Example data table
| Scenario | Inputs | Typical output |
|---|---|---|
| Two-tailed z test | dist=z, alpha=0.05, tail=two | z ≈ -1.96, +1.96 |
| Upper-tail t test | dist=t, df=12, alpha=0.01, tail=upper | t ≈ t0.99(12) |
| Two-tailed chi-square | dist=chi2, df=10, alpha=0.05, tail=two_chi | χ² lower and upper cutoffs |
| Upper-tail F | dist=f, df1=5, df2=20, alpha=0.05, tail=upper | F ≈ F0.95(5,20) |
Formulas used
- Z: critical value is the inverse normal CDF: z = Φ⁻¹(p).
- t: t = T⁻¹(p; df), computed using the incomplete beta relation for the t CDF.
- χ²: x = Χ²⁻¹(p; df), using the regularized incomplete gamma function.
- F: x = F⁻¹(p; df1, df2), using the regularized incomplete beta function.
How to use this calculator
- Select a distribution that matches your statistical test.
- Pick the tail mode for your hypothesis setup.
- Enter alpha, or enable confidence to auto-convert.
- Provide degrees of freedom when required.
- Press “Find Critical Value” to compute cutoffs.
- Download CSV or PDF to document your result.
FAQs
1) What is a critical value?
A critical value is a cutoff on a distribution that separates the rejection region from the non-rejection region for a chosen significance level.
2) When should I use two-tailed mode?
Use two-tailed mode when your alternative hypothesis allows extreme results on both sides, such as “different” rather than “greater” or “less.”
3) Why do t and χ² need degrees of freedom?
Their shapes change with sample size and constraints. Degrees of freedom capture that change, so the critical value adjusts to your dataset.
4) What does “upper tail” mean here?
Upper tail means the area to the right is alpha. The calculator finds the value where cumulative probability equals 1 minus alpha.
5) How does confidence relate to alpha?
For common confidence-interval workflows, alpha is 1 minus confidence. A 95% confidence level corresponds to alpha 0.05.
6) Why can χ² two-tailed values be asymmetric?
The chi-square distribution is not symmetric. Splitting alpha between tails produces two different cutoffs, not a simple plus-minus pair.
7) What is two-tailed F used for?
It is often used in variance comparisons. The right cutoff uses alpha/2, while the left cutoff is obtained through an inverse relationship with swapped degrees.
8) How accurate are the computed results?
The calculator uses established approximations and numerical inversion. For typical academic and applied work, the values match standard tables closely.