Tukey Q Critical Value Calculator

Find the studentized range statistic for post-hoc ANOVA comparisons

Tukey Critical Value
—
Tukey Critical Value
—

Reference Table

Tukey q* critical values for the currently selected α. Your selection is highlighted in green. These table values were calculated with SciPy 1.18.1; they were not transcribed from a publisher table. Between finite rows, results use linear interpolation. Above df = 120, interpolation in 1/df approaches the infinite-df limit. Interpolated results are approximations. Calculation method · Reproduce the table · References and acknowledgements.

How to Use

1
Run your one-way ANOVA and note k (number of groups) and dferror (which is N − k, where N is total sample size).
2
Choose your significance level α. The standard choice is 0.05.
3
Read the q* critical value from the calculator above.
4
Calculate the HSD threshold: HSD = q* × (MSW / n)1/2, where MSW is the mean square within-groups from your ANOVA table and n is the number of observations per group.
5
Compare each pairwise mean difference to HSD. If |xi − xj| exceeds HSD, that pair is statistically significantly different.

Worked Example

Using NorthStar Enterprises employee satisfaction data

NorthStar Enterprises — 4 Division Comparison

k = 4
Groups
df = 36
Error DF
α = 0.05
Significance
MSW = 24.6
Mean Sq Within
n = 10
Per Group

Step 1: Look up q* for k = 4, df = 36, α = 0.05

Since df = 36 is between df = 30 and df = 40 in the table, we interpolate:

q*(30) = 3.845   q*(40) = 3.791

q*(36) = 3.845 + (36 − 30)/(40 − 30) × (3.791 − 3.845) = 3.845 − 0.032 = 3.813

Step 2: Calculate HSD

HSD = q* × (MSW / n)1/2 = 3.813 × (24.6 / 10)1/2 ≈ 3.813 × 1.568 ≈ 5.980 (using unrounded values)

Any two division means differing by more than 5.980 are significantly different at α = 0.05. For example, if Corporate Services (mean 78) vs Manufacturing (mean 65) gives |78 − 65| = 13 > 5.980, that difference is significant.