Paired T-Test Calculator
Test whether the mean of paired differences differs from μ₀ using dependent-samples t-test formulas verified against scipy.stats.ttest_rel.
Independent groups? Use the two-sample t-test. Nonparametric paired alternative: Wilcoxon signed-rank test.
Related Calculators
T-Test Calculator
Compare sample means with common t-test workflows and interpretable outputs.
Two Sample T-Test Calculator
Compare two independent means with Welch (default) or pooled Student t-tests from raw data or summary statistics, with both methods side by side.
Wilcoxon Signed-Rank Test Calculator
Paired or one-sample Wilcoxon signed-rank test with zero handling, tie correction, exact p for small n, and a signed-rank table.
Learn More
T-Test in Excel: T.TEST and the Data Analysis ToolPak
Run a t-test in Excel with T.TEST or the Analysis ToolPak: the type and tails arguments, paired and unequal-variance tests, a checked example and how to read the output.
T-Test in R and Python: t.test() and scipy.stats
Run one-sample, two-sample, Welch and paired t-tests in R and Python side by side, see why the default variance option differs and how one outlier can flip the result — checked against SciPy.
Paired vs independent
Use a paired test when the same subjects are measured twice or subjects are matched. The test statistic depends only on the differences, so order within each pair must be consistent (always before − after).
Assumption
The differences should be approximately normal; the raw measurements need not be. With strong positive correlation, pairing increases power by removing between-subject variation.
Var(D) = σ₁² + σ₂² − 2ρσ₁σ₂
t = (d̄ − μ₀) / (Sd / √n), df = n − 1
Worked example
Before: 5, 6, 7, 8, 9; after: 3, 5, 4, 7, 6. Differences 2, 1, 3, 1, 3 give d̄ = 2, Sd = √2, t ≈ 4.472, df = 4, two-sided p ≈ 0.0111 (matches Load example).
Software
| Software | Command |
|---|---|
| Excel | T.TEST(array1, array2, tails, 1) — type 1 = paired |
| R | t.test(x, y, paired = TRUE) |
| Python | scipy.stats.ttest_rel(a, b) |
| TI-84 | Store differences in L3; 2:T-Test on L3 with μ₀ = 0 |
Related guides and calculators
If the differences are clearly not normal use the Wilcoxon signed-rank test or the sign test instead. The one-sample t-test is the same calculation applied to the differences, the effect size calculator reports Cohen's d, and the Shapiro-Wilk test checks normality. See t-test in Excel and parametric vs nonparametric tests.
Frequently Asked Questions
What is being tested?
Whether the population mean of the pair differences μ_D equals μ₀ (usually 0).
Why report correlation between the two columns?
Large r shows pairing is worthwhile: difference variance shrinks when measurements move together.
Can I enter summary statistics only?
Yes — provide n, mean of differences, and SD of differences from your report.
What if list lengths differ?
The calculator stops and tells you which column is missing values so you can align pairs.
Is this the same as a repeated-measures ANOVA with two levels?
With one within factor and two levels, the F test is equivalent to the paired t test when balanced.
One- or two-tailed p-value?
Your alternative selection controls both p and the confidence interval tail (scipy convention).
What is Cohen's d_z here?
Standardized mean difference using the SD of the differences: d_z = (d̄ − μ₀) / Sd.
When should I use Wilcoxon instead?
When differences are clearly skewed or heavy-tailed and n is small enough that normality matters.
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