What does it mean to have a low quantitative but very high verbal/writing GRE for stats PhD application? I am applying to statistics and biostatistics PhD programs for next year, and I just got my GRE scores back, and they weren't at all what I expected.

I ended up with a 159V/156Q/6.0AW, which corresponds to 83rd/61st/99th percentile in the verbal/quantitative/analytical writing sections, respectively. So, I have a combined score of 315 with a 6.0 writing.

I understand that most statistics programs place a premium on the quantitative score, and so I am a little worried about my low quantitative score, but I am wondering if the relatively high verbal/writing can offset the weakness there.

For some idea of what kind of student I am, I have about a 3.5 GPA with an applied math major, with a full course in real analysis, mathematical statistics and some graduate level regression theory, and potentially an honors thesis (pending). Since my scores are so atypical compared to other applicants (who typically have much stronger quantitative and lower verbal/writing), I am unsure of what my score means for my application as a whole.

I suppose the question of "do stats programs care?" can be generalized to "do quantitative programs care?". While I see there are some threads on low quantitative scores, there aren't too many about a low quant score combined with a high(er) verbal/writing score.
Most programs have a minimum threshold for GRE scores, and I'd be very surprised if you pass that threshold. I'm applying to engineering programs and most Top 25 programs require atleast 165-166 on Quant. For someone who's applying to statistics, I would aim for at least 90% percentile (166). It's very easily doable, just do a few practice exams and practice problems and I'm sure you can pull it off.