Paired t Test Calculator

Calculator

Data mode

Difference direction

Alternative hypothesis

Hypothesized mean difference

Alpha level

Confidence level percent

Missing row rule

Decimal places

Summary pair count

Summary mean difference

Summary SD of differences

Raw paired data

Enter one pair per line. You may separate values with commas, spaces, tabs, semicolons, or pipes.

Example data table

Pair	Before score	After score	Difference
1	82	88	6
2	76	79	3
3	90	93	3
4	68	72	4
5	74	78	4

Formula used

The calculator first creates paired differences. With the default option, each difference is second measure minus first measure.

d_i = X_{2i} - X_{1i}

mean difference = sum(d_i) / n

s_d = sqrt(sum((d_i - mean difference)^2) / (n - 1))

SE = s_d / sqrt(n)

t = (mean difference - hypothesized difference) / SE

df = n - 1

CI = mean difference +/- t critical * SE

Cohen dz = mean difference / s_d

How to use this calculator

Choose raw data when you have both paired columns.
Paste one matched pair on each line.
Select the direction for the paired difference.
Enter the hypothesized mean difference. Zero is common.
Choose alpha, confidence level, and alternative hypothesis.
Press Calculate to show the result above the form.
Use CSV or PDF to save the current report.

Paired t test overview

A paired t test compares two measurements from the same subject. It is used when observations are naturally matched. Common cases include before and after scores, left and right side readings, repeated lab trials, and matched product tests. The method studies the differences inside each pair, not the two columns separately.

Data and input choices

This calculator accepts raw paired data or summary statistics. Raw data is preferred because it allows checks for pair count, missing rows, sample means, standard deviations, and correlation. Summary mode is useful when a report already gives the mean difference and its standard deviation.

Hypothesis meaning

The test asks whether the average paired difference equals a hypothesized value. Most studies use zero. A positive difference may mean improvement, gain, or higher second readings. A negative value may show decline or lower second readings. The selected direction controls that meaning, so label your variables carefully.

Test statistic

The t statistic is the observed mean difference minus the hypothesized difference, divided by the standard error. The standard error uses the sample standard deviation of the paired differences and the square root of the pair count. Degrees of freedom are one less than the number of complete pairs.

Tail selection

A two tailed test checks for any change. A right tailed test checks whether the mean difference is greater than the hypothesized value. A left tailed test checks whether it is smaller. Choose the alternative before reading the p value.

Intervals and effect size

Confidence intervals show a practical range for the true mean paired difference. If a two sided interval excludes zero, the same alpha level usually gives a significant two sided result. The interval also helps judge importance, not only significance.

Effect size is reported with Cohen's dz. It divides the mean paired difference by the standard deviation of the differences. When raw values are supplied, an average standard deviation effect size is also shown. These values help compare results across scales.

Assumptions and review

Good paired analysis depends on suitable data. Pairs should be related by design. Differences should be roughly normal, especially for small samples. Large outliers can change the result. Always inspect the differences, confirm units, and explain the study design. Export the report when you need a record for review, teaching, or documentation. State limitations clearly in final notes.

FAQs

What is a paired t test?

It is a hypothesis test for two related measurements. It tests the mean of the paired differences, not two independent group means.

When should I use this calculator?

Use it for before and after data, matched subjects, repeated measurements, or paired instruments where each row belongs together.

What does the p value mean?

The p value shows how unusual the observed mean difference is if the null hypothesis is true. Smaller values give stronger evidence against the null.

What is the usual null hypothesis?

The usual null states that the true mean paired difference equals zero. You may change it when testing against another target difference.

What direction should I choose?

Choose the direction that matches your research question. The default reports after minus before, which is common for improvement studies.

Can I use summary data?

Yes. Enter pair count, mean paired difference, and standard deviation of differences. Raw data gives more checks and extra effect details.

What assumptions matter most?

The pairs must be related by design. The paired differences should be roughly normal, especially when the sample is small.

What is Cohen dz?

Cohen dz is a paired effect size. It divides the mean paired difference by the standard deviation of the paired differences.