The Tukey HSD ("honestly significant difference" or "honest significant difference") test is a statistical tool used to determine if the relationship between two sets of data is statistically significant – that is, whether there's a strong chance that an observed numerical change in one value is causally related to an observed change in another value. In other words, the Tukey test is a way to test an experimental hypothesis.

The Tukey test is invoked when you need to determine if the interaction among three or more variables is mutually statistically significant, which unfortunately is not simply a sum or product of the individual levels of significance.

Simple statistics problems involve looking at the effects of one (independent) variable, like the number of hours studied by each student in a class for a particular test, on a second (dependent) variable, like the student's scores on the test. In such cases, you usually set your cut-off for statistical significance at P < 0.05, wherein the experiment reveals a greater than 95 percent chance that the variables in question truly related. Then you refer to a t-table that takes into account the number of data pairs in your experiment to see if your hypothesis was correct.

Sometimes, however, the experiment may look at multiple independent or dependent variables simultaneously. For example, in the above example, the hours of sleep each student got the night before the test and his or her class grade going in might be included. Such multivariate problems require something other than a t-test owing to the sheer number if independently varying relationships.

ANOVA stands for "analysis of variance" and addresses precisely the problem just described. It accounts for the rapidly expanding degrees of freedom in a sample as variables are added. For example, looking at hours vs. scores is one pairing, sleep vs. scores is another, grades vs. scores is a third and meanwhile, all of those independent variables interact with one another, too.

In an ANOVA test, the variable of interest after calculations have been run is F, which is the ​found​ variation of the averages of all of the pairs, or groups, divided by the ​expected​ variation of these averages. The higher this number, the stronger the relationship, and "significance" is usually set at 0.95. Reporting ANOVA results usually requires the use of a built-in calculator such as those found in Microsoft Excel as well as dedicated statistical programs such as SPSS.

John Tukey came up with the test that bears his name when he realized the mathematical pitfalls of trying to use independent P-values to determine the utility of a multiple-variables hypothesis as a whole. At the time, t-tests were being applied to three or more groups, and he considered this dishonest – hence "honestly significant difference."

What his test does is compare the differences between means of values rather than comparing pairs of values. The value of the Tukey test is given by taking the absolute value of the difference between pairs of means and dividing it by the standard error of the mean (SE) as determined by a one-way ANOVA test. The SE is in turn the square root of (variance divided by sample size). An example of an online calculator is listed in the Resources section.

The Tukey test is a post hoc test in that the comparisons between variables are made after the data has already been collected. This differs from an a priori test, in which these comparisons are made in advance. In the former case, you might look at the mile run times of students in three different phys-ed classes one year. In the latter case, you might assign students to one of three teachers and then have them run a timed mile.