Critical Value Calculator
Find the critical value (cutoff) for a hypothesis test in seconds — choose Z, t, χ², or F, pick one-tailed or two-tailed, and enter α (and df when needed). Includes optional step-by-step + mini visualization.
Background
A critical value is the test-statistic cutoff that defines the rejection region. For a right-tailed test with significance level α, the critical value c satisfies P(X ≥ c) = α (so F(c) = 1 − α).
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).
- Two-tailed means α is split: α/2 = 0.05/2 = 0.025 in each tail.
- Upper cutoff uses CDF probability p = 1 − α/2 = 1 − 0.025 = 0.975.
- So z = F^{-1}(0.975) ≈ 1.96.
- By symmetry for Z, the lower critical value is −1.96.
- Answer: −1.96 and +1.96.
Example 2 — t, right-tailed
Find the t critical value for α = 0.05, df = 18 (right-tailed).
- Right-tailed means the rejection area is α in the right tail.
- So the target CDF probability is p = 1 − α = 1 − 0.05 = 0.95.
- Compute t = F^{-1}(0.95) with df=18 → t ≈ 1.734.
- 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).
- Right-tailed χ² uses the CDF target p = 1 − α = 1 − 0.05 = 0.95.
- Compute χ² = F^{-1}(0.95) with df=4 → χ² ≈ 9.488.
- 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.
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