Rather than plotting a thick line, use a patch object instead for the standard deviations:
x = 0:99;
y = sin(2*pi*x/25);
sd = rand(size(y));
patch([x, fliplr(x)], [y+sd, fliplr(y-sd)], [1 1 1]*0.8, 'EdgeColor','none')
If you want the ‘y’ line in top of the patch object, plot it after you plot the patch object.