How this calculator works
A one-sample t test compares a mean with a null value. A paired test first subtracts matched observations and tests their mean difference. Independent groups can use Welch or pooled inference; pooling assumes equal population variances. Observations must be independent across sampling units. Small samples need a plausible normal population model without severe outliers. The confidence interval always uses two sides at 1 - alpha, even when a one-sided test is selected.
Formula
One sample or differences: t = (x̄ - μ0)/(s/√n), df = n - 1. Welch uses SE² = s1²/n1 + s2²/n2 and Satterthwaite df. CI = estimate ± t(1 - α/2, df) × SE.
Worked example
A mean of 2, null 0, SD 2 and n = 4 give t = 2, df = 3 and two-sided p about 0.139326. Observations 1, 2, 3 have mean 2, SD 1 and t = 2√3 against zero.
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Frequently asked questions
When should I select paired?
When each observation in one list corresponds to the same subject or matched unit in the other. Complete pairs must be in matching positions.
Does non-significance establish equal means?
No. It indicates insufficient evidence for the selected alternative at this alpha. Equivalence requires a different design.
Why is the interval two-sided for a one-tailed test?
It is a separately labelled central interval for estimation. Its boundaries do not reproduce the one-sided rejection rule.