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In various sources (see e.g. here), the following formula is given for confidence interval for median (especially for the purpose of drawing notches on box-and-whisker plots):

$$ 95\%\ CI_{\rm median} = {\rm Median} \pm \frac{1.57\times IQR}{\sqrt{N}} $$

The magic constant $1.57$ makes me crazy, I cannot figure out how it was obtained. Various approximations (e.g., assume that our distribution is Gaussian and $N$ is big) give no clues — I get different values for the constant.

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That's easy. If we check the original paper where notched box-and-whisker plots were introduced (Robert McGill, John W. Tukey and Wayne A. Larsen. Variations of Box Plots, The American Statistician, Vol. 32, No. 1 (Feb., 1978), pp. 12-16; fortunately, it's on JSTOR), we found section 7 where this formula is justified in the following way:

Should one desire a notch indicating a 95 percent confidence interval about each median, C=1.96 would be used. [Here C is different constant which is related to ours, but the exact relation is of no importance as will be clear later — I.S.] However, since a form of "gap gauge" which would indicate significant differences at the 95 percent level was desired, this was not done. It can be shown that C = 1.96 would only be appropriate if the standard deviations of the two groups were vastly different. If they were nearly equal, C = 1.386 would be the appropriate value, with 1.96 resulting in far too stringent a test (far beyond 99 percent). A value between these limits, C = 1.7, was empirically selected as preferable. Thus the notches used were computed as $M \pm 1.7(1.25R/1.35 \sqrt{N})$.

Emphasis is mine. Note that $1.7\times 1.25/1.35=1.57$, which is your magic number.

So, the short answer is: it is not a general formula for median CI but a particular tool for visualization and the constant was empirically selected to achieve a particular goal.

There's no magic.

Sorry.

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