Critical Value Calculator
Pick a confidence level and tail type to get the critical z value your test statistic must beat — with the significance level and rejection region spelled out.
Example: with Confidence level 95% · Custom confidence level (%) 95 · Test type Two-tailed → Critical z value: ±1.960.
Computed by the calculator below using its default values. Change any input to see your own numbers.
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Check it outHow to find a critical value
A critical value marks the edge of the rejection region: if your test statistic lands beyond it, the result is significant at that level. For a z test, convert the confidence level to α (95% means α = 0.05), split α across the tails, and invert the normal CDF: z* = Φ⁻¹(1 − α/2) two-tailed, or Φ⁻¹(1 − α) one-tailed. That’s where the classics come from: ±1.960 for 95% two-tailed, 1.645 for 95% one-tailed, ±2.576 for 99% two-tailed.
For small samples with an unknown population SD, use the t critical value instead — same logic, but the t distribution with n − 1 degrees of freedom has fatter tails, so its cutoffs are larger: at 95% two-tailed, t = 2.228 with 10 degrees of freedom vs z = 1.960. As the sample grows, t sinks toward z.
How it’s calculated
z* = Φ⁻¹(1 − α/2) for two-tailed tests and Φ⁻¹(1 − α) for one-tailed, where α = 1 − confidence/100. The inverse normal CDF uses Acklam’s rational approximation (relative error below 1.15 × 10⁻⁹), so the three decimals shown are exact. For t or χ² tests, apply the same α logic to those distributions’ tables.
Results update as you type and are estimates, not professional advice — verify important decisions with a qualified professional.
Common mistakes
- Forgetting to split α for two-tailed tests — 95% two-tailed uses 0.025 per tail (z = 1.960), not 0.05.
- Using z when the population SD is unknown and the sample is small — use t with n − 1 degrees of freedom.
- Comparing a negative test statistic to +1.960 — in a two-tailed test, compare |z| to the critical value.
Frequently asked questions
What is a critical value?
The cutoff a test statistic must exceed for significance at a chosen α. Beyond it lies the rejection region; z = ±1.960 is the classic 95% two-tailed cutoff.
How do you find a critical value?
Convert the confidence level to α = 1 − CL/100, split it per tail (α/2 if two-tailed), and read the inverse normal CDF: z* = Φ⁻¹(1 − α/2).
What is the t critical value and when do I need it?
The same cutoff taken from the t distribution with n − 1 degrees of freedom — use it when the population SD is unknown and the sample is small. At 95% two-tailed with df = 10, t = 2.228.
What about chi-square critical values?
χ² tests read the chi-square distribution’s upper tail instead: with 1 degree of freedom at α = 0.05 the cutoff is 3.841 — exactly the two-sided z critical value squared.
What are degrees of freedom?
The number of values free to vary after estimating parameters. For a one-sample t test, df = n − 1, so 25 observations give 24 degrees of freedom.