Yes. There's the difference between what happens in the public perception...and there are also some interesting studies on what informs the public perception of what happens in our justice system. The public may think certain results are unwanted or undesirable, but when properly informed, the public perception often changes.
A minimum sentence is a very blunt way to communicate policy. It's a very blunt way to communicate a position on justice issues. Quite frankly, I think we don't give the public enough credit, and these sorts of policies can be communicated in much more nuanced ways that reflect the evidence and the reality.