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Hello,

It seems like noun after the word their can be singular or plural, but I don't know what is right. Thank you for your advice.

Example 1.
I collected grasshoppers.
Their head width was measured.
Their head widths were measured.

Example 2
Lions change their foraging strategy depending on...
Lions change their foraging strategies depending on...

Example 3
There are three people over there.
Their nose is large.
Their noses are large.

Example 4
They have a head.
They have heads.
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Hi;
Their is a possessive pronoun - the subject case is "they" and the object case is "them."

Examples:
  1. Subject case:
My friends are coming.
They (refers to "my friends") are coming.

2. Object case:
I gave my friends many gifts.
I gave them (refers to "my friends") many gifts.

3. Possessive.
My friends are bringing books. The books belong to them. The books are their (my friends') books.
My friends are bringing a book. The book belongs to them. The book is their book.

Notice that their is used when there are many people (or animals).
The word that follows can be singular or plural, but that object is something that belongs to all of them.

Example 1.
I collected grasshoppers.
Their head width was measured.
Their head widths were measured. (Each grasshopper has one head, but there are many grasshoppers (they). There are many heads.)

Example 2
Lions change their foraging strategy depending on...
Lions change their foraging strategies depending on...

A group of lions (they) can have one strategy that they all use together at the same time, or many strategies that they use.

Example 3
There are three people over there.
Their nose is large.
Their noses are large.

There are three people. There are three noses. All the (their) noses are large.
Comments  
AlpheccaStars,thank you very much!