In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function.
The uncertainty is usually defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage.
Most commonly the error on a quantity, Δx, is given as the standard deviation, σ. Standard deviation is the positive square root of variance, σ2. The value of a quantity and its error are often expressed as x ± Δx. If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one dimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is a 68% probability that the true value lies in the region x ± σ. Note that the percentage 68% is approximate as the exact percentage that corresponds to one standard deviation is slightly larger than this.
If the variables are correlated, then covariance must be taken into account.
Read more about Propagation Of Uncertainty: Linear Combinations, Non-linear Combinations, Caveats and Warnings, Example Formulas, Partial Derivatives
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