You’re right. Studies do show that the outcomes for hip and total knee replacements are better when done by surgeons and hospitals with high-volume. There are fewer problems, fewer readmissions, and fewer (if any) deaths. The same is true for patients hospitalized for hip fractures and total shoulder replacements (TSR).
Low-, medium-, and high-volume may be defined differently from study to study. For example, there are more total hips and total knees done each year across the U.S. compared to total shoulders. The number of cases in a high-volume center will likely be much higher for hips and knees compared to shoulders.
In a recent study of shoulder replacements, fewer than four operations per year is low-volume. More than four per year but less than one each month defines middle-volume. High-volume hospitals did at least one TSR each month.
Since TSRs are rare compared to total hips or total knees, some surgeons may only do one a year. Outcomes are better when surgeons do at least one TSR per quarter. Results are even better when a surgeon does TSRs once per month.