Real analysis is absolutely not an applied subject, by which I mean it's improbable that you'll solve any practical problems, or even impractical models of practical problems, during this course. It's the language in which probability is developed, though, so as a statistician you'll probably be expected to know some probability theory, which is why analysis was recommended to you.
Answer: all of them.
The entire Stein&Shakarchi series is excellent, so if you like their style it's a great addition to your library. Especially if you're later interested in complex analysis, harmonic analysis and analytic number theory. The examples and additional lemmas in their books are a great supplement to your understanding of real analysis.
Royden is an excellent overview of real analysis, providing down-to-earth proofs and examples. It's probably the easiest to read out of the books listed. Make sure to get 3rd edition or higher as earlier ones are riddled with typos.
Rudin's Real&Complex Analysis is a work of art, and I would suggest using it as a supplement to any of the above. His exposition is beautifully terse but concise. The reason I suggest another supplement is its a bit too terse, leaving most pressing questions as an exercise to the reader.
Folland is excellent in his presentation of measure theory, relegated to just 3 chapters, and then moving onto point set topology.
Kolmogorov-Fomin is a more classic text, especially in its notation. It's tiny in frame, but dense on details. Would recommend it as a supplement to any of the above.
One suggestion would be to study real analysis with applications in mind. One example is to simultaneously study advanced probability theory, which will supplement your understanding of sigma algebras and general convergence principles in real analysis. Probability: Theory and Examples by Durrett is exhaustive.
Best Answer
I read first few chapters of Folland's Real Analysis during summer after my senior year at Ohio State to prepare for a rigorous PhD program that I was soon to begin (with one focus being pure / applied econometrics). I had previously also read Baby Rudin, ch. 1-7 for a course (you'll need ch. 7 since uniform convergence of sequences of functions is important for measure theory). Folland is a bit terse, but precise, and it has been the primary text for first year grad analysis at many universities because its coverage/presentation is fairly traditional and complete: measure theory, basic functional analysis, topology that you need for analysis if you haven't taken a course in general topology), Fourier analysis, distributions. And one of the other reviewers is correct: you need to read at least Ch. 1 - 6 (to get Lp spaces), and Ch. 8 (Fourier Analysis) is very helpful for those who will take probability theory like you). Ch. 7 is a "bonus" since almost no text at this level proves the major theorems in locally compact Hausdorff spaces as thoroughly as Folland (and you'll see other measure theory texts refer you to this chapter for its coverage). I agree Folland is a bit dry, but adequacy of coverage makes up for it. A more lively presentation of similar topics is first 3 parts of Serge Lang's "Real and Functional Analysis" (and part 4 on differential calculus in Banach spaces is well done too). I just found about Axler's book, and I'm looking forward to skimming it soon.