# Tukey HSD

Tukey’s Honestly Significant Differences (also known as Tukey’s Whole Significant Differences). The test statistic is:

$t={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{\sqrt {{\frac {\sum _{j=1}^{J}\sum _{i=1}^{n_{j}}w_{ij}(x_{ij}-{\bar {x}}_{j})^{2}}{v}}({\frac {1}{e_{1}}}+{\frac {1}{e_{1}}})}}}$ where:

${\bar {x}}_{1}$ and ${\bar {x}}_{2}$ are the means of the two groups being compared and ${\bar {x}}_{j}$ is the mean of the $j$ of $J$ groups,
when applying the test to Repeated Measures, each respondent’s average is initially subtracted from their data and it is this corrected data that constitutes of $x_{ij}$ ,
$n_{j}$ is the number of observations in the $j$ th of $J$ groups,
$w_{ij}$ is the Calibrated Weight for the $i$ th observation in the $j$ group,
$e_{j}$ is the Effective Sample Size for the $j$ group,
$v=(J-1)(\sum _{j=1}^{J}e_{j}-1)$ for Repeated Measures and $v=\sum _{j=1}^{J}e_{j}-J$ otherwise.
$t$ is evaluated using a Tukey’s Studentized Range distribution with $v$ degrees of freedom for $J$ groups.