Below, I have divided Tillman's career since 2012 in two. The numbers shown are his for his highest velocity fastball games versus his worst velocity games.
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The sample sizes here are somewhat robust being about 150 innings pitched. The suggestion is that higher velocity doubles home run rates while everything else remains about the same. This is likely due to Tillman being able to challenge and live up in the zone a bit more when his velocity is up. It is a bit surprising, but it seems to show that Tillman can adapt to situations and remain as successful as he normally is.
Although sample sizes are inadequate, I have cut Tillman's MLB career in five equal portions of eleven games to see if any potential differences might emerge.
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Three very soft conclusions jump out of this, Tillman struggles in games where his velocity falls below 90.9 MPH, which is below average for a right handed starting pitcher and has been for about a dozen years. Second, when his velocity is hopping above 92.5, Tillman appears to be able to overpower batters. Third, he has absolutely dominated in games where he has experienced 20th to 40th percentile velocity.
The data is not very clean, but I will try to state a very tentative narrative. Tillman has two major styles of pitching. When his velocity is present, he overpowers hitters and challenges them. That results in a lot of solo homeruns, but low run scoring overall. However, when his speed deteriorates he becomes more hittable, giving up more home runs and other meaningful contact. Eventually, this erosion of ability becomes obvious and a second game plan is engaged with more nibbling and less challenging of hitters. With suitable velocity, this approach appears to be very successful. Unfortunately, when his velocity dips further, this more careful strategy is quite unsuccessful. Again, this is a tentative assessment.