Paired t-Test

Statistics
process

Also known as: matched pairs t-test, dependent samples t-test

Grade 9-12

View on concept map

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. Paired designs are powerful because they eliminate between-subject variability.

Definition

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.

💡 Intuition

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.

🎯 Core Idea

The paired t-test reduces the problem to a one-sample t-test on the differences. By pairing, you control for subject-to-subject variation, making it easier to detect a treatment effect.

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).

Formula

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

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.

🌟 Why It Matters

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

💭 Hint When Stuck

When comparing two related measurements on the same subjects, use a paired t-test. First, compute the difference d_i = x_{\text{after}} - x_{\text{before}} for each pair. Then calculate \bar{d} and s_d, and compute t = \bar{d}/(s_d/\sqrt{n}). Finally, compare t to the critical value or find the p-value with n-1 degrees of freedom.

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

🚧 Common Stuck Point

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.

⚠️ 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.

Frequently Asked Questions

What is Paired t-Test in Math?

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.

Why is Paired t-Test important?

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 usually 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 Paired t-Test?

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

Next Steps

two sample tests

How Paired t-Test Connects to Other Ideas

To understand paired t-test, you should first be comfortable with hypothesis testing, confidence interval, sampling distribution, mean and standard deviation. Once you have a solid grasp of paired t-test, you can move on to two sample tests.

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