely to become firefighters (again, assuming all non-physical qualifications as well as interest are identical). A comment like this one would be an illustration of failing to understand this important aspect of averages:
Bjorn, I can think of lots of differences on average. Considering that the traits you mention are all distributions with nearly 100% overlaps between the sexes, considering that job performance never depends on just one trait (or is evaluated on just one), and considering that any differences also result in jobs where women outperform men, I don't see how any of them would lead to an overall gender imbalance in pay.
Apart from the fact that "100% overlaps" doesn't really make any sense whatsoever any which way one tries to understand it, this comment about paying men and women differently misses the point that if physical characteristics make any difference in who gets hired anywhere, then it will also contribute to a difference in salary between men and women overall.

"But my best friend's mother earns twice as much as her father...!" Fascinating. Did I forget to mention that we're talking on average?


  1. I agree with what you're saying - the other comments are meaningless. Of course, I'm the only one backing you up in that whole debate anyhow, so it's not shocking I agree with your point :)

    However, to be fair, I highly doubt that firefighters (or other physically-dominated fields) are tipping the pay scale between men and women. I think the pregnancy/sacrifice is far more likely, though even that doesn't explain why single, childless women only make 90% of their male counterparts. To argue there's no discrimination is silly, seeing as less than 100 years ago women in america couldn't vote - of course there's still progress to me made as far as equal opportunity and employment - but I think we're well on our way to fixing those, and I expect that 90% to approach 100% in the next few decades.

  2. Hi Christie.

    And I agree back at you here as well. I am not ever going to argue that discrimination on the job market doesn't exist. I don't know anything about that, except claims that it does. [I have argued that there is no gender pay gap, based on June O'Neill's analysis, but I fear she's not totally accurate there.]

    The firefighters example is just good to illustrate why there would be any difference coming from the physical traits. Others would be the pregnancy issue (my guess is also that this makes the biggest difference) mental differences, and then some from discrimination. I plan do deal with those later on.

    Btw, Jeff Darcy had a lot of good points about the pregnancy issue on the second thread on Greg's blog, in case you missed it.

  3. Not following the debate on Greg's blog. But isn't the mostly likely reason for the difference in pay (after accounting for different career paths etc) that the importance of wealth for sexual attractiveness is greater for men than for women. So the rewards for being wealthy are greater for men. Since they have a greater incentive then, all other things being equal, you would expect them to earn more.

  4. Tom, I actually argued exactly that on Greg's blog. And I will argue it again here, once I get around to writing something more comprehensive about the gender pay gap. I just wanted to take the related issues one at a time.

  5. I am with you for the most part. Stephanie Z has outed herself as more into the fiction than into the science. My comment was perhaps not so clearly written, but she did not get it at all, not even close.

    And the feminist male is some kind of fashion statement that I fail to appreciate. Thank you for the rational approach!

  6. Like Greg, she just doesn't seem to get the explain/justify difference at all. She has accused me of ignoring answers to my own delight, and it is clear as day that that's what she's doing all the time. Which is why I thought it was pointless (apart from unpleasant) to post on that thread any further.

  7. When I am right in more than 50% of the cases independently on who I argue with (assuming I argue enough and several times with everyone), does this make me the smartest one of all? I betcha! Or at least I hope so :)

    See one way to to turn probabilities into absoluta.

    And I guess many people try the same thing in conversations, but unfortunately fail in statistics.


  8. keep quoting these dead white guys for a reason. We seem to be repeating some particularly nasty history, right now.


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