The back for the dresser is a sheet of 1/4″ maple-faced plywood. I rough-cut it to the approximate size of the dresser and then hand-planed it down to the final fit.
I was a little worried because the plywood is, in reality, under-thickness and would not house properly in the 1/4″ grooves I had made in the carcass. But given that the plywood has a little ‘bend’ to it, it presses against the groove in places which prevents it from rattling around. I’m quite happy with the fit.
T r i a l & E r r o r 
space balls or a dab or two of silicone will help with rattles