How to use it
- Choose P-value from t or Critical t from α.
- Enter the t statistic (it may be negative) or the significance level α, for example 0.05.
- Enter the degrees of freedom. Fractional df from Welch’s test are fine.
- Read the p-values for all three tails, or the one- and two-tailed critical values.
How it works
Student’s t distribution with ν degrees of freedom has the upper tail
P(T > t) = ½ · I(ν / (ν + t²); ν/2, ½) for t ≥ 0,
where I is the regularised incomplete beta function. The left tail is 1 minus the right tail, and the two-tailed p-value is 2 × P(T > |t|). Critical values are found by inverting this tail numerically: the one-tailed t has P(T > t) = α, the two-tailed one has P(T > t) = α/2.
Worked example
With df = 2 the tail has a closed form, P(T > t) = ½(1 − t/√(t² + 2)). For t = 2: 2/√6 = 0.8165, so the right tail is ½ × 0.1835 = 0.091752 and the two-tailed p-value is 0.183503. The calculator shows the same numbers.
For the critical value at α = 0.05 with df = 2 the two-tailed result is 4.3027 and the one-tailed is 2.92. With df = 1 (the Cauchy distribution) they are 12.7062 and 6.3138.
Tips
- Two-tailed is the default in most reports. If a paper says “p < 0.05” without a direction, it is usually two-tailed.
- Do not halve a p-value after looking at the data. Choose one- or two-tailed before running the test.
- Large df: above about 100 degrees of freedom the t and normal results differ only in the second or third decimal.