Calculator Input
Example Data Table
This example compares matched scores from the same subjects under two conditions.
| Pair | Condition A | Condition B | A − B |
|---|---|---|---|
| 1 | 82 | 78 | 4 |
| 2 | 76 | 74 | 2 |
| 3 | 91 | 87 | 4 |
| 4 | 88 | 83 | 5 |
| 5 | 79 | 80 | -1 |
| 6 | 95 | 91 | 4 |
| 7 | 84 | 81 | 3 |
| 8 | 90 | 86 | 4 |
Formula Used
A correlated t test is also called a paired samples t test. It studies the mean of paired differences. First, the calculator creates each difference:
d = X1 − X2
Then it finds the mean difference, sample standard deviation, standard error, t statistic, and degrees of freedom.
t = (d̄ − μ0) / (sd / √n)
df = n − 1
The two tailed p value is calculated from the t distribution. The confidence interval is:
d̄ ± t critical × standard error
Cohen's dz is calculated as d̄ / sd.
Effect size r is calculated as √(t² / (t² + df)).
How to Use This Calculator
- Enter the first matched sample in the Sample A box.
- Enter the second matched sample in the Sample B box.
- Keep both lists in the same paired order.
- Choose the difference direction.
- Enter the hypothesized mean difference, usually zero.
- Set alpha and confidence level.
- Press the calculate button.
- Use the CSV or PDF buttons to export results.
About the Two Tailed Correlated T Test
Purpose
A two tailed correlated t test compares two related measurements. The same person, item, class, machine, or subject appears twice. The goal is to test whether the average paired difference differs from a chosen null value. That value is usually zero.
When It Fits
Use this test for before and after studies. Use it for matched treatment results. Use it when two scores come from the same unit. The test is stronger than an independent t test when pairing is real. Pairing removes subject level noise from the comparison.
What the Result Means
The calculator reports the t statistic, degrees of freedom, and two tailed p value. A small p value suggests that the observed mean difference is unlikely under the null hypothesis. The result is significant when the p value is less than or equal to alpha.
Confidence Interval
The confidence interval gives a practical range for the true mean paired difference. If the interval excludes the null difference, the two tailed test usually becomes significant at the matching alpha level. The interval also shows direction and size.
Effect Size
Statistical significance does not always mean practical importance. Cohen's dz describes the mean difference in standard deviation units. The effect size r gives another simple scale. Larger absolute values suggest stronger paired change. Always interpret effect size with the study field, measurement scale, and sample quality.
Data Quality
The paired t test assumes numeric paired data. It also assumes that paired differences are reasonably normal. The raw scores do not need to be identical in spread. The differences are the main focus. Outliers can strongly affect the mean, standard deviation, and p value.
Good Practice
Check the pairing before calculation. Do not sort one column alone. Review the difference direction. Report the sample size, t statistic, degrees of freedom, p value, confidence interval, and effect size. These details make the analysis easier to verify and reuse.
Frequently Asked Questions
1. What is a two tailed correlated t test?
It is a paired samples test. It checks whether the average difference between two related measurements differs from a chosen null value.
2. When should I use this calculator?
Use it for before and after scores, matched subjects, repeated measures, paired trials, or two related observations collected from the same unit.
3. What does two tailed mean?
Two tailed means the test checks for a difference in either direction. The mean difference may be greater than or less than the null value.
4. What is the usual null difference?
The usual null difference is zero. You can change it when your hypothesis expects another benchmark or meaningful reference difference.
5. Why must both samples have equal length?
Each value in one sample must match one value in the other sample. Unequal lengths break the paired structure required by the test.
6. What is Cohen's dz?
Cohen's dz is a paired effect size. It divides the mean paired difference by the standard deviation of paired differences.
7. What if my p value is below alpha?
A p value below alpha indicates statistical significance. You reject the null hypothesis for the selected two tailed test level.
8. Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a clean printable summary.