how does a designer determine the appropriate x-height.

Does it look good as it is now?
If yes.. leave it as is.
If no... change it.

Hi Sevag! You can tell a question is good when it's hard to answer... :-)

I don't think "if it looks good, it's good" always works. If it did, text fonts would have descenders longer than ascenders (because some of the former need more room to look nice) but the opposite is true... Things beyond looks matter when you're making a text font. Some things to keep in mind:
- More important than the relationship of the x-height to the Em is the relationships between the x-height, ascender height, descender depth, cap height and numeral height(s). The x-height is the main player (at least in the Latin alphabet, but for example not in the Georgian) but it can't ignore the rest.
- In a text font an x-height is usually around 60-65% of the ascender height, but it can go over or under that for specific uses: under, for let's say a large book set in a relatively large point size; over (by a good deal) if it's for a newspaper. The reason for this is that the requirements for reading shift a bit, depending mostly on size, but also other factors: in "choppy" reading like in a newspaper seeing individual letters clearly is more important, whereas when reading an immersive book in comfort the ascenders and descenders matter more (because they help define boumas).
- Cap height is typically a little under ascender height, and descender depth is less than ascender height (sometimes by a good amount - check out Berling). Numerals can have a variety of schemes...
- The big one... The vertical proportions are tied in to two other major parameters: the color (darkness) of the font, and the overall spacing. This is because each one of those three things points to a certain ideal point size range, and they have to be pointing at the same range. Smaller sizes require darker color, a bigger x-height and looser spacing; larger sizes, the opposite. It's hard to get a feel for what's a good balance, but it helps a lot to look at existing fonts by masters like Carter. Remember though that there's no precise formula, and personal taste can shift things quite a bit, especially when it comes to color.

I hope that helps.

hhp

Another factor that Hrant doesn’t mention is the ability to pack lines tighter. This is relevant in newspaper/magazine design because a multi-column text will have shorter lines and thus need less space between the lines* than, say, a book. Also, counter to what Hrant says, in large sizes (well above text size) the same requirement comes into play in, for example, a headline. This ties in with headline faces commonly being spaced tighter horizontally than text faces: they are also “spaced” tighter vertically, using the x-height and descender depth as the main variables.

* I’d love for someone else to explain this in depth, but in general:
1: At a certain distance (and this distance is closely intertwined with the length of the lines and whitespace around the text) lines begin to form a visual “mass” instead of just separate lines.
2: In a book, with longer measure, it’s hard to find the next line if the lines don’t have enough space between them.

Good points - there's certainly a lot more in here... For example that the overall width of a font also plays into what vertical proportions work well; and how the frequency of linebreaks (which is very high in newspapers) affects what role width ends up playing in economy...

Also, even in a book "packing lines tighter" matters, because it's not just a matter of collision. For example when the descenders (which are linguistically infrequent) are too long you're wasting vertical space that could be used to amplify readability in other regions (mostly the x-height, but secondarily the ascenders). I use the term "apparent leading", which is what I think determines ease of "line return" when reading.

One thing you wrote that I'm not getting is that #1. Is that an aesthetic issue?

hhp

Are you asking me, Hrant?

At a certain distance (and this distance is closely intertwined with the length of the lines and whitespace around the text) lines begin to form a visual “mass” instead of just separate lines.

I’m just observing that with too much leading a text block “falls apart”.