# 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.

• Commented Apr 24, 2017 at 14:18

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.