Why does a paired t-test (when appropiate) result in better variance?

Question

I didn't quite get the wikipedia explanation here:

http://en.wikipedia.org/wiki/Paired_difference_test#Use_in_reducing_variance

I agree that both the unpaired and paired means are the same...then I see how

$\text{var}(\bar{Y_2} - \bar{Y_1})$

will include a covariance term..but what is the alternative variance? Take the paired differences first, and then the variance of the result? And how would that be any different?

Thanks.

Answer 1

Let's say we have the two conditions in Table 1. Each condition has a variance of 4 yielding a pooled variance of 4 as well and we double that to get the variance of the effect, 8. What if they were actually paired values and qualify for a paired t-test? We take the variance of the differences, the variance of the actual effect, which can be seen from the table to be 0 because they're all equal. This is the kind of thing that can happen when you have a paired test and how it can be more sensitive with a smaller standard error.

Table 1.

A1  A2  A1-A2
11   5   6
13   7   6
15   9   6
var(A1) = 4
var(A2) = 4
var(A1-A2) = 0