Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I have been recently intrigued by a seemingly simple problem: How to compare the averages of two groups with different sizes.

For example: Suppose you have a driver A who wins 100 out of 200 races, and a driver B who wins 1 out 2 races. It is clear that although the average is the same, driver A's achievement is less likely to occur (so it can be considered more valuable?).

I worked out a solution based on the Binomial distribution with the MLE for each driver as the parameter.

Pr(X = 100|1/2) = 0.0563 (N = 200)

Pr(X = 1|1/2) = 0.5 (N = 2)

The results matches my expectation as it indicates that the first event is less likely to occur. The problem however comes when I have a situation like this:

Driver A wins 65 out of 161 races.

Driver B wins 68 out of 244 races.

By evaluating the probabilities in the same way I got:

Pr(X = 65|65/161) = 0.0640

Pr(X = 68|68/244) = 0.0569

Intuitively, I reject this result because it is clear that driver A did a better job (because both drivers won almost the same number of races). I know it is probably because of the parameter I am using, but I don't know how to fix it.

Any thoughts?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Comparing two averages with different group sizes

**Physics Forums | Science Articles, Homework Help, Discussion**