How this calculator works
A critical value marks the boundary of a rejection region under the null distribution. The normal and t distributions are symmetric about zero; F and chi-square cutoffs are positive and generally unequal in a two-sided test. The tool numerically inverts distribution tail probabilities. Degrees of freedom may be fractional, as in a Welch test, and are limited to 100000 for numerical reliability.
Formula
Right cutoff q satisfies P(T ≥ q) = α. Left cutoff satisfies P(T ≤ q) = α. Two-sided cutoffs use α/2 in each tail.
Worked example
At alpha 0.05 a two-sided normal test uses approximately -1.959964 and 1.959964. A two-sided t test with 10 degrees of freedom uses approximately -2.228139 and 2.228139.
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Frequently asked questions
Why are F cutoffs unequal?
The F distribution is skewed and bounded below by zero. Symmetric tail areas do not imply symmetric statistic values.
Can I enter a confidence level?
Convert confidence to alpha first: a 95% central interval corresponds to alpha 5%.
Can Welch degrees of freedom be decimals?
Yes. Do not round a Welch estimate down before computing the cutoff.