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.

Example:

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?