Example Data Table
| Pair |
Before |
After |
After - Before |
| 1 | 72 | 75 | 3 |
| 2 | 68 | 71 | 3 |
| 3 | 91 | 93 | 2 |
| 4 | 84 | 86 | 2 |
| 5 | 77 | 80 | 3 |
| 6 | 69 | 73 | 4 |
| 7 | 88 | 90 | 2 |
| 8 | 82 | 85 | 3 |
Formula Used
The calculator first converts each pair into a difference. Use \(d_i = x_{2i} - x_{1i}\), unless the reverse direction is selected.
Mean difference: \( \bar d = \frac{\sum d_i}{n} \)
Standard deviation: \( s_d = \sqrt{\frac{\sum(d_i-\bar d)^2}{n-1}} \)
Standard error: \( SE = \frac{s_d}{\sqrt n} \)
Test statistic: \( t = \frac{\bar d-\mu_0}{SE} \)
Degrees of freedom: \( df = n - 1 \)
Confidence interval: \( \bar d \pm t_{critical} \times SE \)
Effect size: \( d_z = \frac{\bar d}{s_d} \)
How to Use This Calculator
Choose raw data when you have every matched pair. Enter one pair per line. Use commas, spaces, tabs, or semicolons between values.
Choose summary statistics when you already know sample size, mean difference, and standard deviation of differences.
Select the difference direction carefully. It controls the sign of the mean difference and t statistic.
Set the hypothesized mean difference. Most paired t tests use zero. Choose the alternative hypothesis before reading the p value.
Press Calculate. The result appears above the form. Use the CSV and PDF buttons to save the output.
About the Matched Paired T Test
A matched paired t test compares two related measurements. It is useful when each subject gives two values. Common cases include before and after scores. It also fits twin studies, repeated lab tests, and matched control designs. The test does not compare two separate groups. It studies the list of differences inside each pair.
Why Pairing Matters
Pairing removes much of the noise between subjects. Each person, item, or location acts as its own control. This often gives better precision than an independent sample test. The method works best when pairs are planned before data collection. It should not be used to force a relationship after results are known.
Main Assumptions
The differences should be measured on a numeric scale. The pairs should be independent from other pairs. The distribution of differences should be roughly normal. Small samples need closer checking. Larger samples are more forgiving. Extreme outliers can change the answer. Always review the difference values before trusting the test.
What the Results Mean
The mean difference shows the average paired change. A positive value means the first column is larger on average. A negative value means the second column is larger. The t statistic compares the observed mean difference to its standard error. The p value estimates how unusual the result is under the null claim.
Confidence and Effect Size
A confidence interval gives a likely range for the true mean difference. A narrow interval suggests better precision. A wide interval shows more uncertainty. Cohen's dz gives a standardized paired effect. It divides the mean difference by the standard deviation of differences. This helps compare results across different units.
Practical Use
Use raw paired data when possible. It gives the strongest audit trail. Summary mode is helpful for reports and textbooks. Check the sample size, mean difference, and standard deviation carefully. Select the alternative hypothesis before calculating. Then compare the p value with alpha. Report the test statistic, degrees of freedom, p value, confidence interval, and effect size together. Good reporting also states the measurement direction. Readers must know which value was subtracted first. This calculator uses first value minus second value. Change the column order when your research question needs it.
FAQs
What is a matched paired t test?
It is a test for two related measurements. It checks whether the average difference between paired values is different from a hypothesized value.
When should I use this calculator?
Use it for before and after data, repeated measurements, matched subjects, paired samples, or the same item measured under two conditions.
What is the usual null hypothesis?
The usual null hypothesis says the true mean difference equals zero. You can change this value when testing a specific expected difference.
What does a small p value mean?
A small p value means the observed paired difference would be unusual if the null hypothesis were true. Compare it with your alpha level.
Why does direction matter?
Direction controls the sign of each difference. It affects the mean difference, t statistic, and one-tailed p value interpretation.
Can I use summary statistics?
Yes. Enter sample size, mean difference, and standard deviation of differences. Use raw data when possible for better checking.
What assumptions should I check?
Pairs should be related within pairs and independent across pairs. Difference values should be roughly normal, especially with small samples.
What is Cohen's dz?
Cohen's dz is a paired effect size. It divides the mean difference by the standard deviation of paired differences.