Paired Data and the Paired t-Test

Paired t Procedures

By the end of this lesson, you will be able to identify paired data, compute the differences, and apply one-sample t procedures to draw conclusions about a mean difference.

What Is Paired Data?

Paired data arise when:

1. Two measurements are taken on each individual (before/after, left/right, two treatments applied to the same subject), OR
2. Each individual in one group is matched to a specific individual in another group based on similar characteristics

Key insight: Once you form the differences, a paired design becomes a one-sample problem — you apply one-sample t procedures to the differences.

Why Pair?

Pairing removes individual-to-individual variability that would otherwise appear as random noise. This makes it easier to detect a real treatment effect.

Example: Testing whether a new drug reduces blood pressure. If you use two separate groups, differences in baseline blood pressure between the groups muddy the results. If you measure each person before and after, only the within-person change matters.

The Paired t Test

Parameter: $\mu_d$ = mean of the differences in the population

Hypotheses:

• $H_0: \mu_d = 0$ (no mean difference)
• $H_a: \mu_d \neq 0$, $> 0$, or $< 0$ (depending on the research question)

Steps:

1. Compute $d_i = x_{i,\text{after}} - x_{i,\text{before}}$ (or whichever order makes sense)
2. Find $\bar{d}$ and $s_d$ from the differences
3. Apply a one-sample t test: $t = \dfrac{\bar{d} - 0}{s_d / \sqrt{n}}$, df = n − 1

Worked Example

A sleep researcher tests whether a relaxation technique improves sleep duration. 8 volunteers record sleep hours before and after the technique:

Person	Before	After	Difference (After − Before)
1	6.5	7.2	+0.7
2	7.0	7.5	+0.5