Yes, Police Killings and Crime Go Together
Campaign Zero has a report claiming the opposite, with this ... interestingly formatted chart:
The problem: The police-killing numbers are calculated per million residents, and most of these cities don't even have a million people. When I matched up Campaign Zero's rates with population data to roughly calculate the raw numbers (spreadsheet here, data mostly from here), I found only three cities that killed even ten people total. More than half killed three or fewer. So even combining three years' worth of data, as the Huffington Post says the activist organization did to "confirm the findings," leaves us with rates that are practically random. A single incident can radically alter the rate.
I would also use the murder or homicide rate instead of the violent-crime rate: Dead bodies are less likely to go unreported, or to be misclassified if they are reported. They also represent the most serious form of violence, which is important considering that deadly force is legal only if an officer reasonably believes there's a serious threat to life or limb.
Fortunately, a better analysis has already been done. The criminal-justice professor Peter Moskos once combined ten years of data and found that "of course a high homicide rate correlates very much [with] more police-involved shootings."
Robert VerBruggen (@RAVerBruggen) is editor of RealClearPolicy.
Note: In response to a comment, I've removed a reference to the chart as a "scatterplot." Technically, it is a two-Y-axis line chart with points instead of lines and the data sorted according to one of the variables.