How to Find a Critical Value

The most common mistake is treating "n ≥ 30 means use Z" as a rule, and understanding how to find critical value begins with the choice between Z and t depending on whether the population standard deviation (sigma) is known, not on sample size alone. Finding a critical value by hand means looking up the right number from a table or computing it with an inverse function. The first step is always the same: decide which distribution your test statistic follows.

If sigma is known, use the Z (standard normal) distribution. If sigma is unknown and estimated from the sample standard deviation (s), use the t distribution. For goodness-of-fit or tests of variance, use the chi-square distribution. For ANOVA or comparing multiple group variances, use the F distribution. Each lookup, the reasoning behind the table setup, and the four ways students most often get the wrong number are explained.

A critical value is the boundary of the rejection region for a given alpha (α), the probability of a Type I error. The first question is: one tail or two?

For a one-tailed test, put all of α into one tail. For a right-tailed test, you need the value that leaves α in the right tail. For a left-tailed test, you need the value that leaves α in the left tail. For a two-tailed test, split α equally: α/2 in each tail. This means the critical value is the quantile at cumulative probability 1 − α (right tail), α (left tail), or 1 − α/2 (two tail).

The relationship is simple: smaller α makes the critical value larger. At α = 0.05 two-tailed, the Z critical value is 1.96. At α = 0.01 two-tailed, it is 2.576. The choice of α is a disciplinary convention, psychology uses 0.05; physics may use 0.01, not a mathematical requirement.

Degrees of freedom (df) determine the shape of the t, chi-square, and F distributions.For a chi-square goodness-of-fit test, df = number of categories minus 1, minus any estimated parameters. For an F test in ANOVA, df₁ is the numerator degrees of freedom (number of groups minus 1) and df₂ is the denominator degrees of freedom (total observations minus number of groups).

More df means the t distribution approaches the Z distribution. For chi-square and F, more df shifts the distribution right and makes it more symmetric. Printed tables list critical values for common df values; software gives exact values for any df.

The standard normal table (Z-table) gives the cumulative probability from negative infinity up to a given Z-score. To find a critical value for a right-tailed test, look up 1 − α. For a two-tailed test, look up 1 − α/2. For a left-tailed test, look up α.

The most common mistake is looking up 1 − α for a two-tailed test instead of 1 − α/2. At α = 0.05 two-tailed, 1 − α/2 = 0.975, which gives Z = 1.96. Using 1 − α = 0.95 gives Z = 1.645, the one-tailed critical value. That difference of 0.315 changes your rejection decision.

NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7, publishes exact Z critical values. Moore, McCabe & Craig, Introduction to the Practice of Statistics (10th ed.), gives the standard normal table with cumulative probabilities to four decimal places.

t tables are laid out the same way as Z tables but require df. Find your df row and your tail probability column. Most printed t tables give values for common alphas at each df. For df = 10 and α = 0.05 two-tailed, the critical value is 2.228 (from OpenIntro Statistics, 4th ed.).

Chi-square tables are asymmetric and always positive. Hypothesis tests use the right-tail critical value. For df = 5 and α = 0.05, the chi-square critical value is 11.07 (NIST/SEMATECH). Do not use the left-tail value, that gives a number near zero, not the threshold.

F tables require two df values: numerator (df₁) and denominator (df₂). For df₁ = 3, df₂ = 20 at α = 0.05, the critical value is 3.10 (rounded from software value 3.098). Most printed F tables list only α = 0.05 and 0.01; for other alphas, use software.

Interpolation

If your exact df or alpha is not in the printed table, interpolate linearly between the two nearest values. Software gives exact quantiles and makes interpolation unnecessary.

Every critical value is a quantile: the value below which a given proportion of the distribution falls. The inverse cumulative distribution function (inverse CDF) returns the quantile for a given probability. For Z, this is written Φ⁻¹(p). For t, it is t⁻¹(p, df). For chi-square, it is χ²⁻¹(p, df). For F, it is F⁻¹(p, df₁, df₂).

Software functions implement these inverse formulas directly. On a TI-84 Plus CE (OS 5.6), use invNorm for Z and invT for t. Excel and Google Sheets use NORM.S.INV for Z, T.INV for t, CHISQ.INV.RT for chi-square right-tail, and F.INV.RT for F right-tail. The TI-84 does not have native chi-square or F inverse functions; use the chi-square cdf and trial-and-error, or use a different tool.

Example 1: Right-Tailed Z Test

A battery manufacturer claims its batteries last more than a certain number of hours. Sample n = 36, population sigma = 10 hours, α = 0.05. Sigma is known, so use Z. Right-tailed test, so use α directly: cumulative probability = 1 − 0.05 = 0.95. The Z-table gives Z = 1.645. If the sample mean is 103, the test statistic is Z = (103 − 100) / (10 / √36) = 3 / 1.667 = 1.80. Since 1.80 > 1.645, reject H₀.

Example 2: Two-Tailed t Test

A diet program claims average weight loss of 5 kg. Sample n = 11, sample mean = 4.2 kg, sample SD = 1.5 kg, α = 0.05. Sigma unknown, so use t. Two-tailed, so use α/2 = 0.025 in each tail. Cumulative probability = 1 − 0.025 = 0.975. df = n − 1 = 10. The t-table gives critical value = 2.228. Test statistic = (4.2 − 5) / (1.5 / √11) = −0.8 / 0.452 = −1.77. |−1.77| = 1.77 < 2.228, so fail to reject H₀.

Example 3: Chi-Square Goodness-of-Fit

A test for equal preference among 4 categories, α = 0.05. df = 4 − 1 = 3. Chi-square test is right-tailed. Look up χ² at α = 0.05, df = 3. NIST/SEMATECH gives 7.815. If the test statistic is 9.2, reject H₀.

Example 4: F Test in ANOVA

Three groups (df₁ = 2), total n = 24 (df₂ = 21), α = 0.05. Right-tailed F test. From NIST/SEMATECH, F = 3.10 (rounded). If the F statistic is 4.5, reject H₀.

• Using Z when sigma is unknown. Result: critical value is too small, inflating Type I error. Use t instead.
• Using the wrong tail for chi-square or F. Result: the critical value is near zero instead of the actual threshold. Hypothesis tests use the right-tail critical value.
• Forgetting to halve alpha for two-tailed tests. Result: critical value is too small, rejecting the null too often.
• Misreading degrees of freedom for F. Result: critical value from the wrong row/column. Numerator df = number of groups − 1. Denominator df = total observations − number of groups.
• Using printed tables without interpolation. Result: approximate critical value. Software gives exact values.