Matched Pairs (Dependent Samples)
What if the two groups are related? Like "Before & After" measurements on the same person? We use a Paired t-Test.
Dependent vs Independent Samples
- Independent: Group A (Men) vs Group B (Women). No relation between individuals.
- Dependent (Paired): Weight Before Diet vs Weight After Diet for the same people. Or Twins. Or Husbands vs Wives.
The Paired t-Test
We calculate the difference ($d$) for each pair, then test if the average difference ($\bar{d}$) is zero.
$$ t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}} $$
$\bar{d}$ = Mean of differences
$s_d$ = Standard deviation of differences
$n$ = Number of pairs
$\bar{d}$ = Mean of differences
$s_d$ = Standard deviation of differences
$n$ = Number of pairs
Example: Weight Loss Program
We measure 5 people before and after a diet.
| Person | Before | After | Diff ($d$) |
|---|---|---|---|
| 1 | 200 | 190 | -10 |
| 2 | 180 | 175 | -5 |
| 3 | 210 | 200 | -10 |
| 4 | 190 | 188 | -2 |
| 5 | 170 | 165 | -5 |
Mean diff $\bar{d} = -6.4$. Std Dev $s_d = 3.58$.
$$ t = \frac{-6.4 - 0}{3.58 / \sqrt{5}} = \frac{-6.4}{1.6} = -4.0 $$
Critical t ($df=4, \alpha=0.05$) is 2.776. Since $|-4.0| > 2.776$, we reject $H_0$. The diet works!
Test Yourself
Q1: In a paired t-test, what are we actually analyzing?