"Bulk density" for tailings piles usually devolves to a simple weight of sample (dried or wet with some estimation of moisture content) and volume of hole that the sample was collected from. This assumes something like auger or sonic drilling and stable hole walls. The methods may vary according to pile thickness and variability of the material in that pile probably no surprises that the material at the bottom of a thick pile will be far more compacted than material at the top. If holes are unstable, then excavated test pit volumes can possibly be used. Test pits have their limitations as they often only test the top-less compacted portion of the pile; base of the pile is under-represented.

As for "statistically significant" you'll probably get a range of answers. More is always better when it comes to data. Honestly it probably depends on how variable the results are that you're getting how accurate your data is, and the expected variability within the pile. Assuming a large area to be covered, representative samples of reasonably homogenous material/consistent population within your dataset, it would be probably be nice to have 100+ samples from a numerical perspective. However, practicalities come into it, and, if doing relatively large volume test pits, you might not be able to justify that many.

In the absence of any knowledge about your project, that's what I'd probably do. Will be interesting to see what the other contributors come back with. 

I can give you an example method for deriving a density for tailings, if you are still after measurement methods.

To estimate friable near-surface material, we used dry running sand of known density (weigh a vessel of known volume), and backfilled auger holes from bags of known mass. The volume of the auger hole was derived from the mass of sand used to back fill it. We augered through a 100mm hole in cardboard to ensure capturing all the spoil including cleaning off the auger flight after each hole, which was collected and weighed. Hence you get a mass and a derived volume to calculate BD. The sample was than dried to derive moisture content.

The issue with this method was limited depth penetration (2m max) does not allow for testing the compaction and variable moisture content with depth.

To get around that we used some of the intact pieces of diamond core (the only sample means available), and assumed no bias in sample selection, and no significant moisture change had occurred during drilling which was reasonable for the specifics of that drill program. That was then air weighed, waxed, air weighed, water weighed to derive a moist BD, and the sample then broken up and dried to derive moisture content. We used this sequence to avoid volume change during drying of the sample.

Both methods are a bit messy and make assumptions about what is happening down the drill holes, but since the tails we measured were a small proportion of a larger project, we could live with the assumptions. I would be cautious of using a BD derived from the BD of ore feed since there are many assumptions (provenance, packing factor, etc), and actual measurements should trump theoretical answers. What is the thickness of the tailings?

Was also faced with the challenge of elevated BD numbers owing to the high Magnetite content of the BIF derived sands quite a monster when it comes to tonnage and mineral accounting reconciliations if one does not do their math right in the first place. Once again, thank you for your invaluable input.