Critical Value Calculator

Find the critical value to compare against your test statistic in any hypothesis test.

📖 What is a Critical Value?

A critical value is the boundary point that separates the rejection region from the non-rejection region in a hypothesis test. If the computed test statistic falls beyond the critical value (into the rejection region), the null hypothesis is rejected at the chosen significance level α.

Critical values depend on three factors: the probability distribution of the test statistic (Z, t, F, or χ²), the significance level α (the acceptable probability of a Type I error - falsely rejecting H₀), and the direction of the test (one-tailed or two-tailed).

The most widely used critical value is z = 1.96, the two-tailed critical value for the standard normal distribution at α = 0.05. This appears in confidence interval formulas (95% CI: estimate ± 1.96 × SE) and in Z-tests for proportions and large-sample means. For the t-distribution, the critical value is higher and decreases as degrees of freedom increase, reflecting the heavier tails and greater uncertainty in small samples.

Critical values are equivalent to the quantile (inverse CDF) of the distribution. For example, the Z critical value at α = 0.05 (two-tailed) is the 97.5th percentile of the standard normal distribution: Φ⁻¹(0.975) = 1.96.

📐 Formulas

Two-tailed: z_α/2 = Φ⁻¹(1 − α/2)

One-tailed right: z_α = Φ⁻¹(1 − α)

One-tailed left: z_α = Φ⁻¹(α) = −Φ⁻¹(1 − α)

t critical value: t_(α/2, df) = quantile of t-distribution with df degrees of freedom

Chi-square critical value: χ²_(α, df) = quantile of chi-square distribution (right-tailed)

F critical value: F_(α, df₁, df₂) = quantile of F-distribution (right-tailed)

Common Z critical values: α = 0.10 → 1.645 | α = 0.05 → 1.960 | α = 0.01 → 2.576 | α = 0.001 → 3.291 (all two-tailed)

📖 How to Use This Calculator

Select the distribution corresponding to your test: Z for large samples/known σ, t for small samples/unknown σ, F for ANOVA/regression, chi-square for categorical data.

Enter the significance level α. The most common is 0.05 (5%). For stricter testing use 0.01; for exploratory work, 0.10.

Enter degrees of freedom if required. For a one-sample t-test: df = n − 1. For two-sample: df = n₁ + n₂ − 2.

Choose the tail type and click Find Critical Value. Compare to your test statistic: if |test stat| > critical value, reject H₀.

📝 Example Calculations

Example 1 - Z Critical Value (α = 0.05, two-tailed)

Standard Z-test at 5% significance, two-tailed: z_crit = ±1.960

Reject H₀ if |Z| > 1.960. Used for large-sample mean tests and proportion tests.

Result = ±1.960

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Example 2 - t Critical Value (α = 0.05, df = 19, two-tailed)

One-sample t-test, n = 20, two-tailed: t_crit = ±2.093

Higher than 1.96 because the t-distribution has heavier tails for small samples.

Result = ±2.093

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Example 3 - Chi-square Critical Value (α = 0.05, df = 4)

Goodness-of-fit test with 5 categories (df = 4): χ²_crit = 9.488

Reject H₀ if χ² > 9.488.

Result = 9.488

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Example 4 - F Critical Value (α = 0.05, df₁ = 3, df₂ = 36)

One-way ANOVA with 4 groups of 10: F_crit = 2.866

Reject H₀ (all means equal) if F > 2.866.

Result = 2.866

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Example 5 - Z Critical Value (α = 0.01, one-tailed right)

Right-tailed Z-test at 1% significance: z_crit = 2.326

Reject H₀ (H₁: μ > μ₀) if Z > 2.326.

Result = 2.326

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❓ Frequently Asked Questions

What is a critical value in statistics?+

A critical value is the threshold that a test statistic must exceed to reject the null hypothesis. It is determined by the distribution of the test statistic, the significance level (α), and the tail direction. For a two-tailed Z-test at α = 0.05, the critical values are ±1.96 - any Z-score outside this range leads to rejection of H₀.

What is the critical value for Z at α = 0.05?+

For a two-tailed Z-test at α = 0.05: z_crit = ±1.96. For a one-tailed right test at α = 0.05: z_crit = +1.645. For a one-tailed left test: z_crit = −1.645. These values come from the inverse of the standard normal CDF (the quantile function Φ⁻¹).

How does the critical value change with sample size?+

For t-tests, the critical value depends on degrees of freedom (df = n − 1 for one-sample). As n increases, df increases, and the t-distribution approaches the normal distribution - so the critical value decreases toward the Z critical value. For n > 30, t and Z critical values are very close.

What is the relationship between critical value and p-value?+

They convey the same information from different directions. If |test statistic| > critical value, then p < α, and you reject H₀. If |test statistic| < critical value, then p > α, and you fail to reject H₀. Both approaches always give the same conclusion.

What critical values are used for confidence intervals?+

Confidence intervals use the same critical values. A 95% CI uses z* = 1.96 (for Z) or t*(df) for t-intervals. A 99% CI uses z* = 2.576. The CI is: estimate ± critical_value × standard_error.

What is the F critical value used for?+

The F critical value is used in ANOVA and regression F-tests. If the computed F-statistic exceeds the critical F value (at df₁, df₂, α), the null hypothesis that all group means are equal (ANOVA) or that all regression coefficients are zero (regression F-test) is rejected.

What is the chi-squared critical value used for?+

The chi-squared (χ²) critical value is used in goodness-of-fit tests, tests of independence, and variance tests. If your computed χ² statistic exceeds the critical value at your chosen significance level and degrees of freedom, you reject the null hypothesis. For example, in a test of independence for a contingency table with df = 4 and α = 0.05, the critical value is 9.488. Unlike Z and t, chi-squared tests are typically one-tailed (right-tailed) because chi-squared values are always non-negative.

What critical value should I use for 95% confidence?+

For a two-tailed test at 95% confidence: use z = 1.96 (normal distribution, large samples), or the t critical value for small samples. For one-tailed at 95%: z = 1.645. At 99% confidence two-tailed: z = 2.576. These values define the rejection regions - test statistics beyond these thresholds lead to rejecting the null hypothesis.

How does degrees of freedom affect the critical value?+

For the t-distribution, lower degrees of freedom produce larger critical values (wider tails). With df = 5, the 95% two-tailed critical value is t = 2.571. With df = 30, it is 2.042. With df infinity (large sample), it converges to z = 1.96. This is why small-sample t-tests are more conservative - they require stronger evidence to reject the null.

What is the critical value for a 95% confidence interval?+

For a 95% confidence interval (alpha = 0.05, two-tailed), the Z critical value is 1.96. For a t-distribution, the critical value depends on degrees of freedom: at 30 df it is 2.042, at 100 df it is 1.984, approaching 1.96 as df increases. Use the Z critical value only when n is large (typically n > 30) and population variance is known.

Tags: Critical Value Z Critical Value T Critical Value F Critical Value Chi-Square Hypothesis Testing Significance Level Rejection Region

Critical Value Calculator

📖 What is a Critical Value?

📐 Formulas

📖 How to Use This Calculator

📝 Example Calculations

❓ Frequently Asked Questions

🔗 Related Calculators

📌 Quick Tips