Critical Value Calculator

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

Use this Critical Value Calculator to find the Z critical values for hypothesis tests and confidence intervals. Enter a significance level (α) and select whether you need a one-tailed or two-tailed test. The calculator returns the critical Z value(s) that define the rejection region boundary.

What Are Critical Values?

Critical values are the boundary points on a probability distribution that separate the rejection region from the non-rejection region in a hypothesis test. If the test statistic falls beyond the critical value (into the rejection region), you reject the null hypothesis. The critical value depends on the significance level (α), which represents the maximum probability of making a Type I error (rejecting a true null hypothesis).

For Z tests, which use the standard normal distribution, the critical values are Z scores. In a two-tailed test at α = 0.05, the critical values are approximately ±1.96. This means that if the test statistic Z is less than -1.96 or greater than 1.96, you reject the null hypothesis. The total area in both tails combined equals α, with α/2 in each tail.

Critical values also define confidence intervals. A 95% confidence interval uses the critical value z* = 1.96 (the Z score that leaves 2.5% in each tail). The confidence level is 1 − α, so a smaller α produces a wider confidence interval with a higher confidence level.

Critical Value Formulas

Two-tailed test: The critical values are ±z_α/2, where:

z_α/2 = Φ⁻¹(1 − α/2)

Right-tailed test: The critical value is z_α, where:

z_α = Φ⁻¹(1 − α)

Left-tailed test: The critical value is −z_α, where:

−z_α = Φ⁻¹(α)

Worked Example

Find the critical values for a two-tailed hypothesis test at the 1% significance level (α = 0.01).

1. Determine tail areas: α/2 = 0.01/2 = 0.005 in each tail.
2. Find the upper critical value: z_0.005 = Φ⁻¹(1 − 0.005) = Φ⁻¹(0.995) ≈ 2.57583
3. State both critical values: The critical values are ±2.57583.
4. Decision rule: Reject H₀ if the test statistic Z < −2.576 or Z > 2.576.

Common Critical Values Reference

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