Paired T Test Input Form
Paste values with spaces, commas, semicolons, or new lines.
Example Data Table
This sample illustrates matched observations, such as scores before and after an intervention.
| Pair | Before | After | Difference (After - Before) |
|---|---|---|---|
| 1 | 54 | 59 | 5 |
| 2 | 47 | 50 | 3 |
| 3 | 61 | 65 | 4 |
| 4 | 58 | 62 | 4 |
| 5 | 63 | 68 | 5 |
| 6 | 49 | 53 | 4 |
| 7 | 57 | 60 | 3 |
| 8 | 60 | 64 | 4 |
Formula Used
The paired t test works with differences between matched observations.
Difference for each pair: di = Yi - Xi or the reverse direction you select.
Mean difference: d̄ = Σdi / n
Standard deviation of differences: sd = √[Σ(di - d̄)² / (n - 1)]
Standard error: SE = sd / √n
Test statistic: t = (d̄ - μ0) / SE
Confidence interval: d̄ ± t* × SE
Main assumptions:
- Each value in the first sample is naturally paired with one value in the second sample.
- Pairs are independent from other pairs.
- The distribution of paired differences is roughly normal, especially for small samples.
- Measurements are numeric and recorded on a consistent scale.
How to Use This Calculator
- Enter a label for each matched sample, such as Before and After.
- Paste the same number of values into both text areas.
- Select the difference direction, hypothesis type, significance level, and confidence level.
- Set the hypothesized mean difference, usually 0, then click the calculate button.
- Review the t statistic, p value, confidence interval, effect size, pair table, and graph. Use the CSV or PDF buttons to export the report.
Frequently Asked Questions
1. What does a paired t test compare?
It compares the average difference between two related measurements. Common examples include before-and-after scores, repeated tests, or matched observations from the same subjects.
2. When should I use a paired t test?
Use it when each value in one sample belongs directly with one value in the other sample. The method is designed for repeated measurements or matched pairs.
3. Can the two samples have different lengths?
No. A paired t test requires the same number of observations in both samples because every value must have one matching partner.
4. What does the p value mean?
The p value measures how unusual your observed mean difference would be if the null hypothesis were true. Smaller values indicate stronger evidence against the null.
5. Why is the confidence interval useful?
It gives a plausible range for the true mean difference. The interval also shows the direction and practical size of the observed change.
6. What if the paired differences are not normal?
For small samples with clearly non-normal differences, consider a nonparametric alternative such as the Wilcoxon signed-rank test. Larger samples are often more robust.
7. What happens if every pair changes by the same amount?
Then the difference variance becomes zero. The calculator still reports the result, but the test statistic can become infinite because the standard error collapses to zero.
8. How is this different from an independent t test?
An independent t test compares unrelated groups. A paired t test removes between-subject variation by analyzing within-pair differences instead.