Paired T Test Mean Confidence Interval Calculator

Calculator Input

Input mode

Confidence level (%)

Hypothesized mean difference

Alternative test

Difference direction

Decimal places

Summary pair count

Summary mean difference

Summary standard deviation

Sample one values

Sample two values

Example Data Table

Pair	Before	After	After minus before
1	82	86	4
2	79	83	4
3	91	95	4
4	88	90	2
5	76	80	4
6	85	87	2

Formula Used

For each pair, calculate d = sample two - sample one, unless you choose the reverse direction.

mean difference = sum(d) / n

standard error = standard deviation of d / sqrt(n)

confidence interval = mean difference ± t critical × standard error

t statistic = (mean difference - hypothesized difference) / standard error

Cohen dz = mean difference / standard deviation of d

How to Use This Calculator

Select raw data or summary statistics.
Enter paired values in the same order for both samples.
Choose the confidence level and difference direction.
Enter a hypothesized difference if you want a test.
Press Calculate to see the result above the form.
Use the CSV or PDF button to save the report.

Paired T Tests for Matched Mean Change

A paired t test studies two related measurements. The values may come from people after training. They may also come from machines, twins, stores, or trials. The key input is the difference inside each pair. That makes the method different from an independent two sample test.

This calculator focuses on the confidence interval for the mean difference. It also reports the test statistic, standard error, margin of error, degrees of freedom, p value, and effect size. These details help you judge precision and importance.

Why the Paired Method Matters

Pairing removes variation that belongs to each subject or matched unit. For example, a patient may have a high baseline score. Another patient may have a low baseline score. The paired method compares each person with their matching result. This can make the estimate more sensitive when the pairing is valid.

The method works best when pairs are planned before analysis. Each first value must match one second value. Missing or crossed pairs can distort the result. The differences should be roughly normal when the sample is small. Large samples are forgiving, but outliers still matter.

Understanding the Interval

The confidence interval gives a likely range for the true average difference. A narrow interval shows higher precision. A wide interval shows more uncertainty. If the interval is fully above zero, the average change is positive at that confidence level. If it is fully below zero, the average change is negative. If it crosses zero, the data do not clearly separate the mean difference from no change.

Use this tool for teaching, audits, quality checks, and business tests. Enter raw paired values when possible. Use summary mode when you already know the mean difference, standard deviation, and sample size. Export results for records, reports, or review. Always inspect the pair differences before making a final decision.

Good Reporting Practice

Report the sample size, mean difference, confidence level, interval, and units. Mention the pairing design. State which order you used for the difference. For example, after minus before has a different sign than before minus after. Clear wording prevents wrong conclusions. Also note outliers, because one extreme pair can shift the mean and enlarge the interval.

FAQs

What is a paired t confidence interval?

It is a range for the true mean of paired differences. Each difference comes from two linked observations, such as before and after values for the same subject.

When should I use this calculator?

Use it when every value in one sample has a natural match in the other sample. Common cases include repeated tests, matched subjects, and before after studies.

What does the mean difference mean?

It is the average of all pair differences. Its sign depends on the chosen direction, so check whether the calculator subtracts first from second or second from first.

Why is degrees of freedom n minus one?

The paired test analyzes one list of differences. The sample standard deviation uses one estimated mean, so the degrees of freedom are pair count minus one.

Can I use summary statistics?

Yes. Enter the number of pairs, mean difference, and standard deviation of differences. Do not enter separate sample deviations for the two original columns.

What if the interval includes zero?

An interval crossing zero suggests the average paired change is not clearly different from zero at the selected confidence level.

What is Cohen dz?

Cohen dz is the paired mean difference divided by the standard deviation of differences. It gives a standardized effect size for matched data.

Are outliers important here?

Yes. Extreme pair differences can change the mean, standard deviation, margin of error, and p value. Review the differences before reporting results.