UX research: qualitative, quantitative, both?

I’m trying to get a better understanding of how qualitative UX research can incorporate quantitative work, the kinds of quantitative work that folks have found themselves using in UX, and where they see this going in the future?

Myself, I’ve done mostly qualitative work within the UX field, but much of my psychology research was a combination of both. I find qualitative work makes more sense to me than qualitative, perhaps because there are always so many hidden assumptions in quantitative work.

At least in qualitative work, some of the explicit work around it includes hunting down and stating one’s assumptions.
In qualitative work, I’ve found it’s much easier to lose track of the assumptions necessary for a statistical analysis to make sense to use.

Is your underlying population on a normal curve? Is it one-sided? Are p-values appropriate? Are regression tests? What about primary component analysis?

I have found it difficult to correctly identify the appropriate statistical method to apply to problems at hand.

Does anyone else have thoughts on this? Maybe pointers to suggest for more on how to best incorporate quantitative research into UX without making too many unfounded assumptions in the process?

I’ve had at least one hiring manager mention that quantitative research is going to be a bigger part of UX practice in the future. This makes total sense to me, and at the same time, I can’t figure out what that means - and math and statistics have never been difficult for me. Figuring out which method apply when, that’s hard.

If you were trying to measure success quantitatively, how would you? If you were trying to determine if the methods you’d been using - personally or in a company - were actually the right ones to answer the question of measuring success, what would you do?