Critical Values Table

How to find a critical value

• Start with four choices. Pick the distribution that matches your test: z for a mean with known standard deviation or a large sample proportion, t for a mean with an estimated standard deviation, chi-square for variance and goodness-of-fit or independence tests, F for ANOVA and variance ratios, and Pearson r for a correlation. Pick α. Pick the direction of the alternative hypothesis, which decides whether the test is two-tailed or one-tailed. Last, work out the degrees of freedom (df).
• Then use the calculator above or the quick tables below. For a two-tailed z test at α = 0.05 the value is ±1.960. For a t test with 10 df at the same α it is ±2.228. For chi-square with 5 df and a right-tailed α of 0.05 it is 11.070. For F with 3 and 20 df it is 3.098. As df grows, t approaches z: with infinite df the two-tailed t value is 1.960.
• Confidence level to α: α = 1 − confidence level. For a 90% confidence interval α = 0.10, so the two-tailed z critical value is ±1.645; at 95% it is ±1.960 and at 99% it is ±2.576. The same α with a t distribution gives a larger value for small df.
• In a standard z table, the body holds cumulative areas, so look up 1 − α for a right-tailed test (0.95 for α = 0.05) or 1 − α/2 for a two-tailed test (0.975), then read z from the row and column headers. The area 0.975 sits at z = 1.96.

One-tailed or two-tailed

A two-tailed test looks for a difference in either direction, so α is split between the two tails and each tail holds α/2. A one-tailed test puts all of α in one tail, so the cut-off moves closer to zero: 1.645 instead of 1.960 for z at α = 0.05. Decide the direction from your research question before you see the data, not afterwards. Chi-square and F tests are usually right-tailed because only a large statistic counts as evidence against the null hypothesis. The calculator also returns both cut-offs of a two-tailed chi-square or F test when you need them.

The Pearson r critical values table

• The Pearson r table answers a different question: how large must a sample correlation be before it is unlikely to come from a population with no correlation? The degrees of freedom are n − 2, where n is the number of pairs. Each value is r = t / √(t² + df), with t the critical value of the t distribution at the same df and α. The test assumes a roughly linear relationship and independent pairs, and, for small samples, roughly normal data.
• Take 12 pairs of measurements. Then df = 10. At two-tailed α = 0.05 the table gives 0.576, so a correlation with |r| above 0.576 is significant. At α = 0.01 with 22 pairs (df = 20) the value is 0.537. The same r from a larger sample is easier to call significant, which is why a significant r is not automatically a strong one.

Missing rows and rounding

Printed tables skip some df. For t, F and Pearson r, the cautious habit is to use the next smaller df listed, which gives a slightly larger critical value. Chi-square works the other way: its critical value grows with df, so use the next larger df listed to stay on the safe side, or interpolate between the two rows. The calculator removes the guesswork: type any whole df and it returns the value directly. All values here are rounded to 3 decimals with ordinary rounding, and the calculator also shows more digits. Printed tables can differ in the last digit because of rounding, so check how a textbook rounds and which df it uses.