Enter matched observations or summary statistics
Use the same row for each matched pair. Summary mode requires the standard deviation of the paired differences.
Example data table
These eight matched scores use the second measurement minus the first measurement.
| Pair | Before | After | Difference |
|---|---|---|---|
| 1 | 82 | 86 | 4 |
| 2 | 76 | 80 | 4 |
| 3 | 91 | 94 | 3 |
| 4 | 68 | 72 | 4 |
| 5 | 74 | 79 | 5 |
| 6 | 88 | 92 | 4 |
| 7 | 79 | 81 | 2 |
| 8 | 85 | 89 | 4 |
Formula used
For each pair, compute a difference: di = secondi − firsti. Reverse the order when you select the other difference direction.
Here, d̄ is the mean difference, sd is the sample standard deviation of differences, n is the pair count, and t is the critical t value. A one-sided interval uses tconfidence, n − 1.
How to use this calculator
- Choose raw paired data or summary statistics.
- Name both measurements and add units when helpful.
- Choose the difference direction before interpreting signs.
- Select a confidence level and interval type.
- Enter matched values on matching lines, or add summary values.
- Calculate the interval, then inspect its limits and precision.
- Download the result as CSV or PDF for documentation.
Paired confidence intervals explained
A paired t confidence interval estimates the average change between two related measurements. The observations must come from the same people or items. Common designs include before-and-after studies, repeated laboratory tests.
The calculator first creates one difference for every pair. You choose whether that difference is after minus before or before minus after. Direction changes the sign, not the size of uncertainty. A positive result reflects the direction you selected.
Next, the calculator measures variation among those differences. The standard deviation describes how differently pairs changed. The standard error equals the standard deviation divided by the square root of the pair count. A smaller standard error produces a narrower interval.
Reading the interval
The t distribution supplies the critical value. Its degrees of freedom equal the number of pairs minus one. Higher confidence levels use larger critical values. Therefore, a 99 percent interval is wider than a 95 percent interval. One-sided intervals may be useful.
The interval is mean difference plus or minus the critical value times standard error. It gives a range for the population’s average paired difference. A two-sided 95 percent interval follows a repeated-sampling coverage rule. It does not mean there is a 95 percent probability that one fixed parameter lies inside this completed interval.
Inspect the lower and upper limits carefully. An interval excluding zero supports a nonzero average difference at the matching two-sided significance level. An interval containing zero does not prove no effect exists. It shows that the data remain compatible with zero under the chosen confidence level.
Data checks and reporting
Use raw paired values whenever possible. They allow checking pair alignment, omitted records, and unusual differences. The summary mode is useful when only the pair count, mean difference, and difference standard deviation are available. Do not enter separate group standard deviations in summary mode. Paired analysis needs the standard deviation of differences.
The method assumes independent pairs and approximately normal differences. The individual before and after columns need not be normal. With small samples, inspect a difference plot and investigate extreme values. Keep the matching rule consistent. Do not mix unrelated observations.
Report the confidence level, direction, sample size, interval, and units. Also report the mean difference and standard deviation of differences. These details make results easier to verify. The exports preserve the displayed values for review or sharing.
Frequently asked questions
1. What is a paired t confidence interval?
It estimates a plausible range for the population mean difference between matched observations. It analyzes the within-pair changes rather than treating both measurements as independent groups.
2. When should I use paired data?
Use paired data when each first measurement has a specific matching second measurement. Examples include the same participant before and after treatment, matched twins, or repeated machine readings.
3. Why does difference direction matter?
The direction controls the sign of the mean difference and interval. Switching it reverses positive and negative values, but it does not change the interval width, standard error, or evidence strength.
4. What does an interval containing zero mean?
At the selected confidence level, the observed data remain compatible with no average change. This is not proof that the true difference is zero or that the measurements are identical.
5. Can I use separate group standard deviations?
No. Summary mode needs the standard deviation of the pairwise differences. Separate standard deviations discard the matching information needed for a paired t interval.
6. What are the degrees of freedom?
For a standard paired t interval, degrees of freedom equal n minus one, where n is the number of complete matched pairs used in the calculation.
7. Why is a higher confidence interval wider?
Higher confidence requires a larger critical t value. The margin of error grows, so the interval must become wider to provide the requested repeated-sampling coverage.
8. What is Cohen dz?
Cohen dz divides the mean paired difference by the standard deviation of differences. It is a standardized effect size for paired designs. Interpret it using subject knowledge, not universal thresholds alone.
9. Are the original measurements required to be normal?
No. The key distributional condition concerns the differences between paired values. With small samples, inspect those differences for severe skewness or extreme outliers.
10. Can I use a one-sided interval?
Yes, when a directional lower or upper bound was scientifically planned before examining the data. Do not choose one-sided reporting merely because a two-sided result was inconvenient.
11. What do the download buttons include?
CSV and PDF exports include the selected settings, calculated statistics, interval, and paired differences when raw data were used. They support transparent reporting and later checking.