Developing both Applied and Theoretical Interests I've seen profiles of researchers who, over time, come to work in both theoretical and applied areas, although I suspect (having seen profiles of other researchers who recently finished their PhD) that these researchers had one forte to begin with.

As an undergraduate hoping to become a researcher, I'd like to develop both interests one day.

Is that possible in both directions? Can I go to school to become a theoretician and later develop applied interests; or go to school to become an applied researcher but later learn and pursue more theoretical interests?

As an example, I'd like to one day both deeply understand statistical theory and yet have the skillset to work with data. Is it better to start off in one area?
Short answer is yes you can. Now, how/why do researchers change directions? Many reasons; 


Chasing different funding/proposals venues.
Being part of a school or department where interaction of professors and available lab equipment can lead (with time) to you redirecting your research.
When advising students for MS/PhD or being a part of search/research committees, one tend to think outside the box (outside of his research area) and incorporate different methods/ideas
You learn more new stuff with time! Remember that a PhD in subject X does not mean that you get to work in X for the rest of your life. It simply means that you know X (in details) much more than other people (or professors).