Error propagation equation

The error propagation equations (Mandel 1984, Bevington and Robinson 1992) can be used to estimate the standard deviation in the outcome of a calculation if the standard deviations of the inputs are known. For the most common operations, the error propagation rules are summarized in box 1.

The conditions imposed for use of the error propagation equation are:

  • The uncertainties are relatively small, the standard deviation divided by the mean value being less than 0.3;
  • The uncertainties have Gaussian (normal) distributions;
  • The uncertainties have no significant covariance.

Under these conditions, the error propagation equations can be applied. The method can be extended to allow non-Gaussian distributions and to allow for covariances (see e.g.: http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm).

 

Sorts and locations of uncertainty addressed by the error propagation equations
TIER 1 addresses statistical uncertainty (inexactness) in inputs and parameters and estimates its propagation in simple calculations. It does not treat knowledge uncertainty separately from variability related uncertainty. It provides no insight in the quality of the knowledge base or in issues of value loading.
 
Required resources:
The error propagation equations require no specific hardware or software and can typically be applied on the back of the envelope or on an ordinary scientific calculator, or using a spreadsheet.
Most of the time will be consumed by quantifying the uncertainties in the parameters and inputs, which can be derived from statistics if available or otherwise can for instance be obtained by means of expert elicitation.
 
Strengths and limitations
Typical strengths are:
  • Requires very little resources and skills (but the choice of the aggregation level for the analysis is an important issue that does require skills)
  • Quick (but can be dirty) 
Typical weaknesses are:
  • Has a limited domain of applicability (e.g. near-linearity assumption)
  • The basic error propagation equations cannot cope well with distributions with other shapes than normal (but the method can be extended to account for other distributions).
  • Leads to a tendency to assume that all distributions are normal, even in cases where knowledge of the shape is absent and hence a uniform distribution would be reflecting better the state of knowledge.
  • Can not easily be applied in complex calculations
 
Guidance on application
Do not use the error propagation equation if you do not have good reasons to assume that parameter uncertainty is distributed normally. Use Monte Carlo analysis instead.
 
For further guidance we refer to standard handbooks on statistics and measurement error analysis.
 
Pitfalls
Typical pitfalls in the use of the error propagation equation are:
  • Forgetting that this appraoch takes the model structure and boundaries for granted
  • Bias towards assuming all parameter uncertainty to be distributed normally.
  • Ignoring dependencies and covariance 

 

References
 
Bevington, P. R. and D.K. Robinson, D. K. (1992) Data Reduction and Error Analysis for the Physical Sciences, WCB/McGraw-Hill Boston USA, p.328.
 
IPCC, Good Practice Guidance and Uncertainty Management in National Greenhouse Gas Inventories, IPCC, 2000. Annex 1 Conceptual basis for uncertainty analysis
 
Harry Ku (1966). Notes on the Use of Propagation of Error Formulas, J Research of National Bureau of Standards-C. Engineering and Instrumentation, Vol. 70C, No.4, pp. 263-273.
 
Mandel, J. (1984) The Statistical Analysis of Experimental Data, Dover Publications New York, USA, p.410.