Calculator
Enter matched observations, prepared differences, or summary statistics. The default setting calculates a two-sided 90% confidence interval.
Example Data Table
This example compares a first measurement with a second measurement. The calculator uses second minus first as the difference direction.
| Pair | First value | Second value | Difference |
|---|---|---|---|
| 1 | 76 | 82 | 6 |
| 2 | 80 | 83 | 3 |
| 3 | 70 | 74 | 4 |
| 4 | 65 | 70 | 5 |
| 5 | 88 | 90 | 2 |
| 6 | 92 | 96 | 4 |
| 7 | 75 | 78 | 3 |
| 8 | 84 | 85 | 1 |
| 9 | 79 | 83 | 4 |
| 10 | 68 | 73 | 5 |
Formula Used
A paired confidence interval is calculated from the sample of paired differences. The data are first converted into differences.
For a 90% two-sided interval, α equals 0.10. The calculator uses the Student t distribution with n − 1 degrees of freedom.
How to Use This Calculator
- Select raw pairs, differences only, or summary statistics.
- Keep the confidence level at 90 for a standard 90% interval.
- Choose the direction of the paired difference.
- Enter a null mean difference if you want a paired t test result.
- Click calculate to show the interval below the header.
- Use the CSV or PDF button to save your result.
Paired T Confidence Interval Guide
Purpose
A paired t 90% confidence interval is used when two measurements belong to the same subject, item, or matched unit. It is useful before and after training, left and right measurements, repeated lab readings, and matched experimental designs.
Why Differences Matter
The method starts by converting every pair into one difference. This removes much of the person to person variation. The interval then estimates the true average difference in the population. A 90% level gives a balanced interval. It is narrower than a 95% interval. It is still useful when the study needs moderate confidence and faster screening.
Input Choices
This calculator accepts three input styles. You can enter two matched lists, a single list of differences, or summary statistics. Raw pairs are best when you have original data. Difference mode is useful when differences are already prepared. Summary mode helps when a paper gives only sample size, mean difference, and standard deviation.
Output Details
The output includes the mean difference, standard deviation, standard error, degrees of freedom, critical t value, margin of error, lower bound, and upper bound. It also reports a paired t statistic against a chosen null difference. That helps compare the interval result with a hypothesis test.
Assumptions
A valid paired interval needs matched observations. Each first value must belong to the same unit as the first second value. Missing or mismatched rows can distort the answer. The differences should be reasonably continuous. Strong outliers can widen the interval and shift the mean. For small samples, a quick dot plot of differences is helpful.
Interpretation
Use the interval as an estimate, not a promise. If the full interval is above zero, the average change is positive at the selected confidence level. If it is below zero, the average change is negative. If it crosses zero, the data do not show a clear mean change at this level. Always combine the result with study design, sample quality, and practical importance.
Reporting
Keep the direction consistent. If after minus before is selected, a positive value means an increase after treatment. If before minus after is selected, the sign reverses. Record that choice in reports so readers understand the conclusion. Save exports for audits, lessons, reviewers, and later checks.
FAQs
What is a paired t 90% confidence interval?
It is an estimated range for the true mean difference between two matched measurements. The 90% level means the method captures the true mean difference in about 90% of repeated similar studies.
When should I use a paired t interval?
Use it when each observation in one group is naturally linked to one observation in another group. Common examples include before and after scores, repeated measurements, twins, matched patients, or paired product tests.
What does it mean if the interval includes zero?
If the interval includes zero, the data do not show a clear average paired difference at the selected confidence level. The result may still be practically useful, but statistical evidence is weaker.
Why does this calculator use differences?
A paired method works on the differences within each pair. This controls for variation between subjects or matched units. The interval is then built from the mean and spread of those differences.
Can I enter summary statistics only?
Yes. Use summary mode when you know sample size, mean difference, and standard deviation of differences. This gives the same interval as raw data when those summary values are accurate.
What is the standard error?
The standard error estimates how much the sample mean difference may vary from sample to sample. It equals the standard deviation of differences divided by the square root of sample size.
Does a 90% interval mean 90% of values are inside it?
No. It refers to confidence in the estimation method, not the percentage of individual values covered. The interval estimates the true population mean difference, not the spread of all differences.
Why is the t distribution used?
The t distribution accounts for uncertainty when the population standard deviation is unknown. That is the usual case in paired studies, especially when sample sizes are small or moderate.