I have the impression that current error bars on 1D plots of S(k) are computed as the std of the values of S falling in each bin. But we are rather interested in the standard deviation of the empirical mean, which is the same value but divided by the sqrt of the number of points that fall in each bin. A quick sanity check is that the error bar in a bin should get smaller as the number of points in that bin increases.
I have the impression that current error bars on 1D plots of S(k) are computed as the std of the values of S falling in each bin. But we are rather interested in the standard deviation of the empirical mean, which is the same value but divided by the sqrt of the number of points that fall in each bin. A quick sanity check is that the error bar in a bin should get smaller as the number of points in that bin increases.