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Tip: Z/t are symmetric around 0. χ² and F are right-skewed and typically used in variance tests & ANOVA.

Two-tailed splits α across both tails (α/2 each). χ² and F are most often right-tailed.

Two-tailed → α/2 = 0.025 per tail.

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How to use this calculator

  • Choose the distribution (Z, t, χ², or F).
  • Pick the tail based on your alternative hypothesis (H₁).
  • Enter α (and degrees of freedom if needed).
  • Click Calculate to get the critical value(s) and a shaded rejection region.

How this calculator works

  • Right-tailed: find c such that P(X ≥ c)=α → F(c)=1−α.
  • Left-tailed: find c such that P(X ≤ c)=α → F(c)=α.
  • Two-tailed: split α: lower cutoff at α/2 and upper cutoff at 1−α/2. (For Z/t you’ll see ±c.)
  • Critical values are computed by inverting the CDF (finding F^{-1}) using stable special functions + a safe bisection search.

Formula & Equation Used

Right-tailed: c = F^{-1}(1−α)

Left-tailed: c = F^{-1}(α)

Two-tailed: c_L = F^{-1}(α/2),   c_U = F^{-1}(1−α/2)

Example Problems & Step-by-Step Solutions

Example 1 — Z, two-tailed

Find the Z critical values for α = 0.05 (two-tailed).

  1. Two-tailed means α is split: α/2 = 0.05/2 = 0.025 in each tail.
  2. Upper cutoff uses CDF probability p = 1 − α/2 = 1 − 0.025 = 0.975.
  3. So z = F^{-1}(0.975) ≈ 1.96.
  4. By symmetry for Z, the lower critical value is −1.96.
  5. Answer: −1.96 and +1.96.

Example 2 — t, right-tailed

Find the t critical value for α = 0.05, df = 18 (right-tailed).

  1. Right-tailed means the rejection area is α in the right tail.
  2. So the target CDF probability is p = 1 − α = 1 − 0.05 = 0.95.
  3. Compute t = F^{-1}(0.95) with df=18 → t ≈ 1.734.
  4. Answer: Reject H₀ if t ≥ 1.734.

Example 3 — χ², right-tailed

Find the χ² critical value for α = 0.05, df = 4 (right-tailed).

  1. Right-tailed χ² uses the CDF target p = 1 − α = 1 − 0.05 = 0.95.
  2. Compute χ² = F^{-1}(0.95) with df=4 → χ² ≈ 9.488.
  3. Answer: Reject H₀ if χ² ≥ 9.488.

Note: Example values are common textbook cutoffs (rounded). The calculator will compute and display more digits.

Frequently Asked Questions

Q: What is a critical value?

It’s the cutoff that separates the rejection region from the non-rejection region at a chosen α.

Q: What happens in a two-tailed test?

You split α into α/2 on each tail, producing two cutoffs (lower and upper).

Q: Why are χ² and F usually right-tailed?

Those statistics are nonnegative and “more extreme” evidence against H₀ is usually in large values.

Q: What’s the difference between critical value and p-value?

Critical values are cutoffs computed from α. A p-value is computed from your observed statistic.

Q: Can I enter a custom α?

Yes — choose “Custom…” and enter any α between 0 and 1.