I have come accross this sentence:
There is a 2.5% probability that whenever we measure a woman, their height will be less than 142 centimeters.
Is the use of their correct here? Shouldn't it be rather her?
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"Correct" is opinion as English is a natural language, and is therefore off topic.
However, "usual" and "accepted" are measurable and therefore valid questions. My experience suggests that the use of the singular they is spreading at an accelerating rate down this list :
Gender unknown (the doctor I will see tomorrow)
Gender hidden (my partner if I choose to hide their gender)
Gender irrelevant (the doctor I saw yesterday)
Unspecified person of known gender (the woman in question)
Specified person of known gender (my girlfriend)
My experience is that as people are criticised more and more for politically incorrect use of he or she but almost never criticised for they except by pedants, people (especially children) are increasingly using they as it's easier than thinking which is correct.
"Correct" is a social judgment, not a linguistic one.
The OED has examples of singular their going back to 1382, but there are still people around who think there is something wrong with it.
If you care about being judged by people who think that, then don't use it. The rest of us will carry on using this useful English word with a long history.
Insofar as anything is right or wrong in grammar, 'their' is wrong here, and 'her' is right. Imagine you were learning English as a foreign language; I'm sure you'd be told that (unless there is gender ambiguity) a female individual requires the possessive adjective "her".
The only reason I can think of for using 'their' in a case like this would be to tease those who are annoyed by such solecisms.
To be short: it depends who you ask. The singular pronoun "them" has been in use since the 14th century, so it would be hard for someone to claim that there's no precedent for using it. Not only is there; it's actually rather popular.
But should you use it? That's where it depends who you ask. The Wikipedia article on the subject has an excellent, well-referenced section describing the advice of various style guides. Suffice to say they disagree.
If the experts don't agree, who are we to answer your question? Certainly singular "they" appears in most any modern dictionary. But some style guides prohibit its use, while other style guides encourage it. As with all decisions to be predicated on opinion in writing, when deciding whether to use it, take the rules of the organization on whose behalf you are writing into account, and choose whichever option is most appropriate for your target audience.
Perhaps we should start from a simple point, which is at the root of your question about the use of the plural pronoun to refer to a singular noun. It all comes down to gender. The writer has got into the habit of using the plural, genderless ‘their’ in preference to the singular gender feminine ‘her’. Why?
The personal pronoun is a rare English word that inflects by gender. You could have an ethical/social argument about whether it would be proper to bring an end to this form of gender discrimination.
There is even a linguistic case for suggesting this anomaly in our language would be better replaced. We don’t seem to need a gender in the plural Of course the only possible neutral pronoun is the neuter ‘it’. To start with, at least, it would be hard to swallow people being referred to as ‘it’, but we might get used to it eventually.
However, it is not, as has been pointed out, for students of usage to legislate. The usage cited here of the (neutral) plural pronoun in relation to a singular feminine noun is, I think a symptom of transitional unease.
It goes back to the usage, generally accepted well into the 1960s, that, given a singular noun which refers to a class including both female and male members, the masculine form should be used.
This is not just a feature of English. It applies even more to romance and Teutonic languages, where the articles and adjectives all inflect in this way. Greek even retains a neuter form. In all of them the masculine will be used for adjectives agreeing with words like doctor and feminine for words like nurse for obvious (but now discredited) reasons that these avocations are historically associated with men and women, respectively. But it is harder to change in these languages, partly because the ‘gender’ inflexions are not solely indicators of sex. In French window as well as woman (une fenêtre, une femme) is feminine and monster (un monstre) as well as man (un homme) is masculine. In almost gender uninflected English nouns are grammatically genderless, with a few exceptions, such as the ship, as in ‘steady as she goes’.
In all inflected languages I know, where masculine and feminine are conjointly qualified by the same adjective, the adjective must be in the masculine form (as in Handsel and Gretel are lost - Handsel et Gretel sont perdus - not perdues). Thus the precedence of the masculine over feminine is much harder to break in French than in English.
But for English the problem arises with ‘singular plural’ adjectives like each, every, any, when qualifying a singular noun that refers to a plurality.
Each to his own?
Every student must study as hard as he can?
Don’t tell anybody how he should behave?
Thanks to the feminist movement a majority (I hope) of Brits, especially younger folk, have grown uncomfortable with this use of the masculine form of the personal pronoun. This is especially true when we speak of presigious occupations.
Every doctor does the best he can for each of his patients no matter what his needs may be.
We are on the horns of a dilemma: do we continue to ‘mis-assign’ gender (masculine for both) or person (plural for singular)?
We can escape by having it both ways, by using a cumbersome disjunction of both genders
Every doctor does the best he or she can for each of his or her patients no matter what his or her needs may be.
Or we can duck the dilemma by using the neutral plural their.
Every doctor does the best they can for each of their patients no matter what their needs may be.
Over time this strategy has become common enough to displace the old rule of number consistency to a large extent
There is another way off the two horns without ‘infringing’ the old number rules by avoiding the singular/plural collective adjectives altogether.
All doctors do the best they can for all their patients no matter what their needs may be.
But at times we really want ‘each’ and ‘every, because it makes the relationship personal rather than general. In that case there is no escape from the uncomfortable ‘he or she’ (I use ‘s/he’) without committing the discredited gender bias.
Which brings me to your original question and my transitional stress. What I mean is that speakers and writers of English have become so accustomed to the shift of number, replacing he/him/his with they/them/their that some overshoot the reason for number shift and use it automatically in the absence of the original reason.
The shift from her to (what I should in my distant youth have said was the incorrect) their in your quoted example is for that speaker/writer becoming a fixed speech habit. I have never seen it before, and I doubt whether it is common enough to be becoming standard usage - yet. But it might do so. At which point, if ever, a ‘neogrammatism’ becomes widely enough used to be accepted as a standard usage nobody can say. For the present I don’t recommend it.