An ANOVA test is used to find out if there is a significant difference between three or more group means. However, the ANOVA analysis simply indicates there is a difference between two or more group means, but it does not tell you what means there is a significant difference between. For example, in the table below there are five groups listed. An ANOVA indicated there is a significant difference! Is it between mean I & II; means II & IV? Means III and IV? In order to find out what means there is a significant difference between, a post hoc test needs to be done. The Tukey Test is a post hoc test designed to perform a pairwise comparison of the means to see where the significant difference is!

The first step in doing a Tukey Test is to arrange the means in ascending order in a comparison table and to calculate the difference between each pair of means.

Means in Ascending Order
		Grp 4	Grp 5	Grp 3	Grp 2	Grp 1
		17.8	32.5	33.4	46.6	70.2
Grp 4	17.8	----	14.7	15.6	28.8	52.4
Grp 5	32.5	----	----	0.9	14.1	37.7
Grp 3	33.4	----	----	----	13.2	36.8
Grp 2	46.6	----	----	----	----	23.6
Grp 1	70.2	----	----	----	-----	----

The next step is to calculate the minimum pairwise difference needed. Use the following formula.

D_min= Q_t (sq rt(MS_{within group}))/(sq rt(S))

where

D_min is the minimum pairwise difference needed for significance

Q_t is the t statistic for d_f within group, number of means being tested and alpha level




MS is the Mean Square Error(within group) from ANOVA

S is the group sample size (number of data point in each cell of the ANOVA)

Example

D_min= Q_t (sq rt(MS_{within group}))/(sq rt(S))

D_min= 4.89(sq rt(113.4))/(sq rt(36))

D_min= 4.89(10.65)/(6)

D_min= 8.68

The final step is to compare the difference between the means in the table you constructed to the minimum pairwise difference. The ones that are larger than the minimum are the means pairs that are significantly different.