Statistical formula for the Tukey test

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The following table shows the notation used in this topic.

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

Note Homogenous subsets are only available via 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.