Paired t-Test Formula

The Formula

t = \frac{\bar{d} - \mu_{d_0}}{s_d / \sqrt{n}}

When to use: You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after - before) for each student, you eliminate individual variation and focus purely on the change.

Quick Example

5 students' scores before and after tutoring: differences d = +8, +3, +12, +5, +7. \bar{d} = 7, \quad s_d = 3.39, \quad t = \frac{7 - 0}{3.39 / \sqrt{5}} = \frac{7}{1.52} \approx 4.61 \quad (df = 4) Strong evidence that tutoring improved scores (p < 0.01).

Notation

\bar{d} = mean of differences, s_d = standard deviation of differences, n = number of pairs, df = n - 1. Usually H_0: \mu_d = 0.

What This Formula Means

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences d_i = x_{1i} - x_{2i} as a single sample.

You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after - before) for each student, you eliminate individual variation and focus purely on the change.

Formal View

t = \frac{\bar{d} - \mu_{d_0}}{s_d / \sqrt{n}} with df = n - 1 where d_i = x_{1i} - x_{2i} and \bar{d} = \frac{1}{n}\sum d_i

Worked Examples

Example 1

medium
Students' scores before and after tutoring: Before: \{70, 65, 80, 75, 60\}, After: \{75, 70, 85, 80, 70\}. Conduct a paired t-test at \alpha=0.05 to test if tutoring improved scores.

Solution

  1. 1
    Differences d_i = \text{After} - \text{Before}: 5, 5, 5, 5, 10
  2. 2
    \bar{d} = (5+5+5+5+10)/5 = 30/5 = 6; s_d = \sqrt{\frac{\sum(d_i-\bar{d})^2}{n-1}} = \sqrt{\frac{4+1+1+1+16}{4}} = \sqrt{4.25} \approx 2.06
  3. 3
    t-statistic: t = \frac{\bar{d}}{s_d/\sqrt{n}} = \frac{6}{2.06/\sqrt{5}} = \frac{6}{0.921} \approx 6.51
  4. 4
    df=4; t^*_{0.05, 4} = 2.132 (one-tailed); 6.51 > 2.132 → Reject H_0

Answer

t \approx 6.51 > 2.132. Reject H_0. Tutoring significantly improved scores.
A paired t-test analyzes differences between matched pairs (before-after, or twins). By working with differences, we eliminate between-subject variability, increasing power. The test becomes a one-sample t-test on the differences.

Example 2

hard
Explain when to use a paired t-test vs. a two-sample t-test. If shoe comfort is measured on the same subjects wearing Brand A and Brand B, which test is appropriate and why?

Common Mistakes

  • Using a two-sample t-test on paired data—this ignores the pairing and loses statistical power.
  • Computing the mean of each group separately and testing the difference, instead of computing individual differences first and testing those.
  • Forgetting to check conditions: the differences (not the original data) must be approximately normal, or the sample of pairs must be large enough.

Why This Formula Matters

Paired designs are powerful because they eliminate between-subject variability. Before/after studies, twin studies, and crossover trials all use paired analysis.

Frequently Asked Questions

What is the Paired t-Test formula?

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences d_i = x_{1i} - x_{2i} as a single sample.

How do you use the Paired t-Test formula?

You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after - before) for each student, you eliminate individual variation and focus purely on the change.

What do the symbols mean in the Paired t-Test formula?

\bar{d} = mean of differences, s_d = standard deviation of differences, n = number of pairs, df = n - 1. Usually H_0: \mu_d = 0.

Why is the Paired t-Test formula important in Math?

Paired designs are powerful because they eliminate between-subject variability. Before/after studies, twin studies, and crossover trials all use paired analysis.

What do students get wrong about Paired t-Test?

Students use a two-sample t-test when they should use a paired t-test. The giveaway: if the same subjects appear in both groups, or subjects are deliberately matched, it's paired.

What should I learn before the Paired t-Test formula?

Before studying the Paired t-Test formula, you should understand: hypothesis testing, confidence interval, sampling distribution, mean, standard deviation.

← Back to Paired t-Test See All Examples Practice Problems