My favourite example of why common sense is pretty useless in maths is an example of comparing averages;
There are two shops, each with 5 salespeople. One shop performs very well, each of the salespeople selling 50, 40, 40, 40 and 30 widgets respectively. The other performs very badly, as the staff selling 20, 10, 10, 10 and 0 widgets each.
The average monthly sales for the first shop is (50+40+40+40+30)/5 = 40 widgets.
The average monthly sales for the second shop is (20+10+10+10+0)/5 = 10 widgets.
As an experiment the worst salesperson from Shop 1 is moved to Shop 2.
The average monthly sales for the first shop is (50+40+40+40)/4 = 42.5 widgets.
The average monthly sales for the second shop is (30+20+10+10+10+0)/6 = 13.3 widgets.
Common sense says that both stores are now performing 'better', but quite obviously that is nonsense.
The point , I think is that it's an easy way to mislead, "Average employee productivity is up in all our stores!". Of course in reality it's a good idea to mix high performers in with low performers to teach them to improve.
Each store is performing better: the successful store lost its weakest sales person, the unsuccessful store gained a (comparatively) excellent sales person.
Conversely, this also illustrates a fallacy in many arguments against immigration. Suppose Mexicans/Indians/whoever are all (or predominantly) worse than Americans on whatever scale you choose to measure them. When you allow immigration you'll cause averages to go down in both countries by the converse of the mechanism you describe above.
Various people (Vinod Khosla, Steve Sailer) fallaciously use this statistical truism as an argument against immigration by people without a PhD.
The only way I can think of to steelman arguments like this are by combining Steve Sailer style nationalism with fairly extreme "redistribute all the things" liberalism.
I don't get the fallacy. If you believe that there are positive externalities from living in a country with people who are better on whatever scale you choose to measure them, then migration benefits a (relatively) small number of migrants and hurts everyone else, both in their country of origin and destination.
Whether this should prevail over freedom of individuals is another matter.
Externalities is a separate argument, and requires doing more than simply looking at average qualities.
If you want to make that argument you need to start by drawing the line between people who are positive and people who are negative. This is a more compelling argument, but it's not the one I described.
Khosla is a PIO, but as I recall from his debate with Bryan Caplan (they were both supposed to be on the same side!) he only favors immigration of highly skilled individuals. I'm having trouble finding it, but he had very little substance behind his arguments - just a cheap slogan ("import prosperity, not poverty" or something like that). Overall it was very disappointing.
There are good arguments to be made both for and against immigration - the particular one I describe here just isn't one of them.
There are two shops, each with 5 salespeople. One shop performs very well, each of the salespeople selling 50, 40, 40, 40 and 30 widgets respectively. The other performs very badly, as the staff selling 20, 10, 10, 10 and 0 widgets each.
As an experiment the worst salesperson from Shop 1 is moved to Shop 2. Common sense says that both stores are now performing 'better', but quite obviously that is nonsense.