Presets
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Lot and sample
A bigger sample lowers the error, but only until it approaches the lot mass.
Ore and particles
Coarse particles and rare, heavy minerals make the sample grade scatter much more.
Model checks
The science
Why sampling scatters
A lot is made of discrete particles, either ore mineral or waste. A sample picks a random fraction p = Ms/ML of them, so the number of ore particles in the sample fluctuates and the measured grade scatters around the true grade. Pierre Gy's fundamental sampling error for a liberated mixture of identical particles is σ² = f d³ c (1/Ms − 1/ML) with the shape factor f = π/6 for spheres, the particle diameter d and the sampling constant c = ((1 − a)/a) [(1 − a) ρm + a ρg], where a is the mass fraction of mineral and ρm, ρg are the densities of mineral and gangue.
What it means
The error falls as 1/√Ms, grows with d³ (eight times the sample for double the particle size) and explodes for rare minerals (small a) and heavy ones (large ρm). Inverting the formula gives the sample mass needed for a target precision. The page compares the formula with repeated random draws of the real particle numbers.
Honest simplifications
All particles have the same size and are fully liberated (ore and waste are separate grains), the selection of every particle is independent with probability p, and only the fundamental error is modelled: no segregation, delimitation, extraction or weighing errors, which in practice often dominate. The falling dots are a schematic of the cutter, not a particle-by-particle simulation of the lot, and the formula needs enough ore grains in the sample to be accurate.
Checks in the page
Check 1 compares the seeded Monte Carlo spread for a reference lot with the formula. Check 2 tests that the sample grade is unbiased. Check 3 draws samples at the mass computed for a target precision and verifies the precision is reached.
What to try
- Load Fine crushed ore after Iron ore, coarse: the scatter shrinks by a large factor because the error scales with the cube of the particle size.
- Load Rare mineral (gold-like 0.5 %): even a 10 kg sample from a 2 tonne lot gives a very wide spread.
- Drag the sample mass slider and watch the dot on graph 2 follow the theory line.
- Read the sample mass for the target precision and enter it: the measured sigma should land close to the target.