grammatical number - Plural or not after "they differ by ..."

There are a few constructions in English for which I often have a hard time deciding wether I should use the plural form or not. In this case it's with the verb "differ".
Which sentences are correct in these examples?

John and Bob differ by the size of their heads.

or: They differ by the sizes of their heads.

and

They differ by their height.

or: They differ by their heights.

On Google I found both. In French it would be all singular (the size of their head). I suspect that in English the correct form is the plural one, because in this language you tend to use the plural, e.g. "They touch their noses" (correct me if I'm wrong) and not "They touch their nose" (which is what you'd say in French, even if there are several noses – just one per person).

But here, I have the vague feeling that (maybe you could somewhat argue that) John and Bob differ because of one common attribute – which takes different values... so maybe, singular? I think I just prefer the French way!

Answer

If you are using their then it definitely should be plural. "They differ by the size of their head" would suggest that the two of them shared a single head.

But you can, and should use the singular if you say "they differ by the size of the head".

Similarly "their heights differ", or "...by their heights", but "they differ in height".