Paired T-Test Calculator

What Is a Paired T-Test?

The paired t-test (dependent samples t-test) compares means from two related measurements on the same subjects.

Question answered: "Is there a significant change between two conditions for the same subjects?"

Examples:

• Before vs. after treatment on the same patients
• Test vs. retest scores for the same students
• Left eye vs. right eye measurements
• Monday vs. Friday performance for the same workers

Why "Paired"?

Each observation in one condition is matched with a specific observation in the other:

The test analyzes the differences, not the raw values.

The Hypotheses

Null hypothesis (H₀): μ_d = 0 (mean difference is zero)

Alternative hypothesis (H₁):

• Two-tailed: μ_d ≠ 0 (there is a difference)
• Right-tailed: μ_d > 0 (positive change)
• Left-tailed: μ_d < 0 (negative change)

The Test Statistic

$t = \frac{\bar{d}}{s_d / \sqrt{n}}$

Where:

• $\bar{d}$ = mean of differences
• $s_d$ = standard deviation of differences
• $n$ = number of pairs

$df = n - 1$

This is essentially a one-sample t-test on the differences.

Example Calculation

Scenario: Blood pressure before and after a relaxation intervention.

Data: 15 patients

Summary: d̄ = -6.5, s_d = 4.2

Calculation: t = -6.5 / (4.2 / √15) t = -6.5 / 1.08 t = -6.02

P-value: For df = 14 (two-tailed), p < 0.001

Conclusion: Blood pressure decreased significantly after the intervention.

Why Paired Tests Are More Powerful

Paired designs control for individual differences:

Unpaired: Between-subject variability obscures effect Paired: Each subject serves as own control

The difference scores often have much smaller variance than the raw scores, making real effects easier to detect.

Applications

Pre-Post Studies

Measuring the same subjects before and after intervention.

Crossover Trials

Each subject receives both treatments (in different periods).

Matched Pairs

Pairs matched on confounding variables (twins, matched controls).

Repeated Measures

Same subjects measured under different conditions.

Within-Subject Experiments

Comparing responses to different stimuli within the same person.

Assumptions

Paired Observations

Each measurement in one group corresponds to a specific measurement in the other.

Normality of Differences

The differences should be approximately normally distributed, OR n should be large (n > 30).

Check with histogram or QQ-plot of differences.

Independence of Pairs

Different pairs should be independent (one subject's difference shouldn't affect another's).

When Not to Use Paired T-Test

Different subjects in each group: Use two-sample t-test.

More than two conditions: Use repeated measures ANOVA.

Non-normal differences with small n: Use Wilcoxon signed-rank test.

Effect Size: Cohen's d

For paired data:

$d = \frac{\bar{d}}{s_d}$

In the example: d = -6.5 / 4.2 = -1.55 (large effect)

Confidence Interval for Mean Difference

$CI = \bar{d} \pm t^* \times \frac{s_d}{\sqrt{n}}$

95% CI for example: -6.5 ± 2.14 × 1.08 = (-8.8, -4.2)

The interval excludes 0, confirming significant decrease.

Carryover Effects

In crossover designs, order matters. If Treatment A affects response to Treatment B:

• Include washout periods
• Randomize order
• Test for carryover effects
• Consider parallel design instead

Paired vs. Independent: Key Differences

Common Mistakes

Using Independent Test for Paired Data

Ignoring pairing wastes information and reduces power.

Ignoring Carryover in Crossover

Treatment order effects can bias results.

Assuming Normality of Raw Scores

What matters is normality of differences, not original measurements.

Reporting Results

APA format:

"A paired-samples t-test revealed a significant decrease in blood pressure following the intervention, t(14) = -6.02, p < .001, d = -1.55. Mean blood pressure decreased by 6.5 mmHg (SD = 4.2)."

Related Calculations

• Two-Sample T-Test — Independent groups
• One-Sample T-Test — Compare to value
• Wilcoxon Signed-Rank Test — Non-parametric paired
• Repeated Measures ANOVA — More than two conditions
• CI for Difference in Means — Interval estimate

Summary

The paired t-test compares two related measurements on the same subjects by analyzing the differences. This design controls for individual variability, increasing power to detect effects. The test is equivalent to a one-sample t-test on the differences, testing whether the mean difference is zero. Assumptions include normality of differences (not raw scores) and independence of pairs. Report the mean difference, standard deviation of differences, t-statistic, p-value, and effect size for complete interpretation.