Table 10.3 of the data quality guideline provides for values, depending on the type of exchange and process modeled. This value reflects the fact that even “perfect” data is uncertain: there are fluctuations over time, errors in measurements, etc. The deterministic value is also called “Geometric mean” in those tools. In ecoEditor and ecoQuery, mu is called “Arithmetic mean of log-transformed data”. Going from the deterministic value to mu is straightforward: this value is taken as equal to mu*. Three inputs are necessary from the data provider to determine the parameters of the lognormal distribution: the deterministic value, the basic uncertainty and the pedigree matrix. The larger the standard deviation, the larger is the skewedness and the further apart those three quantities will be. The mode (the most likely value) is found at a lower value, exp(mu – sigma2). The arithmetic mean is found slightly higher than the geometric mean, at exp(mu + sigma2/2). In the lognormal distribution, the median corresponds to the geometric mean, and is found at exp(mu). The quantity sigma* is useful to calculate intervals of confidence: The median and standard deviation of x, noted mu* and sigma*, can be obtained through the following equations: Where x is the random variable, mu and sigma are the median and standard deviation of the distribution of ln(x) (sometimes called “the underlying normal distribution). The probability density function (PDF) of the lognormal is: 51, No.5.ĭefinition and basic properties of the lognormal distributionĪ variable is lognormally distributed when the logarithm of the sample is normally distributed. As a primer, we recommend “ Log-normal Distributions across the Sciences: Keys and Clues” by Eckhard Limper et al, in BioScience, May 2001, Vol. The lognormal is not as intuitive as the normal distribution and is often confusing to new users. It has the advantage of not being defined in the negative domain, so credits do not accidentally happen during a Monte Carlo simulation. The lognormal is the most common distribution chosen to describe the uncertainty in ecoinvent.