About

Hypothesis testing hinges on comparing a test statistic against a critical value derived from the relevant sampling distribution. An incorrect critical value means either a false rejection of a true null hypothesis (Type I error at rate α) or a failure to detect a real effect. This calculator implements inverse cumulative distribution functions for four core distributions: Standard Normal (Z), Student's t, Chi-Square (χ²), and Fisher's F. It supports one-tailed (left or right) and two-tailed tests. All computations use numerical algorithms (Acklam's rational approximation for Z, regularized incomplete Beta and Gamma functions for t, χ², and F) with iterative refinement to at least 8 significant digits.

Limitations: the t and F distributions assume normally distributed populations. The χ² approximation loses accuracy below df = 1 combined with extreme α values (< 0.0001). For practical research work, confirm results against published tables (e.g., NIST/SEMATECH) when operating near distribution tails beyond the 99.99% confidence level.

Formulas

The critical value c satisfies the equation where the cumulative distribution function equals the target probability. For a right-tailed test at significance level α:

P(X ≥ c) = α ⇒ c = F⁻¹(1 − α)

For a left-tailed test:

P(X ≤ c) = α ⇒ c = F⁻¹(α)

For a two-tailed test, the area α is split equally into both tails:

c_lower = F⁻¹(α2) c_upper = F⁻¹(1 − α2)

The inverse standard normal Φ⁻¹(p) uses Acklam's rational approximation with six coefficients per numerator and denominator polynomial, yielding maximum relative error < 1.15 × 10⁻⁹. The inverse t-distribution leverages the relationship t = sign(p − 0.5) ⋅ √df ⋅ 1 − I_x⁻¹I_x⁻¹ where I_x⁻¹ is the inverse regularized incomplete Beta function I⁻¹(df2, 12; q). The χ² inverse uses χ² = 2 ⋅ P⁻¹(df2, p) where P⁻¹ is the inverse regularized lower incomplete Gamma function.

Where α = significance level, df = degrees of freedom, F⁻¹ = inverse CDF of the chosen distribution, p = cumulative probability, I_x = regularized incomplete Beta function, P = regularized lower incomplete Gamma function.

Reference Data

Distribution	Typical Use	α = 0.10	α = 0.05	α = 0.025	α = 0.01	α = 0.005	α = 0.001
Z (right-tail)	Large-sample proportion/mean tests	1.2816	1.6449	1.9600	2.3263	2.5758	3.0902
t (df=5, right)	Small-sample mean test	1.4759	2.0150	2.5706	3.3649	4.0322	5.8934
t (df=10, right)	Small-sample mean test	1.3722	1.8125	2.2281	2.7638	3.1693	4.1437
t (df=20, right)	Small-sample mean test	1.3253	1.7247	2.0860	2.5280	2.8453	3.5518
t (df=30, right)	Moderate-sample mean test	1.3104	1.6973	2.0423	2.4573	2.7500	3.3852
χ² (df=1, right)	Goodness-of-fit, independence	2.7055	3.8415	5.0239	6.6349	7.8794	10.8276
χ² (df=5, right)	Goodness-of-fit, independence	9.2364	11.0705	12.8325	15.0863	16.7496	20.5150
χ² (df=10, right)	Goodness-of-fit, independence	15.9872	18.3070	20.4832	23.2093	25.1882	29.5883
χ² (df=20, right)	Goodness-of-fit, independence	28.4120	31.4104	34.1696	37.5662	39.9968	45.3147
F (df1=1, df2=10, right)	ANOVA, regression F-test	3.2850	4.9646	6.9367	10.0443	12.8265	21.0398
F (df1=2, df2=10, right)	ANOVA	2.9245	4.1028	5.4564	7.5594	9.4270	14.9098
F (df1=5, df2=20, right)	ANOVA	2.1582	2.7109	3.2891	4.1027	4.7616	6.4612
F (df1=3, df2=30, right)	ANOVA	2.2758	2.9223	3.5894	4.5097	5.2388	7.0527
F (df1=10, df2=50, right)	Multi-factor ANOVA	1.7459	2.0261	2.3055	2.6985	2.9930	3.6710

Frequently Asked Questions

A one-tailed test places the entire rejection region α in one tail of the distribution. A right-tailed test rejects when the test statistic exceeds c = F⁻¹(1 − α); a left-tailed test rejects below c = F⁻¹(α). A two-tailed test splits α equally, placing α/2 in each tail, producing two critical values. For example, a two-tailed Z test at α = 0.05 gives critical values of ±1.9600, while a one-tailed test at the same α gives +1.6449 (right) or −1.6449 (left).

Use the t-distribution when the population standard deviation is unknown and estimated from the sample. With small samples (n < 30), the t-distribution has heavier tails than the standard normal, producing larger critical values that account for the additional estimation uncertainty. As degrees of freedom increase beyond approximately 120, the t-distribution converges to the Z-distribution. At df = ∞, they are identical.

The Chi-Square distribution is right-skewed with its shape governed entirely by df. As df increases, the distribution shifts rightward and becomes more symmetric (approaching normal by the Central Limit Theorem). For a fixed α = 0.05, the right-tail critical value increases roughly linearly with df: χ²(df=1) ≈ 3.84, χ²(df=10) ≈ 18.31, χ²(df=50) ≈ 67.50. Each additional degree of freedom adds approximately 1.0 to the critical value at moderate df.

In one-way ANOVA, the F-statistic equals the ratio of between-group variance to within-group variance: F = MS_between / MS_within. The critical value F_crit depends on df1 (number of groups minus 1) and df2 (total observations minus number of groups). If the computed F exceeds F_crit at your chosen α, you reject the null hypothesis that all group means are equal. The F-distribution is always right-tailed in ANOVA; a left-tail critical value is used only in specialized variance ratio tests.

The choice of α depends on the cost of a Type I error (false positive). In medical trials, α = 0.01 or 0.005 is common because falsely approving an ineffective drug has severe consequences. In social sciences, α = 0.05 is conventional. In particle physics, the 5σ standard corresponds to α ≈ 2.87 × 10⁻⁷. Some researchers advocate moving the default from 0.05 to 0.005 (Benjamin et al., 2018) to reduce irreproducibility. Always report exact p-values alongside critical value comparisons.

Textbook tables are truncated to 3-4 decimal places due to print constraints and were computed from approximations available decades ago. This calculator uses Acklam's rational approximation for the normal distribution (relative error < 1.15 × 10⁻⁹) and continued fraction expansions for Beta and Gamma functions with convergence to machine precision. The extra digits matter when computing p-values near decision boundaries or when cascading calculations (e.g., Bonferroni-corrected thresholds for multiple comparisons) amplify rounding errors.

Yes. Select "Left-tailed" to find the value below which a specified proportion α of the χ² distribution falls. Left-tail χ² critical values are used in confidence interval construction for population variance: the lower bound uses χ²(α/2) and the upper bound uses χ²(1 − α/2). For example, with df = 10 and α = 0.05 (left-tail), the critical value is approximately 3.9403, meaning only 5% of the χ² distribution with 10 degrees of freedom lies below this value.