Runs Batted In and Earned Run Average

Date: 10/26/2000 at 12:21:40
From: Tara Kuty
Subject: Runs Batted In

How do you calculate RBI for a player on a baseball team?

Also, how do you find an ERA for a pitcher?

Date: 10/27/2000 at 12:01:30
From: Doctor TWE
Subject: Re: Runs Batted In

Hi Tara - thanks for writing to Dr. Math.

Runs Batted In (or RBIs) is a "counting stat" like home runs or hits. 
Every time a batter gets a hit or makes an out and a runner scores on 
the play, (s)he gets an RBI - except in a few special cases. 

  A batter is NOT awarded an RBI when (s)he bats into a double play 
(even if a run scores on the play). 

  A batter is NOT awarded an RBI on an error unless, in the official 
scorekeeper's opinion, the run would have scored even without the 
error. For example, if with less than two outs, the batter hits a deep 
fly ball to the outfield and the outfielder bobbles and drops the ball 
(the batter is safe on first base), the official scorekeeper may 
charge the outfielder with an error, but award the batter an RBI, 
deeming that the play would have been a sacrifice fly if the error had 
not occurred.

A pitcher's Earned Run Average (or ERA) is the average number of 
earned runs the pitcher gives up per full game (9 innings). It is 
calculated as:

     ERA = 9*ER / IP

where ER is the number of earned runs the pitcher has allowed and IP 
is the number of innings the pitcher has pitched. For example, Andy 
Pettitte pitched 13 2/3 innings and allowed 3 earned runs in his two 
starts in the 2000 World Series, so his ERA for the series was:

     ERA = 9*3 / (13+2/3) = 1.9756... or 1.98

Al Leiter pitched 15 2/3 innings and allowed 5 earned runs in his two 
starts in the World Series, so his ERA for the series was:

     ERA = 9*5 / (15+2/3) = 2.8723... or 2.87

A run is considered unearned (and thus doesn't count against the 
pitcher's ERA) when an error occurs in an inning, and the run would 
not have scored had the error not occurred. For example, suppose the 
following sequence occurs in an inning:

     Batter 1: Single
     Batter 2: Strikeout
     Batter 3: Safe at first on an error (batter 1 to 2nd base)
     Batter 4: Home Run (1, 3, and 4 score)
     Batter 5: Triple
     Batter 6: Sac Fly (5 scores)
     Batter 7: Walk
     Batter 8: Home Run (7 and 8 score)

Six runs have scored (batters 1, 3, 4, 5, 7, and 8), but let's 
evaluate which are earned and which are unearned. The easiest way to 
do this is to "replay" (on paper) the inning, replacing the error with 
an out. Here's what we would have:

     Batter 1: Single
     Batter 2: Strikeout
     Batter 3: Out (instead of error)
     Batter 4: Home Run (#1 and #4 score)
     Batter 5: Triple
     Batter 6: Fly out (Since this is the 3rd out, 5 doesn't score)
     Batters 7 and 8: never got to bat...

Two runs would have scored in this scenario, so 2 of the 6 runs are 
earned, and the other 4 are unearned. 

Note that any run scored by a batter who reaches base due to an error 
is automatically unearned. Thus, batter 3's run is unearned no matter 
what else happens. Note that because batter 6's fly out would have 
been the third out of the inning, batter 5 would not have scored on a 
"sac fly." So that run, too, is unearned. (The batter still gets 
credit for the sac fly and the RBI, but the run is not charged against 
the pitcher's ERA.) Finally, since batter 6's fly out would have ended 
the inning, batter 8's home run would not have happened, so that run - 
and all subsequent runs in the inning - are also unearned. If the 
pitcher gives up additional runs in an inning after what would have 
been the third out, all of them are unearned. (We don't assume that 
the batter would have still hit the home run next inning because the 
game situation would have changed; the pitcher would have had a chance 
to rest a bit, the score might be different, etc.)

If a relief pitcher enters the game, any runners on base at the time 
(s)he enters the game are charged against the pitcher who allowed them 
to reach base, and whether they are considered earned or unearned is 
determined as above. However, the slate is considered "clean" for the 
relief pitcher entering the game. Any runners the relief pitcher 
allows to reach base and score are considered earned unless more 
errors occur. So if, for example, a relief pitcher entered the game 
after batter 7 in our example above, 7's run would still be considered 
unearned (charged as a run allowed by the original pitcher, but not an 
earned run), but batter 8's home run would be charged as an earned run 
against the relief pitcher.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/