Sena

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.

Worth adding the number that will end this permanently: over a session with say 60 rolls, if somebody calls their number each time, the chance of at least one hit is about 95%.

So it is not merely possible, it is close to expected. If it were not happening every few sessions, that would be the surprising outcome.

Small framing that helped me: ask what each failure would cost the person using the output.

An underfitted forecast is useless in a predictable direction, and people route around it. An overfitted one is confidently specific and occasionally absurd, and people act on it before they learn not to trust it.

"Closer to zero is more likely, and only differences mean anything" is the sentence I needed. I had been reading large negative numbers as failures.

The log-sum-exp warning is timely too — I have exactly that expression in a script and it has only ever run on toy data.

The clean way I have heard it put: statistics asks what the world is like, machine learning asks what happens next. Assumptions are how you make a claim about the world with a straight face; held-out data is how you check what happens next.

Both courses were teaching linear regression. Only one of them was teaching it as a claim.

The reason the two collapsed into one phrase historically is worth a line: for the linear model with normally distributed errors, the least squares estimates are exactly the maximum likelihood estimates.

So under the standard assumptions, the criterion everyone reaches for is not just convenient, it is the principled one. That coincidence is why the pairing became the default and why the words drifted together.

Also worth knowing why nobody sees this in practice — every library returns the log, and reports differences. You ran into it because you computed the raw product yourself, which is exactly the thing the tooling is arranged to avoid.