clock menu more-arrow no yes

Filed under:

Mean, Median, but Sadly No Mode

As you've probably noticed, I've washed my hands clean of the daily roundups. Since they didn't offer anything that couldn't be found by visiting any number of other sports sites, I felt that they were simply taking up space. Now, on to a random musing about the eerie dangers of earned run average...

Normally, the amount of earned runs a pitcher allows per nine innings is conveyed through his earned run average (ERA). However, I thought it'd be interesting to consider a pitcher's earned run median. Consider, for example, a pitcher with an ERA of 3.00. Single-game ERAs below that average could only go down to 0.00 for a single game, whereas ERAs above 3.00 could go all the way up to infinity. Therefore, the higher single-game ERAs push much harder on the mean than do the lower ones, so there isn't an equal possibility of outliers on both sides. That means ERAs may not convey a pitcher's true performance over the course of a season.

Of course, since earned run medians only take single-game ERAs into account, that means they should only be used to assess full-season performances, otherwise the sample size would simply be too small. Also, it leads to the exclusion of relief pitchers, whose single-game ERAs are much too extreme to take into consideration. In the end, this is mostly for fun, since I'm genuinely interested to see if any large discrepancies exist between a pitcher's ERA and his earned run median.

By the way, I'm simply arranging single-game ERAs from lowest to highest and selecting the middle value. If the sample set is even-numbered, I'll use the average of the two middle numbers.

The 2005 Blue Jays' starting rotation:

Player         2005 ERA    2005 Earned Run Median    Unearned Runs    Quality Starts*  Quality Start %
Roy Halladay   2.41        1.29                       1               14                73.7%
Josh Towers    3.71        3.38                      15               21                63.6%
Gustavo Chacin 3.72        3.23                       9               18                52.9%    
Ted Lilly      5.56        6.00                       1                9                36%
Dave Bush      4.49        4.08                       5               12                50%

Note: a quality start is defined as a start in which a pitcher throws at least six innings while allowing three earned runs or fewer. It was created by sporstwriter John Lowe.

I included unearned runs and quality starts in order to encompass a broader picture.

One general trend is that the median is lower than the average, which makes sense considering the aforementioned unequal effect of outliers.

What immediately sticks out is how incredibly low Doc Halladay's earned run median was last season. That means that he gave up 1.29 ER/9IP or lower in more than half his starts last season, which is truly remarkable. The reason his ERA wound up at 2.41 (which is still rather low, mind you) is because of his five non-quality starts. In those five starts, his combined ERA was 7.07. In his 14 quality starts, however, his combined ERA was an incredible 1.26. So, in approximately 3/4 of his starts in 2005, Roy Halladay was next to unbeatable.

Josh Towers is an interesting case because of how many unearned runs he gave up. His run average (both earned and unearned runs) last season was 4.36, which is noticeably worse than his 2005 ERA. That partly accounts for the fact that he posted more quality starts than Gustavo Chacin. If we use RA rather than ERA to calculate quality starts, his total would be 17, or 51.5% of his total starts. Chacin, on the other hand, would still have 18 quality starts, or 52.9% of his total starts, if RA were used.

Gustavo Chacin was awfully unpredictable last season. Only 4 of his 34 starts included single-game ERAs between 3.00 and 5.00. In fact, he had single-game ERAs of 0.00 in five of his starts, but that was counterposed by the six single-game ERAs above 8.50 he posted, as well.

Ted Lilly was reliably putrid last season. In over half his starts, his single-game ERA was higher than 6.00, and in six of his 25 starts, it was higher than 10.00.

David Bush's ERA was noticeably higher than his earned run median because of a few unfortunate starts. On April 18, 2005 against the Red Sox, he gave up 7 ER in 2 innings. Also, on August 8, 2005 against the Tigers he allowed 7 ER in 4.2 innings. Furthermore, on September 16, 2005 against the Yankees, he allowed 6 ER in 2.2 innings. If those three starts are discounted from the 24 he made all season, his ERA drops from 4.49 to a very respectable 3.40.

Obviously, poor performances should not simply be discounted. However, when judging a pitcher's value, it's important to note that his ERA is highly vulnerable to the nasty effects of outliers.