Tukey's Q Table: Studentized Range Distribution Critical Values (α = 0.05 & 0.01)

Tukey's Q Table provides critical values for the Studentized Range statistic q. Researchers use it to carry out Tukey's HSD post-hoc test after a significant one-way ANOVA — identifying which specific pairs of group means differ, while keeping the family-wise error rate at the chosen significance level across all comparisons simultaneously.

Choose your α level (0.05 or 0.01). Count k, the total number of groups in your study. From your ANOVA table, read the within-group df = N − k. Find the column for your k and the row for your df — their intersection is the critical q value. Multiply q by √(MSE / n) to compute the HSD decision threshold.

A t-table supplies a critical value for comparing two means at a fixed per-comparison error rate. Tukey's Q table supplies a critical value for the maximum spread across k ≥ 3 means, holding the family-wise error rate at α regardless of how many pairs you test. Running multiple t-tests inflates the Type I error rate; Tukey's Q table is specifically designed to prevent that inflation.

k is the total number of groups or treatment means being compared — read across the column headers. df (also written ν) is the within-group degrees of freedom, computed as N − k, where N is the total sample size — read down the row index. Both values come directly from the one-way ANOVA summary table.

Use Tukey's HSD when testing all possible pairwise comparisons among three or more means — it has more statistical power than Bonferroni in this scenario because it exploits the correlation structure of the comparisons. Use Bonferroni when you have a small, pre-determined subset of comparisons to test rather than the full set of all pairs.

Yes. For unequal group sizes, apply the Tukey-Kramer modification: replace the single n in the denominator with a pair-specific standard error: SE = √(MSE/2 × (1/n₁ + 1/n₂)). The same critical q value from this table is used; only the SE calculation changes for each pair. Most statistical software applies this correction automatically.

The table begins at df = 1, but q critical values spike at very low degrees of freedom — making the test impractical. Studies with df < 10 will find the HSD threshold so wide that only very large mean differences reach significance. Most researchers aim for df ≥ 20 to achieve reasonable statistical power. The Penn State STAT 503 course materials recommend at least 20 error degrees of freedom for reliable inference.

The table is named after the Studentized Range statistic q — the ratio of the range of a set of independent normal random variables to an independent estimate of their standard deviation. "Studentized" honors William Sealy Gosset (publishing pseudonymously as "Student"), who developed the concept of standardizing by an estimated standard deviation. Tukey extended this into the HSD test in 1949.

In R: qtukey(p = 0.95, nmeans = k, df = df_within) returns the exact critical q at α = 0.05. In Python: from scipy.stats import studentized_range; studentized_range.ppf(0.95, k, df). Both functions compute exact values from the Studentized Range CDF and match the table values here.

A family-wise error rate of 0.05 means there is a 5% chance of incorrectly flagging at least one comparison as significant across the entire set of pairwise tests — not per individual comparison. Without this correction, running 10 independent t-tests at α = 0.05 each carries a ~40% chance of at least one false positive. Tukey's table sets the critical threshold so the overall chance stays at exactly 5%.