Calculator
Example Data Table
| Pair | Before Score | After Score | Difference |
|---|---|---|---|
| 1 | 85 | 88 | 3 |
| 2 | 88 | 90 | 2 |
| 3 | 90 | 93 | 3 |
| 4 | 86 | 89 | 3 |
| 5 | 91 | 94 | 3 |
Formula Used
The paired sample t test uses differences between matched pairs.
Difference: d = second value - first value
Mean difference: d̄ = sum of differences / n
Standard deviation: sd = sqrt(sum((d - d̄)²) / (n - 1))
Standard error: SE = sd / sqrt(n)
Test statistic: t = (d̄ - μ0) / SE
Degrees of freedom: df = n - 1
Confidence interval: d̄ ± t critical × SE
How to Use This Calculator
- Enter the first matched data column in Sample A.
- Enter the second matched data column in Sample B.
- Keep both columns in the same pair order.
- Select the subtraction direction.
- Enter the hypothesized mean difference. Use zero for most tests.
- Choose the alternative hypothesis.
- Set the confidence level and decimal precision.
- Press Calculate. The result appears above the form.
- Use CSV or PDF download for saved reports.
Understanding Paired Sample T Tests
A paired sample t test checks whether two related measurements differ. The same person, part, site, or subject is measured twice. The data may be before and after values. It may also be two matched methods used on the same item. Because the observations are linked, the test studies the differences inside each pair.
Why This Test Matters
Independent tests ignore the match between rows. That can waste information. A paired test removes much of the person to person variation. It focuses on the change that belongs to each matched record. This makes the method powerful when the pairing is valid. It is useful for training studies, medical follow ups, process checks, finance comparisons, and classroom assessments.
Main Calculation Steps
The calculator first subtracts one value from the other. The chosen direction controls the sign. It then finds the average difference, standard deviation of differences, and standard error. The t statistic compares the observed mean difference with the hypothesized mean difference. The degrees of freedom equal the number of valid pairs minus one. The p value depends on the selected alternative hypothesis.
Reading The Output
A small p value suggests the average paired difference is unlikely under the null hypothesis. A confidence interval shows a range of likely mean differences. If this interval excludes zero, a two sided test at the matching confidence level is usually significant. The effect size dz explains the size of the change in standard deviation units. Larger absolute values mean stronger paired change.
Practical Advice
Use pairs that truly belong together. Do not mix unrelated groups. Keep units consistent in both columns. Check for entry mistakes before trusting the result. Extremely unusual differences can strongly affect the t statistic. For small samples, inspect the differences carefully. The paired t test assumes the differences are approximately normal, not that the original columns are normal. With larger samples, the test is usually more stable. Use the export buttons to save results for reports, homework, audits, or review. Always explain the direction of subtraction, because it controls whether the result is positive or negative. When assumptions are doubtful, compare results with a nonparametric signed rank test and report both methods for transparency clearly.
FAQs
What is a paired sample t test?
It is a test for two related measurements. It checks whether the mean difference between paired observations differs from a chosen hypothesized difference, often zero.
When should I use this calculator?
Use it for before and after scores, matched subjects, repeated measurements, paired lab readings, or two methods tested on the same items.
What does the p value mean?
The p value shows how likely the observed mean difference is under the null hypothesis. Smaller values give stronger evidence against the null hypothesis.
What is the null hypothesis?
The null hypothesis says the true mean paired difference equals the hypothesized value. In most practical cases, that value is zero.
Why must both samples have equal length?
Each value in one sample must match one value in the other sample. Unequal lengths break the pair structure needed for this test.
What is Cohen dz?
Cohen dz is a paired effect size. It divides the mean difference by the standard deviation of differences, giving a standardized change measure.
Does the paired t test require normal data?
It assumes the paired differences are approximately normal. The original Sample A and Sample B columns do not each need separate normal distributions.
Can I download my results?
Yes. After calculation, use the CSV button for spreadsheet work or the PDF button for a simple printable report.