Same 2 calculations 2 different results?

[Riis]

Why is ((16/41)+(32/68))/2 not the same as (16+32)/(41+68)??

There is a difference on 0,00995? 1%

 

[tkhunny]

Because it's not?

No doubt you can find examples that are as close as you like. Some are even exact. Try 6/3 and 4/2.

This is of no consequence. In general, it's just not the case that they are the same.

It is a common error to try to average averages. It just doesn't work as hoped.

 

[pka]

If your question is \(\dfrac{\frac{16}{41}+\frac{32}{68}}{2}\)
then that equals \(\dfrac{16}{82}+\dfrac{32}{136}\)

 

[Riis]

Because it's not?

No doubt you can find examples that are as close as you like. Some are even exact. Try 6/3 and 4/2.

This is of no consequence. In general, it's just not the case that they are the same.

It is a common error to try to average averages. It just doesn't work as hoped.

Thanks

 

[Steven G]

1/2 + 1/2 = 1

Using your method, 1/2 + 1/2 = 2/4 which reduces to 1/2. So adding 1/2 to 1/2 does nothing? I guess 1/2 = 0??

 

[tkhunny]

I like the batting average example.

You have two players. One has the following, 285/1000 and the other 4/4. First, a little thinking. The pair's average batting average should be in the neighborhood of the more prolific player. The player who barely gets in the game should have very little influence.

If we average the averages (your first method), we get (285/1000 + 4/4)/2 = 0.643 -- This should not make sense to anyone.

If we examine the totals, both hits and at-bats (your second method), we get (285+4)/(1000+4) = 289/1004 = 0.288 -- Much more reasonable.