Understanding “The flaw of averages in a processing plant”
- 30 abr
- 3 Min. de lectura
By Dr. Luis A. Martínez Tipe, PhD Director General & Principal Researcher, CAIDTech Originally published: February 14, 2026
In a previous paper, I discussed “The Flaw of Averages in Mine Project Evaluation” (read it here). The main issue with traditional evaluation methods is that they often rely on single estimated values (averages), which are inadequate when dealing with uncertainty and non-linear processes like mine planning and design optimization. In mining, using average values for uncertain inputs such as grade, recovery, throughput, availability, costs, and prices leads to systematic errors—this is known as the Flaw of Averages. It assumes the outcome reflects expected performance, but in reality, due to the non-linear and constrained nature of mining systems, results from average inputs do not match the true average outcomes under uncertainty.
Jensen’s inequality lies at the heart of this flaw.
To show how quantitative risk analysis (QRA) can address the Flaw of Averages in mine project evaluations, we provide an (simple) example involving its application in a processing plant.
QRA Example – Two SAG Mills with Discrete Availability
This section presents a simplified Quantitative Risk Analysis (QRA) example for a concentrator plant with two SAG mills operating in parallel. The objective is to quantify throughput risk, NPV distribution (P5/P50/P80), probability of meeting plan, and Value-at-Risk (VaR). The example also illustrates the Flaw of Averages. 1. Input data
Two SAG mills in parallel (SAGs are independent).
Nominal capacity: 1 SAG at full runtime = 25 Mt/y ore.
Discrete availability (A) for each SAG (independent) – here we assume Utilization = 1:
Copper grade: 1.2% Cu.
Copper recovery: 90%.
Copper price: 6,000 US$/t Cu.
Variable ore cost: 15 US$/t ore.
Fixed annual opex: 300 MUS$/y.
Discount rate: 10%.
NPV horizon: 10 years, constant annual performance.
2. Production model
Ore processed (Mt/y): Ore = 25 × (A1 + A2)
Copper produced (t/y): Cu = Ore × Grade × Recovery
Revenue (US$/y): Revenue = Cu × Price
Costs (US$/y): Costs = Ore × Variable Cost + Fixed OPEX
Cash Flow (US$/y): CF = Revenue − Costs
3. Discrete operating states
4. Economic outcomes by state
5. QRA results
Throughput (Ore Mt/y) distribution:
Mean = 43.75 Mt/y
P50 = 43.75 Mt/y
P80 = 50.00 Mt/y
P5 = 37.50 Mt/y
Probability of meeting 50 Mt/y plan = 25%
NPV (10y @ 10%) distribution:
Mean NPV = 11,544.1 MUS$
NPV P50 = 11,544.1 MUS$
NPV P80 = 13,456.6 MUS$
NPV P5 = 9,631.6 MUS$
VaR95 (Mean − P5) = 1,912.5 MUS$
6. The Flaw of Averages (explanation)
As stated at the beginning, The Flaw of Averages is the mistake of planning using average inputs as if they represented a typical outcome. That is, The Flaw of Averages is the error of assuming that average inputs lead to an average outcome.
Here, each SAG has E[A] = 0.5×1.00 + 0.5×0.75 = 0.875.
A deterministic estimate would say Ore = 25×(E[A1]+E[A2]) = 25×(2×0.875) = 43.75 Mt/y.
While that equals the mean, it hides the fact that the plant does not operate at the average state throughout the LOM, but:, 25% of the time it produces 37.50 Mt/y , 50% of the time it produces 43.75Mt, and 25% of the time it produces 50.00 Mt/y.
With material fixed costs, downside states reduce revenue while fixed OPEX remains constant, so decision-making must be based on the full distribution (P5/P50/P80), plan compliance probability, and VaR—not on averages.
8. Conclusions
Discrete availability (1.00 / 0.75) yields ore outcomes of 37.50, 43.75, and 50.00 Mt/y with probabilities of 25%, 50%, and 25%, respectively.
Mean ore throughput is 43.75 Mt/y; probability of meeting a 50 Mt/y plan is 25%.
With Fixed OPEX = 300 MUS$/y, the NPV distribution is P5=9,631.6 MUS$, P50=11,544.1 MUS$, P80=13,456.6 MUS$.
VaR95 relative to the mean NPV is 1,912.5 MUS$, quantifying downside exposure beyond point estimates.
QRA solves the Flaw of Averages by making plan compliance and downside risk explicit via probability distributions and sensitivity, enabling bankable planning and mitigation focus.
If this example is relevant to your mine's life-of-mine (LOM) operations and you're curious about how it could be applied to your project alongside other uncertainties, feel free to reach out for a discussion. Send us a message to: lmartinez@randoanalytics.com
