= The derivation of the multiplier used for constructing group mean confidence intervals for which non-overlap corresponds to an inference of a difference =

It is important to understand the motivation behind the use of the $$(\sqrt{2}/2)=1/\sqrt{2}$$ multiplier (See Baguley (2012b, p.23)).

Although the ratio of the width of the CI for a difference in a pair of individual means from groups containing different subjects is

V(diff) = $$ \sqrt{Va + Vb} = \sqrt{\mbox{2(Pooled Variance)}} \approx \sqrt{2 V(a)} = \sqrt{2} se(a)$$

for two group means having variances of V(a) and V(b) respectively assuming the group means have similar variances (and hence similar standard errors, se(a) and se(b)). V(diff) will be smaller for groups with the same subjects as the covariance term then needs to be is subtracted.

V(diff), therefore, represents the variance of the difference between __two__ independent group means (or the upper bound for the difference between two related group means) and therefore needs to be halved to represent a __single__ group mean (for later comparisons with another group mean).  

__Reference__

Baguley, T. (2012b). Calculating and graphing within-subject confidence intervals for ANOVA. ''Behavior Research Methods'' 44 158-175.