  Problem Roulette (PR), an online study service at the University of Michigan,
offers points-free formative practice to students preparing for examinations in
introductory STEM courses. Using four years of PR data involving millions of
problem attempts by thousands of students, we quantify benefits of increased
practice study volume in introductory physics. After conditioning mean final
grade on standardized (ACT/SAT) math test score, we analyze deviations based on
student study volume. We find a strong effect; mean course grade rises
quadratically with the logarithm of the total number of PR questions
encountered over the term ($N_{\rm Q,tot}$), with an overall gain of $0.77 \pm
0.12$ grade points between $1 < N_{\rm Q,tot} < 1000$. The gains are persistent
across the range of math test score represented in our sample. While $N_{\rm
Q,tot}$ surely correlates with other study habits, the benefits of increased
study in general still hold. A model for final grade using test score and study
volume largely accounts for demographic stratification, including by sex,
parental education level, number of parents at home, nationality /
underrepresented minority status, and regional income level, with two
significant exceptions: students whose parents did not earn a college degree,
who earn $-0.27 \pm 0.04$ grade points ($6.1\sigma$) below expectations and
underrepresented minority students at $-0.14 \pm 0.04$ points ($3.6\sigma$).
Residual scatter in final grade remains comparable to the maximal study gains,
implying that the model is far from deterministic: individual variation trumps
mean trends. Our findings can help motivate students to study more and help
teachers to identify which types of students may especially need such
encouragement.
