Paired T Test Calculator
Run a paired t-test on before and after measurements to test whether the mean difference between them differs significantly from zero.
Paired T Test Calculator
Calculator
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Formula: t = d̄ / (s_d / √n)
Worked example — t = 5.477, df = 4, two-tailed p = 0.0054 | Mean drop of 15 units is statistically significant
Formula
t = d̄ / (s_d / √n)
Tests whether the mean difference between paired observations is significantly different from zero.
Worked Examples
Example 1: Cholesterol before and after treatment
Problem:Before: 200, 210, 190, 220, 205 | After: 180, 195, 185, 200, 190
Solution:Work with the five paired differences (before - after): 20, 15, 5, 20, 15. Mean difference d-bar = 75 / 5 = 15. Deviations from 15: 5, 0, -10, 5, 0, whose squares sum to 150. s_d = sqrt(150 / 4) = sqrt(37.5) = 6.1237. Standard error = 6.1237 / sqrt(5) = 2.7386. t = 15 / 2.7386 = 5.477 with df = n - 1 = 4. The two-tailed 5 percent critical value for df = 4 is 2.776.
Result:t = 5.477, df = 4, two-tailed p = 0.0054 | Mean drop of 15 units is statistically significant
Example 2: A difference too small to detect
Problem:Before: 50, 52, 49, 51, 53 | After: 49, 53, 49, 50, 54
Solution:Differences (before - after): 1, -1, 0, 1, -1. d-bar = 0 / 5 = 0. With a mean difference of exactly zero the numerator of t is zero, so t = 0 regardless of s_d. s_d = sqrt(4 / 4) = 1.0 and the standard error is 1 / sqrt(5) = 0.447. t = 0 / 0.447 = 0 with df = 4, nowhere near the 2.776 critical value.
Result:t = 0.00, df = 4, p = 1.00 | No evidence of any before-and-after change
Frequently Asked Questions
What does the paired t-test measure?
The paired t-test determines whether the mean difference (d̄) between two related measurements is significantly different from zero. By working on the differences rather than the raw values, it controls for between-subject variability, making it more powerful than an independent t-test when the same subjects or matched pairs are measured twice.
How do you interpret the t-statistic and p-value in a paired t-test?
Compare the absolute value of t to the critical value from a t-distribution with df = n − 1 at your chosen significance level α. If |t| exceeds the critical value (e.g., approximately 2.0 for df > 20 at α = 0.05, two-tailed), you reject the null hypothesis H₀: μ_d = 0 and conclude that there is a significant mean difference between the paired measurements.
When should I use a paired t-test vs. an independent t-test?
Use the paired t-test when each observation in one group is directly linked to a corresponding observation in the other group — such as before/after measurements on the same subjects, or matched participants. Use the independent t-test when the two groups consist of entirely different, unrelated subjects. Incorrectly using the independent t-test on paired data wastes statistical power.
What are the assumptions of the paired t-test?
The differences between paired observations must be approximately normally distributed (not the raw data themselves); each pair must be independent of other pairs; the data must be measured on a continuous interval or ratio scale; and pairs should be meaningful (i.e., each before value is truly related to its corresponding after value). For non-normal differences, use the Wilcoxon signed-rank test instead.
How do I build a confidence interval for the mean difference?
Use d̄ ± t* × (s_d/√n), where t* is the critical value of the t-distribution with df = n − 1 at your confidence level. With n = 5, df = 4 and t* ≈ 2.776 for 95 percent confidence. If the resulting interval excludes zero, the two-tailed test at α = 0.05 is significant, so interval and test always agree while the interval also shows how large the change is.
What effect size accompanies a paired t-test?
Report Cohen's d_z = d̄ / s_d, the mean difference expressed in units of the standard deviation of the differences. Unlike t, it does not grow as the sample grows, so it describes the magnitude of the change rather than the strength of the evidence. Values near 0.2, 0.5 and 0.8 are conventionally labelled small, medium and large, though sensible benchmarks vary by field.
What does the sign of t tell me about the direction of change?
The sign follows the subtraction order. Paired T Test Calculator forms each difference as before minus after, so a positive d̄ and a positive t mean the after values are lower on average. Swap the two columns and t flips sign while its magnitude and p-value stay identical. Always state the subtraction order when reporting, because the sign on its own is ambiguous.
How does a one-tailed paired t-test differ from a two-tailed one?
A two-tailed test asks whether the mean difference departs from zero in either direction and splits α between both tails. A one-tailed test puts all of α in one tail and can only detect change in the direction you named in advance. Choosing the direction after seeing the sign of d̄ inflates the false-positive rate, so the decision must be made before the analysis.
Why does pairing increase statistical power?
Working with differences removes every stable characteristic of a subject, such as baseline level, individual habits and instrument offsets, from the comparison. What is left in s_d is only the variability of the change itself, which is usually far smaller than the spread of the raw values. The more strongly the before and after measurements correlate, the bigger the gain over an independent-samples test on the same numbers.
What should I do when one pair has a missing measurement?
A paired t-test uses complete pairs only, so a pair missing either value is dropped, reducing n and therefore df = n − 1 by one. Do not borrow a value from the other column and do not pad the shorter list, since that invents a difference of zero and biases d̄ toward the null. If dropouts are related to the outcome itself, report that as a limitation of the analysis.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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