Accurate mineral sampling is essential for reliable metallurgical accounting, process control, and mineral processing efficiency. Modern mineral sampling systems and ore sampling systems are designed to reduce sampling errors and ensure representative sampling across mining operations.
At the core of modern sampling practices, Pierre Gy’s Theory of Sampling (TOS) provides the scientific framework for evaluating sampling precision, heterogeneity, sample preparation, and representative sampling in mineral processing plants and bulk material handling systems.
THEORETICAL FRAMEWORK: GY’S THEORY OF SAMPLING (TOS)
The Theory of Sampling establishes that total error is composed of:
Where:
- FSE: Fundamental Sampling Error
- GSE: Grouping and Segregation Error
- DE: Delimitation Error
- EE: Extraction Error
- PE: Preparation Error
- ME: Measurement Error
This classification matches the conceptual framework developed by Gy and further expanded by Francis Pitard.
FUNDAMENTAL SAMPLING ERROR (FSE)
Full mathematical development
According to Gy:
Where:
C= constitutional heterogeneity constantfg= granulometric factord= maximum particle size- ms= sample mass
In a more developed form:
Where:
p= fraction of valuable mineralc= grade of the valuable mineral
FSE cannot be eliminated—only reduced by increasing mass or reducing top size. This framework is consistent with ISO ISO 11648-1 regarding the statistical treatment of sampling variances.
General concept of TOTAL ERROR in Pierre Gy’s Theory of Sampling
In Gy’s theory, sampling error is the difference between:
“Lot value” − “Value estimated from the sample”
This error arises because natural materials are heterogeneous; therefore, any sample represents only a part of the lot. Gy shows that total sampling error (TE) can be decomposed into components associated with:
- material heterogeneity,
- the increment-selection process,
- sample preparation,
- process variability over time.
In simplified form:
where each term corresponds to a specific error type.
Error classification according to Pierre Gy
Errors are grouped into two broad categories:
- Errors associated with short-range heterogeneity (related to increment selection).
- Errors associated with long-range heterogeneity (related to temporal variations in the process or material).
The theory considers a total of nine main error types.
The 9 sampling errors according to Pierre Gy
1. Fundamental Error (EF)
The unavoidable error caused by differences among particles in: size, shape, density, and mineralogical composition. It always exists because the material consists of distinct particles.
It depends mainly on:
- maximum particle size,
- sample mass,
- degree of heterogeneity.
It can be estimated using Gy’s Fundamental Error equation.
2. Segregation and Grouping Error (ESG)
Occurs when particles are not uniformly distributed. Examples: size segregation, density segregation, clusters of valuable mineral.
It depends on transport, vibration, free fall, and movement on belts or in chutes.
3. Increment Delimitation Error (ED)
Occurs when the sampling device does not correctly define the increment volume. Examples: cutters that are too small, incorrect cutter geometry, partial stream interception.
This error creates bias.
4. Increment Extraction Error (EE)
Occurs when the increment is not fully extracted from the stream. Examples: cutter too slow, rebound/losses, incomplete cuts.
It also violates equiprobability.
5. Sample Preparation Error (EP)
Occurs during crushing, splitting, pulverizing, and handling. It may be caused by fines loss, contamination, oxidation, or material degradation.
6. Weighting Error (EW)
Occurs when:
“increment mass” ≠ “proportional to the sampled flow”
Typical when the flow varies and the sampling system does not maintain a constant mass ratio.
7. Nugget Effect (EN)
Occurs when the valuable mineral is present as coarse particles (“nuggets”), nuggets/veinlets, etc. Typical example: gold and precious metals.
This generates high variability between samples.
8. Long-range heterogeneity fluctuation error (HFE2)
Represents trends in material quality over time (e.g., changes between mine faces, trucks, geological benches). It is non-random.
9. Periodic fluctuation error (HFE3)
Caused by cyclic process variations (e.g., grinding cycles, periodic operational changes, shift-based blending). Also non-random.
Eliminable vs. unavoidable errors
Unavoidable errors (can only be reduced): EF, ESG, EN, HFE2, HFE3.
Errors eliminable through good design: ED, EE, EW, EP.
These are eliminated by:
- correct sampler design,
- compliance with standards (e.g., ISO 13909),
- controlled preparation protocols.