Critical Value Calculator

A few Z cutoffs appear so often they are worth memorizing:

The two-tailed value at α = 0.05, ±1.96, doubles as the multiplier in 95% confidence intervals.

A critical value is the boundary of the rejection region in a hypothesis test. It is chosen so that, if the null hypothesis were true, the probability of the test statistic landing beyond the boundary equals the significance level α. Observing a statistic past the cutoff is therefore rare enough — by your own chosen standard — to justify rejecting the null.

Formally, the critical value is a quantile of the test statistic's sampling distribution. The calculator inverts the cumulative distribution function to find it:

• Z: Use when the population standard deviation is known or the sample is large (roughly n ≥ 30), and for proportion tests. The distribution never changes, which is why Z tables are so compact.
• t: Use for means when the population standard deviation is estimated from the sample. It has one parameter, degrees of freedom (df = n − 1 for a one-sample test); smaller df means heavier tails and larger critical values.
• Chi-square: Use for goodness-of-fit and independence tests (df tied to the number of categories) and for tests about a single variance. Values are always positive and the distribution is right-skewed.
• F: Use for comparing variances and for ANOVA. It carries two df parameters — numerator (between groups) and denominator (within groups) — and order matters: F(3, 12) is not F(12, 3).

The critical value approach and the p-value approach always reach the same accept/reject decision; they just phrase it differently. The critical value fixes a threshold on the statistic scale before the test: reject if the statistic is more extreme than the cutoff. The p-value works on the probability scale after the test: reject if the tail probability of the observed statistic is below α.

Critical values shine when the same threshold is reused many times — control charts, acceptance sampling, published tables — while p-values communicate how strong the evidence is, not merely whether it crossed the line. Reporting both is common in modern practice.

For the t distribution, critical values shrink toward the Z value as the sample grows. At α = 0.05 two-tailed, the cutoff is 2.5706 with df = 5, 2.0860 with df = 20, 2.0423 with df = 30, and 1.9623 with df = 1000 — essentially the normal 1.96. The extra margin at small df is the price of estimating the standard deviation from limited data.

Chi-square and F critical values behave differently because their distributions change shape with df. A right-tailed chi-square cutoff at α = 0.05 grows with df (18.3070 at df = 10) since the whole distribution shifts right, while F cutoffs generally fall as the denominator df increases and the estimate of within-group variance stabilizes.

A researcher compares a sample of n = 21 reaction times against a published benchmark using a two-tailed one-sample t-test at α = 0.05. The critical value comes from three steps:

1. Degrees of freedom: df = n − 1 = 21 − 1 = 20.
2. Split α across both tails: α/2 = 0.025 in each tail, so the cutoff sits at cumulative probability 1 − 0.025 = 0.975.
3. Invert the t CDF: t₀.₉₇₅,₂₀ = 2.0860, giving critical values of ±2.0860.

The decision rule is: reject H₀ if |t| > 2.0860. If the study produces t = 2.30, that exceeds 2.0860, so the null hypothesis is rejected at the 5% level. Note the same statistic compared against the Z cutoff of ±1.96 would also reject — but with df = 5 the t cutoff would have been ±2.5706, and t = 2.30 would not reject. Small samples demand stronger evidence.

For comparison, selecting Chi-Square with α = 0.05 and df = 10 returns 18.3070, and selecting F with α = 0.05, df₁ = 3, df₂ = 12 returns 3.4903 — the threshold an ANOVA F statistic from four groups of four observations would need to clear.