Which Critical Value Does Your Test Need?

You know which statistical test to run, but you are stuck at the table: Z or t? Chi-square or F? What degrees of freedom? This is the single most common bottleneck in hypothesis testing, and the answer is not "n ≥ 30" but rather which critical value to use. The correct critical value depends on your test type, the known or unknown population standard deviation, and the degrees of freedom your data create. Match your test to a distribution, then to the right df formula, then to the correct tail.

Ignore the old advice that organises critical values by academic level. An ANOVA does not care if you are an undergraduate or a postdoc. What matters is the distribution your test statistic follows under the null hypothesis. Below is the chooser table that maps the five most common test families to their critical-value distribution, df formula, and tail convention. Keep it open while you work.

This table is the core of the page. For each test, it tells you which critical-value distribution to use, how to calculate degrees of freedom, and whether the test is one-tailed or two-tailed. Print it or bookmark it.

Test: One-sample mean, known σ, Distribution: Z, Df: none, Tails: two-tailed for ≠, one-tailed for < or >. Example: comparing a batch mean to a specification when the process sigma is known from historical data.

Test: One-sample mean, unknown σ, Distribution: t, Df: n − 1, Tails: two-tailed for ≠, one-tailed for < or >. This covers nearly every textbook t-test and every confidence interval for a mean when you use sample SD.

Test: Two-sample means, independent, Distribution: t, Df: Welch's approximation or pooled n₁ + n₂ − 2, Tails: two-tailed for ≠. Use the pooled df when you can assume equal variances; otherwise use the smaller of n₁ − 1 and n₂ − 1 as a conservative df.

Test: One proportion, Distribution: Z, Df: none, Tails: two-tailed for ≠. The proportion test uses the normal approximation; check that n·p₀ and n·(1−p₀) are each ≥ 10.

Test: Two proportions, Distribution: Z, Df: none, Tails: two-tailed for ≠. Same normal approximation rule applies.

Test: One variance (chi-square), Distribution: χ², Df: n − 1, Tails: right-tail for hypothesis tests. The chi-square critical value for a variance test always comes from the right tail; the left tail is used only for confidence intervals on variance.

Test: Two variances (F-test), Distribution: F, Df: numerator = n₁ − 1, denominator = n₂ − 1, Tails: right-tail. Always put the larger sample variance in the numerator so the F-statistic is ≥ 1; the critical value is from the right tail of the F-distribution.

Test: ANOVA (one-way), Distribution: F, Df: numerator = k − 1 (number of groups minus 1), denominator = N − k (total observations minus groups), Tails: right-tail. Same logic extends to factorial and repeated-measures ANOVA: numerator df is the number of levels minus 1 for each factor.

Test: Chi-square goodness-of-fit, Distribution: χ², Df: k − 1 − m (number of categories minus 1 minus number of estimated parameters), Tails: right-tail. The most common mistake here is forgetting to subtract the number of parameters estimated from the data (e.g., for a normal distribution, subtract 2 for μ and σ).

Test: Chi-square test of independence, Distribution: χ², Df: (rows − 1)(columns − 1), Tails: right-tail. Works for any contingency table.

Test: Regression slope, Distribution: t, Df: n − 2, Tails: two-tailed for ≠. The test of whether the population slope β₁ is zero uses a t-distribution with df = n − 2; the critical value is the same as for a two-sample t-test with that df.

Test: Correlation coefficient, Distribution: t, Df: n − 2, Tails: two-tailed for ≠. Testing whether Pearson's r is significantly different from zero uses the same t-distribution and df as the regression slope test.

The most persistent error in introductory statistics is the claim that n ≥ 30 justifies the Z distribution. The OpenIntro Statistics textbook (4th ed.) and Moore, McCabe & Craig's Introduction to the Practice of Statistics both state the correct rule: use Z when the population standard deviation (σ) is known; use t when σ is unknown and estimated by the sample standard deviation (s). Sample size does not change this. A sample of n = 5 with known σ uses Z; a sample of n = 500 with unknown σ uses t.

The t-distribution converges to Z as degrees of freedom increase. At df = 30 the critical values are very close, which is why many textbooks present the n ≥ 30 shortcut as a pedagogical convenience. But the shortcut breaks when you need precision: at α = 0.05 two-tailed, Z gives 1.96, t gives 2.042 at df = 30. That 0.082 difference can change a borderline rejection decision. Use the correct distribution; let software handle the df.

Proportion tests use the Z-distribution regardless of sample size, because the test statistic is based on the normal approximation to the binomial. There is no t-distribution for proportions. The only requirement is that the normal approximation is valid: n·p₀ and n·(1−p₀) must each be at least 10 for a one-sample test. For a two-proportion test, the condition applies to each sample separately using the pooled proportion.

The critical value for a proportion test at α = 0.05 two-tailed is 1.96, same as for a mean with known σ. The Z critical value is always the same; it does not vary with sample size. If your sample is too small for the normal approximation, you need Fisher's exact test or a binomial test, not a Z critical value.

Tests for variances use the F-distribution. The F critical value requires two degrees of freedom: numerator df and denominator df. For a one-way ANOVA, numerator df = k − 1 (number of groups minus 1), denominator df = N − k (total observations minus groups). The test is always right-tailed because the F-statistic is a ratio of variances and the larger variance goes in the numerator.

The NIST/SEMATECH e-Handbook of Statistical Methods (section 1.3.6.7) provides F critical value tables for α = 0.05 and α = 0.01. For non-standard alpha levels, use the F.INV.RT function in Excel or Google Sheets. The TI-84 does not have a native F inverse function; you must use a different tool or interpolate from printed tables. When reporting results in a paper, use the software value to the full precision it provides, the difference between 3.10 and 3.098 matters for exact reporting.

Chi-square tests for goodness-of-fit and independence use the χ²-distribution, which is asymmetric and always positive. The critical value always comes from the right tail. A common failure mode is using a left-tail critical value, which returns a number near zero, that is the wrong threshold for hypothesis testing. Left-tail chi-square critical values are used only for confidence intervals on a single variance.

Degrees of freedom depend on the test. For a goodness-of-fit test, df = k − 1 − m, where k is the number of categories and m is the number of parameters estimated from the data. For a test of independence in a contingency table, df = (rows − 1)(columns − 1). The chi-square critical value at α = 0.05 for df = 5 is 11.07, from the right tail. Software gives 11.070; the difference is negligible. The NIST/SEMATECH e-Handbook tables are the source for exact values at standard alpha levels.

Testing whether a regression slope is zero or whether a correlation coefficient is significant both use the t-distribution with df = n − 2. The critical value is the same as for a one-sample mean test with unknown σ at that df. For α = 0.05 two-tailed and df = 20, the critical value is 2.086. For df = 100, it is approximately 1.984, approaching the Z critical value of 1.96 as sample size increases.

The test statistic for a regression slope is the estimated slope divided by its standard error. For a correlation coefficient, the test statistic is t = r√((n−2)/(1−r²)). Both are compared to the t critical value. The TI-84 invT function works for these tests; enter the left-tail area (α/2 for two-tailed tests, α for one-tailed) and the df.

These critical values cover Z, t, chi-square, and F distributions for parametric tests. They do not apply to non-parametric tests such as the Mann-Whitney U or Wilcoxon signed-rank test, which have their own tables. They also do not apply when you need a p-value directly rather than a threshold comparison, use a p-value calculator instead. And they do not replace effect-size reporting: as the ASA Statement on Statistical Significance and P-Values (2016) makes clear, statistical significance is not equivalent to practical importance.

The single thing that most often goes wrong: using the wrong tail for chi-square or F tests. Every hypothesis test for variance, goodness-of-fit, independence, and ANOVA uses the right-tail critical value. The left-tail is for confidence intervals only. If your critical value looks suspiciously small (e.g., close to zero for chi-square), you have used the left tail. Recompute with the right-tail inverse function: CHISQ.INV.RT in Excel or Sheets.