Playoff Bullpen ERA Rank by WS Champion:
2021: 3rd
2020: 9th
2019: 5th
2018: 2nd
2017: 7th
2016: 7th
2015: 4th
2014: no data available
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Ehhh, from this data, all I see is that to win the WS, your bullpen merely shouldn't suck. Those two 7th-best bullpens in '16 and '17 were out of 10 playoff teams.
This data tells me that bullpen ERA isn't especially vital to winning the WS. Now, it's not super safe to proclaim that, as I have said before, this type of data isn't really useful unless we compare it to other aspects vs team success.
So let's look at overall team pitching ERA:
2021: 4th
2020: 4th
2019: 5th
2018: 4th
2017: 5th
2016: 5th
2015: 4th
2014: 2nd
See, here we have consistency. That makes me think this data point is more valid, regardless of whether the average ranking is either or lower than the bullpen ERA number. The bullpen ERA rankings are all over the place. The overall playoff pitching ERA, while worse than I would have guessed, is at least consistent.
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2021: 3rd
2020: 2nd
2019: 5th
2018: 2nd
2017: 6th
2016: 2nd
2015: 1st
2014: 6th
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Kind of all over the place, like bullpen ERA, huh? Well now let's see the average for each (2021-2015):
5.29 Bullpen ERA
4.43 Overall pitching
3.00 Hitting (OPS)
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I would not take away from this that bullpen ERA is the least important aspect, just based on this. But look at it. It's definitely not the most important. It's not even close. The data does not support it. It suggests the opposite, but this is fast-and-dirty, so I don't think any meaningful proclamations can be made from it.
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If I did want to make any tentative proclamations, it'd be these:
-fewer runs are scored in playoff games than regular season games
-bullpen ERA is important
-overall pitching is more important
-in a run-scarce environment, each run scored and run prevented is more important than in a higher run environment
-you have to score at least 1 run in order to win a game (this may be a biggie)