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Statistical formula for the Tukey test
Notation
Description Number of levels for an effect. j-th observation at i-th level. Harmonic mean of the sample size. Number of observations at level i, i=1,...,k. Mean at level i, i=1,...,k. Sum of squares at level i, i=1,…k. Pooled standard deviation from all levels. Degrees of freedom for the within-groups mean square. Experimentwise error rate under the complete null hypothesis. Number of steps between means.     The upper-ε critical point of the Studentized range distribution.
Range values for all values of r.
When finding the critical value R ε,r,f for any pair of means r columns apart, the value S ε,k,f is always used where k is the total number of columns. The lookup table for ε is queried to find the value in the appropriate row for f and the appropriate column for k, regardless of the value of r.
The confidence intervals of the mean difference are calculated using Q i,j instead of Q h.
For all pairs of means   For any two means i and j being tested Constructing homogeneous subsets
Homogenous subsets are only available in diagnostics (see Diagnostics information: Tukey test).
The outermost pair of means have a significant range if: If so, test whether:  Once a whole set is found to be nonsignificant testing stops.
Each time a range proves nonsignificant, the means involved are included in a single group (homogeneous subset). This mean that the columns within a nonsignificant range should be combined into a single column and the test reapplied with the collapsed sets of columns.
Multiple comparisons test for all possible pairs for all i < j.