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A Simple Formula for Calculating the “Mass Density” of a Lognormally Distributed Characteristic: Applications to Risk Analysis

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  • Adam M. Finkel

Abstract

Statements such as “80% of the employees do 20% of the work” or “the richest 1% of society controls 10% of its assets” are commonly used to describe the distribution or concentration of a variable characteristic within a population. Analogous statements can be constructed to reflect the relationship between probability and concentration for unvarying quantities surrounded by uncertainty. Both kinds of statements represent specific usages of a general relationship, the “mass density function,” that is not widely exploited in risk analysis and management. This paper derives a simple formula for the mass density function when the uncertainty and/or the variability in a quantity is lognormally distributed; the formula gives the risk analyst an exact, “back‐of‐the‐envelope” method for determining the fraction of the total amount of a quantity contained within any portion of its distribution. For example, if exposures to a toxicant are lognormally distributed with σin x= 2, 50% of all the exposure is borne by the 2.3% of persons most heavily exposed. Implications of this formula for various issues in risk assessment are explored, including: (1) the marginal benefits of risk reduction; (2) distributional equity and risk perception; (3) accurate confidence intervals for the population mean when a limited set of data is available; (4) the possible biases introduced by the uncritical assumption that extreme “outliers” exist; and (5) the calculation of the value of new information.

Suggested Citation

  • Adam M. Finkel, 1990. "A Simple Formula for Calculating the “Mass Density” of a Lognormally Distributed Characteristic: Applications to Risk Analysis," Risk Analysis, John Wiley & Sons, vol. 10(2), pages 291-301, June.
    • Handle: RePEc:wly:riskan:v:10:y:1990:i:2:p:291-301
    DOI: 10.1111/j.1539-6924.1990.tb01050.x
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    References listed on IDEAS

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    Cited by:

    1. Adam M. Finkel & George Gray, 2018. "Taking the reins: how regulatory decision-makers can stop being hijacked by uncertainty," Environment Systems and Decisions, Springer, vol. 38(2), pages 230-238, June.
    2. Carl F. Cranor & Adam M. Finkel, 2018. "Toward the usable recognition of individual benefits and costs in regulatory analysis and governance," Regulation & Governance, John Wiley & Sons, vol. 12(1), pages 131-149, March.
    3. LECLUYSE, C. & VAN WOENSEL, Tom & PEREMANS, Herbert, 2007. "Vehicle routing with stochastic time-dependent travel times," Working Papers 2007018, University of Antwerp, Faculty of Business and Economics.

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