I don't think this is quite the right assessment of Dozier's defensive contributions when it comes to the statistic side of it. The part about him converting plays on the ones he gets to is accurate but I think the defensive metrics paint him in a different light.
I'll start by saying I do not have a complete trust in the advanced defensive metrics. Each of them have a different flaw while all of them have the flaw of sample size. That being said, when you look at them in aggregate, the picture painted is not one of a league average defender but rather a below-to-slightly below average one.
1. Dozier had -4 Defensive Runs Saved (one of the lowest among starting second basemen) with only Brad Miller, Neil Walker, Starlin Castro, Brandon Phillips, Scooter Gennett, Joe Panik and Daniel Murphy as starters with lower DRS.
2. Ultimate Zone Rating has him at -1.3 runs, or 13th among all second basemen with a minimum of 700 innings.
3. His Revised Zone Rating (how many balls in the second base zone he converts into outs) was .774, 8th.
4. Inside Edge metrics say he's great at the 100%-ers but OK at both the 50-90%-ers (79.3% conversion rate) and the 40-60%-ers (52.4% conversion rate).
In terms of #2-#3 the amount of times Dozier shifts has an influence on where those numbers go. For instance, for RZR out-of-zone plays Dozier leads the league with 99 plays made but that doesn't take into account a defensive shift. Inside Edge's stats do which shows that he's not making those plays with his feet.
I think Dozier does a lot of things right that don't get measured by these stats (makes some of the best forehand/backhand plays in the game) but, in sum, we need to accept that he's not great at getting to plays.
I think this is kind of picking nits, Parker. While some overrate Dozier's defensive value, it should also be noted that he played more innings than a couple of the guys who were in the negative but "above" Brian in those listings.
When we're talking the difference of 1-2 runs over the course of 1300 innings, whether a guy finishes with -1.3 or -0.6 doesn't really matter (even more so when the lower player put in more innings at the position).
Overall, I think Dozier falls into the nebulous area of "average". Sure, he was slightly lower than Joe Panik in UZR but also played 120 more innings. He was slightly below Hernandez and Lowrie in DRS but played up to 220 more innings of defense to get that lower rating.
I think too much attention is given to the exact placement of a player and whether the number is negative or positive. A 3 run swing in DRS or a 2 run swing in UZR doesn't really amount to much in a full season.
It should also be noted that Dozier's overall Fangraphs' Def rating was slightly in the positive (0.9).