Page 299 - Materials Chemistry, Second Edition

11 Uncertainty Management and Sensitivity Analysis 285

this distribution is right skewed. For a log-normally distributed input parameter, the
corresponding distribution and the 95% uncertainty interval are depicted in
Fig. 11.7b. The 95% uncertainty interval (approximately) corresponds to the inte-
2r
gration over the range m=e 2r to m e . The exponential term is thereby defined as
the squared geometric standard deviation:
2r
2
GSD , e : ð11:1Þ
2
With that, the GSD is used to define the 2.5th and 97.5th %iles, i.e. the 95%
uncertainty interval bounds, of a log-normal probability distribution around the
median m of x as

m 2
Probability \x\m GSD 0:95: ð11:2Þ
GSD 2
The uncertainty intervals as discussed above should be distinguished from the
confidence intervals. In practice, a population parameter (mean, median or standard
deviation) is often unknown. In statistical data analysis, confidence intervals are
usually calculated, that is the estimated range of values that frequently contains the
“true” value of the unknown population parameter, if the sampling procedure is
repeated. We need here to clarify some common misconceptions around the in-
terpretation of confidence intervals. For our example on body weight, suppose a
95% confidence interval for the unknown true mean weight that ranges from a to
b (a < b). The statements “95% of the population weighs between a and b kilo-
grams” or “There is a 95% chance that the mean weight of the population lies
between a and b kilograms” are false. The correct interpretation is “If we were to
repeat the weight measurement over and over, then 95% of the time, on average, the
confidence intervals contain the true mean weight.” The latter does not refer directly
to a property of the population parameter, but a property of the procedure itself.
Two useful further readings on common misconceptions and misinterpretations of
confidence intervals and other statistical methods and parameters are the papers
from Greenland et al. (2016) and Hoekstra et al. (2014). For a further study of
confidence intervals, and all the concepts presented in this section as well, the
reader can refer to bibliography in probability and statistics, e.g. Walpole et al.
(2012).
The type of distribution is an important element to precisely describe the
uncertainty of a parameter. The simplifying assumption of a certain type of dis-
tribution (in LCA typically log-normal), instead of attempting to identify the exact
distribution, is very useful when little or no information is available about a
parameter or when using simplified, approximate analytical uncertainty propagation
methods. However, this is sometimes met with criticism by practitioners who would
like to integrate uncertainty information into their LCA studies using the exact
distribution type. While from a purely statistical point of view this is the ideal, the
very large number of variables and their distributions for individual inventory data
and characterisation factors used to quantify the uncertainty of an impact score, will