How do you answer the student who objects that nobody has 1.5 children?

The other reliable move is analogy to a number they already accept without complaint.

A batting average, a goals-per-game figure, a grade point average of 3.4 when no single course awards a 3.4, a household with 2.3 cars. They have lived comfortably with all of these for years. Nobody has ever objected that you cannot score 1.7 goals.

That usually gets a laugh and does real work, because it relocates the confusion. The problem is not the arithmetic and never was — it is that "children" feels like it should be a whole thing in a way "goals per game" does not. Naming that is more useful than defending the mean, and it opens the genuinely interesting question underneath: when should you distrust an average?

The move that works for me is to concede the point completely and immediately, because the student is right and pretending otherwise is what makes the rest sound evasive.

"You are correct. No household has 1.5 children. The mean is not a prediction about any household, and it does not have to be a value anything can take."

Then give them the reason it is still worth computing: it is the number that reproduces the total. 1.5 children per household across 200 households means 300 children. That is a real, physical, countable quantity, and the mean is the per-household rate that recovers it. It answers "how many schools do we need", not "how many children does the family next door have".

Once they see the mean as a rate rather than a portrait of a typical household, the objection dissolves — because rates routinely take values no individual takes.

I would use the moment for the follow-up rather than closing it down, because the student has stumbled onto something real.

Ask them: given the mean is 1.5, how many households do you think have exactly one child? Let them guess, then show the distribution. Usually there is a large group at 0 and 2 and the mean is sitting in a gap.

That sets up the actual lesson — that the mean does not tell you the shape, and for skewed or lumpy data the median or the whole distribution says more. Income is the example everyone remembers afterwards: mean income and median income can be far apart, and which one gets quoted is often a rhetorical choice rather than a statistical one.

The student objected to a number that describes nobody. That is precisely the instinct you want to encourage, just aimed at the right target.

Concede first is the key. Every year I have watched people lose the room by defending the number, when the student's objection is not to the number at all — it is to being told that a summary is a description of an individual.