# Probability that A is smaller than B?

1. Sep 30, 2015

### wavingerwin

Given two data sets A and B, we can, say, conduct ANOVA to see if the average is statistically different.

Is there a way to determine what is the probabilty that A is smaller than B?

Let's say that we can NOT assume anything about A and B e.g. if they follow a normal distribution.

2. Sep 30, 2015

### Hornbein

? Do you mean the probability that the average of A is smaller than the average of B? If we know nothing at all, then we can conclude nothing at all.

3. Sep 30, 2015

### wavingerwin

Yes, something to that effect. Is it possible to say "A is smaller than B X% of the time" ?

4. Sep 30, 2015

### Hornbein

If we know that A and B are iid (independent identically distributed) random variables then we can say that avg of A is smaller or equal to avg of B at least 50% of the time.

Proof: Since they are iid, P[ avg(A)<= avg(B) ] = P[avg(B) <= avg(A)].

P[ avg(A)<= avg(B) ] + P[avg(B) <= avg(A)] >=1

P[ avg(A)<= avg(B) ] + P[ avg(A)<= avg(B) ] >=1

2P[ avg(A)<= avg(B) ] >=1

P[ avg(A)<= avg(B) ] >=1/2

5. Sep 30, 2015

### pwsnafu

You can do non-parametric analysis, such as the Wilcox test.

6. Oct 1, 2015

### Staff: Mentor

I think Wilcoxon signed rank requires the samples be from a single population. Does that fit what we are looking at here?

7. Oct 1, 2015

### pwsnafu

I linked to wrong test. Here's the two sample version.

8. Oct 1, 2015

### Staff: Mentor

Much better - thank you. I thought maybe I had lost my last remaining brain cell.