Propagation of Error

In statistics, propagation of error (or propagation of uncertainty) 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. 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, σ ². 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 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 .

If the variables are correlated, then covariance must be taken into account.

Error estimates for non-linear functions are biased on account of using a truncated series expansion. The extent of this bias depends on the nature of the function. For example, the bias on the error calculated for log x increases as x increases since the expansion to 1+x is a good approximation only when x is small.

In data-fitting applications it is often possible to assume that measurements errors are uncorrelated. Nevertheless, parameters derived from these measurements, such as least-squares parameters, will be correlated. For example, in linear regression, the errors on slope and intercept will be correlated and the term with the correlation coefficient, Δ , can make a significant contribution to the error on a calculated value.

In the special case of the inverse 1 / B where B = N(0,1), the distribution is a Cauchy distribution and there is no definable variance. For such ratio distributions, there can be defined probabilities for intervals which can be defined either by Monte Carlo simulation, or, in some cases, by using the Geary-Hinkley transformation.