Studentized Range (Tukey q) Table

k is the number of group means in the whole family of comparisons, not the number of pairs. If your ANOVA compared four groups, use k = 4 even when you are only looking at one particular pair.

Use the error (within-groups) degrees of freedom from the ANOVA table, which is the total number of observations minus the number of groups, N − k.

Use the next smaller df that is listed, which gives a slightly larger, more conservative critical value. Between 24 and 120 the values change slowly, so linear interpolation on 1/df is also accurate enough for coursework.

With one error degree of freedom the estimate of the standard deviation is extremely unreliable, so an observed range has to be enormous relative to it before it counts as significant. The values fall quickly as df grows.

Each value is the upper α quantile of the studentized range distribution, found by numerically integrating the distribution of the range of k normal variables against the distribution of the sample standard deviation, then solving for q. Every one of the 1,482 cells was checked against SciPy's studentized_range and agrees to three decimal places.