I think John Tukey's idea was that this formula (or just the fact ofusing median and quartiles) is still often approximately correct for quite a few kinds of moderate contaminations...
It may be approximately correct for the width of a CI (and when I checked it was only appproximately correct for a normal), but I would seriously doubt if it were approximately correct for a significance level of 5%.
Remember how fast the tails of the asymptotic normal distribution decay: a 20% error turns 5% into 2%.
BTW, if there is a precise reference for this it would be good to add it to boxplot.stats.Rd, as the confidence limits are unexplained there.
The factor 1.58 for H-spr/\sqrt{n} comes from the product of three approximations going from a 95%
confidence interval for a difference in means, to one for a difference in medians, using the H-spr=IQR
instead of the standard deviation:
H-spr/1.349 \approx \sigma in a N(0,1) dist/n
\sqrt{ \pi / 2} \approx std error of a median
1.7 / sqrt{n} is the average of 1.96 and 1.39=1.96/\sqrt{2}, factors for the standard error of the difference
between two means, in the cases where one variance is tiny, and where both are equal.
I believe this is explained in
@Article{McGill-etal:78,
author = "R. McGill and J. W. Tukey and W. Larsen",
year = "1978",
title = "Variations of Box Plots",
journal = TAS,
volume = "32",
pages = "12--16",
}--
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