Calculator
Enter alpha (or confidence), choose a tail, then calculate the critical value.
Example data table
These examples are computed by the same routines used in the calculator.
| Distribution | Tail | Alpha | df / df1 | df2 | Critical value |
|---|---|---|---|---|---|
| Z | Two | 0.05 | — | — | ±1.96 |
| t | Two | 0.05 | 10 | — | ±2.228 |
| Chi-square | Right | 0.05 | 5 | — | 11.07 |
| F | Right | 0.05 | 5 | 10 | 3.326 |
Formula used
- Z: right tail z = Φ⁻¹(1 − α), left tail z = Φ⁻¹(α), two tails ±Φ⁻¹(1 − α/2).
- t: same tail logic using t = T⁻¹(p; df), where T is the t CDF.
- χ²: right tail x = G⁻¹(1 − α; df), left tail G⁻¹(α; df), two tails [G⁻¹(α/2), G⁻¹(1 − α/2)].
- F: critical values from the inverse F CDF with df1, df2 and the chosen tail.
Φ⁻¹, T⁻¹, G⁻¹ denote inverse CDF (quantile) functions.
How to use this calculator
- Select a distribution that matches your test.
- Pick left, right, or two-tailed based on your hypothesis.
- Enter alpha, or switch to confidence level.
- Provide degrees of freedom when required.
- Click Calculate to see critical value and rejection rule.
- Optionally enter a test statistic for a decision.
- Use the export buttons to save your result.
FAQs
1) What is a critical value?
A critical value is a cutoff on the test statistic scale. If your statistic passes the cutoff, it falls in the rejection region for the chosen alpha and tail.
2) Which tail option should I choose?
Choose right-tailed for “greater than,” left-tailed for “less than,” and two-tailed for “not equal.” The tail choice changes which probability mass defines the cutoff.
3) What does alpha mean?
Alpha is the significance level: the maximum probability of rejecting a true null hypothesis. Common choices are 0.10, 0.05, and 0.01, depending on required strictness.
4) Can I enter confidence level instead of alpha?
Yes. Confidence level converts using α = 1 − confidence. For example, 95% confidence corresponds to alpha 0.05.
5) What are degrees of freedom?
Degrees of freedom measure how much independent information is available. For t and chi-square, df controls the shape of the distribution and therefore the location of critical cutoffs.
6) Why does the F distribution need df1 and df2?
F compares two scaled variances. df1 relates to the numerator variance estimate, and df2 relates to the denominator estimate. Both influence the curve and its quantiles.
7) How is the p-value shown here computed?
If you enter a test statistic, the calculator evaluates its cumulative probability. It then reports a tail-based p-value, using two times the smaller tail for two-tailed tests.
8) Are results exact?
They are high-accuracy numerical approximations computed from standard distribution functions. Small differences can occur versus printed tables due to rounding and implementation details.
Tip: Use Ctrl/Cmd + F to quickly find formulas or FAQs.