# Understanding Tukey’s q & p

For a planned SINGLE comparison, the CI of $\mu_2-\mu_1$ can be interpreted as $(M_2-M_1)\pm qt_{1-\frac{\alpha}{2},df}\times SE$, in which $SE=s\times \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}$. The $df$ and $s$ should be defined according to the denominator of $F$.

For post hoc multiple comparisons, the CI of $\mu_2-\mu_1$ can still be interpreted as $(M_2-M_1)\pm \frac{qtukey_{1-\alpha,nmeans,df}}{\sqrt{2}}\times SE$, wherein $nmeans$ indicates the number of sub-groups whose means are compared. Note that qtukey(...) and ptukey(...) is defined with a two-tailed probability while qt(...) and pt(...) with a single-tailed one.

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