With Johan Santana pitching last night and Cole Hamels tonight, there’s been a lot of talk about the pitchers’ differential between the speed of their fastball and their change-up helping a pitcher strike out more batters. Seems like common sense, but I decided to see how much of a relationship there actually was, so off to FanGraphs I went. This is your warning to exit now if you don’t like math.

I sorted the starting pitchers first by innings pitched over the past three calendar years. I got their K/9 rates, fastball and change-up usage rates, and fastball and change-up average speeds. In another column, I subtracted the average change-up speed from the average fastball speed, giving us the differential. Then, I eliminated all starters who used their change-up less than 10% of the time. It’s an arbitrary cut-off point, but we don’t want to study pitchers who never use a change-up like Tim Wakefield.

We’re left with 30 pitchers, which adequately satisfies our sample size requirement.

Next, I plugged the differentials and the K/9 rates into a scatter plot, added a trend line with a *y=mx+b* equation and an R-square. I don’t actually use the equation for anything but I put it up in case anyone wants to toy with it. In case you’re unfamiliar, the R-square tells us how much of the change in X can be explained by the change in Y.

(*Click the image for a larger version. I have no idea why the images are blurry — anyone have any ideas as to why this is? The pictures look fine when viewed on my computer so I think it’s ImageShack‘s fault.*)

We have an R-square of 0.22, which means that about 22% of the change in a pitcher’s strikeout rate can be explained by the difference in speed between his fastball and change-up. The relationship is positive, meaning that the wider the gap between the FB and CH speeds, the more strikeouts he’ll achieve.

It will start to get real math-heavy here, so brace yourselves.

Next, we want to run a test to see if the correlation coefficient (.47) is reliable using the null hypothesis that there is no correlation between FB/CH differential and K/9. In a two-tailed test using a 95% level of confidence with 28 degrees of freedom (30 players minus 2), we find that our test statistic must lie between negative 2.048 and positive 2.048 to conclude that there is no significant correlation.

A t-test for a correlation coefficient gives us a test statistic of 2.83 which falls in our region of rejection. Thus, we conclude that there is, in fact, a statistically significant correlation between FB/CH differential and strikeout rate based on the group I’ve selected. Not a surprising conclusion, but interesting nonetheless!

I’m sure that if I lowered the threshold below 10% CH usage, or if I used pitchers over two calendar years instead of three (or pitchers over a three-year period other than present minus three years), we’d see less of a correlation. Unfortunately, I don’t have the database skills to extract that.

This is a rough and incomplete look at the relationship, but it’s something I threw into a spreadsheet on a whim. Much better than watching the Mets BABIP-luck their way into a win against Cole Hamels (as I was writing this article, BABIP started to swing back in the Phillies’ favor in the top of the seventh — hey, Paul Bako sighting!).

Feel free to add your thoughts, questions, and criticisms. I especially want criticism, so have at it. I’m not aware of any other studies that have been done on this relationship, but if you know of any, please let me know.

**Addendum**: What this “study” ignores, among other factors, is pitch sequencing, which obviously would have a large effect on k-rates. If you have two pitchers who are 75% FB, 25% change-ups, the pitcher who goes FB, FB, CH, FB, CH, FB, FB, FB will most likely be more successful than the pitcher who goes FB, FB, FB, FB, FB, FB, CH, CH. The pitcher vs. batter match-ups exemplify the Nash equilibrium, but that makes my head explode.