The Washington Post has compiled a database of fatal shootings by police in 2015 and 2016. Breaking it down by state seems like an obvious thing to do, but I haven't seen anyone do it, so here are the rates (per 100,000) by state.
New Mexico 1.6
New Jersey 0.21
Rhode Island 0.19
New York 0.15
The general pattern is clear: fatal shootings by police are most common in the West and least common in the Northeast. The South may be a little higher than average, but doesn't stand out: southern states range from 10th (Louisiana) to 35th (Virginia). Of course, there are racial and ethnic differences: 27% of those killed were black and 18% were Hispanic. But controlling for population composition would make the West stand out even more, since the black population is relatively small in those states.