Batting Averages

Date: 05/07/99 at 08:11:48
From: Caleb M.
Subject: How to average a batting average

Dear Dr. Math,

I've been wondering how to find a batting average. Do you have to 
average the new hit into all the previous hits to come up with the 
correct average? Same with test scores over the semester. Can you 
tell me?

Caleb M.

P.S. I really like the St. Louis Cardinals and Mark McGwire.

Date: 05/07/99 at 10:01:13
From: Doctor Rob
Subject: Re: How to average a batting average

A batting average is the total number of hits divided by the total
number of official at-bats. 

There are two ways you can compute the batting average when the data 
change. One is to do the division over again. The other is to use the 
previously computed average in the following way. Let H be the total 
number of hits up through yesterday, and h the number of hits today, 
A the total number of at-bats up through yesterday, and a the number 
of at-bats today.

The old batting average was H/A. The new batting average is 
(H+h)/(A+a) = (H/A)*(A/(A+a)) + (h/a)*(a/(A+a)). That means that you 
multiply the old batting average by A/(A+a), and multiply today's 
batting average (which is h/a) by a/(A+a), and add them together.

Example: 

Suppose Mark McGwire had 23 hits in 70 at-bats through yesterday, and 
today he was 2-for-5. Then H = 23, A = 70, h = 2, a = 5, and 
H/A = 0.32857, h/a = .40000. Then A/(A+a) = 70/75 = 14/15, and 
a/(A+a) = 5/75 = 1/15, so the new batting average is 0.32857*14/15 + 
0.40000*1/15 = 0.30667 + 0.02666 = 0.33333, and McGwire has raised his 
batting average by 0.00476. Or you could divide H+h = 25 by A+a = 75, 
and get 0.33333 for his new batting average.

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