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Methods for Calculating the Mean of a Lognormal Distribution

All of the following methods are based upon a distribution in which n = 9, and the constituent values are:

  1. Arithmetic Mean
    graphic
     
    formula
  2. Swanson's Mean (Msw): 
    formula
  3. Statistical Mean (Mst): assumes a continuous distribution, (i.e., n = ∞),with contribution from values greater than P1% 
    formula
    Where μ = In median (P50%) value

    σ = In standard deviation (P16% ÷ P50%)

  4. Statistical Mean, truncated above Pl% (Mst [t > 1%]): 
    formula

NOTE: Upper truncation is also discussed on pages 14 and 20.

NOTE: The statistical mean is theoretically the preferred expression. However, I recommend against its general use, for the following reasons:

  1. For field-size distributions, we do not ordinarily anticipate that a given trend or basin will have an infinite number of fields; instead we generally find tens to perhaps a few hundreds of fields. Accordingly, the continuous distribution, when η = °°, seems inappropriate, and leads to an inflated mean field size.

  2. For prospect parameters such as Area, Average Net Pay, Gross Rock Volume, HC-recovery in barrels per acre-foot (bbl/af) or thousand cubic feet per acre-foot (mcf/af), Prospect Reserves, Initial Production Rates, etc., the recommended probabilistic estimating connotations of P99%, P90%, P50%, P10%, and Pl% treat those very large outcomes greater than Pl% as practically and geologically impossible. Because the statistical mean includes contributing values larger than Pl%, such untruncated means are unrealistically large. The P99% and Pl% estimates then become very useful as lower and upper plausibility or credibility checks that encourage iterations leading to greatly improved estimates.

  3. When such distributions are truncated at Pl% and the truncated mean is calculated by incremental summing and averaging, such truncated mean values approach values obtained by using Swanson's Mean. Truncation at the small end of reserves distributions, reflecting commercial or economic thresholds, reinforces the practical utility of Swanson's Mean.

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