Why is the 95% CI for the median supposed to be $±1.57*IQR/\sqrt{N}$? 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.
 A: 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.
