Extraction of error spectrum in Sloan Digital Sky Survey
The variance is the square of the standard deviation, so you can calculate 1$\sigma$ errors of the spectrum by taking the square root of one over the inverse variance of HDU1 (https://data.sdss.org/datamodel/files/SPECTRO_REDUX/RUN2D/PLATE4/spPlate.html#hdu1head). 

(You will get infinity for bad pixels since their inverse variance is set to zero.)