The uncertainty u can be expressed in a number of ways.
It may be defined by the absolute errorΔx. Uncertainties can also be defined by the relative error(Δx)/x, which is usually written as a percentage.
Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval x ± u. 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 approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases.
If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.
Then, the variance–covariance matrix of f is given by
In component notation, the equation
This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are uncorrelated, the general expression simplifies to
where is the variance of k-th element of the x vector.
Note that even though the errors on x may be uncorrelated, the errors on f are in general correlated; in other words, even if is a diagonal matrix, is in general a full matrix.
The general expressions for a scalar-valued function f are a little simpler (here a is a row vector):
Each covariance term can be expressed in terms of the correlation coefficient by , so that an alternative expression for the variance of f is
In the case that the variables in x are uncorrelated, this simplifies further to
In the simple case of identical coefficients and variances, we find
When f is a set of non-linear combination of the variables x, an interval propagation could be performed in order to compute intervals which contain all consistent values for the variables. In a probabilistic approach, the function f must usually be linearized by approximation to a first-order Taylor series expansion, though in some cases, exact formulas can be derived that do not depend on the expansion as is the case for the exact variance of products. The Taylor expansion would be:
where J is the Jacobian matrix. Since f0 is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, Aki and Akj by the partial derivatives, and . In matrix notation,
That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument.
Note this is equivalent to the matrix expression for the linear case with .
Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:
where represents the standard deviation of the function , represents the standard deviation of , represents the standard deviation of , and so forth.
It is important to note that this formula is based on the linear characteristics of the gradient of and therefore it is a good estimation for the standard deviation of as long as are small enough. Specifically, the linear approximation of has to be close to inside a neighborhood of radius .
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(1+x) increases as x increases, since the expansion to x is a good approximation only when x is near zero.
In the special case of the inverse or reciprocal , where follows a standard normal distribution, the resulting distribution is a reciprocal standard normal distribution, and there is no definable variance.
However, in the slightly more general case of a shifted reciprocal function for following a general normal distribution, then mean and variance statistics do exist in a principal value sense, if the difference between the pole and the mean is real-valued.
This table shows the variances and standard deviations of simple functions of the real variables , with standard deviations covariance, and correlation .
The real-valued coefficients a and b are assumed exactly known (deterministic), i.e., .
In the columns "Variance" and "Standard Deviation", A and B should be understood as expectation values (i.e. values around which we're estimating the uncertainty), and should be understood as the value of the function calculated at the expectation value of .
If A and B are uncorrelated, their difference A-B will have more variance than either of them. An increasing positive correlation () will decrease the variance of the difference, converging to zero variance for perfect correlated variables.
The autocorrelation of A is denoted ; only if the variate is exact (), its self-subtraction has zero variance .
For uncorrelated variables (, ) expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation, gives
For the case we also have Goodman's expression for the exact variance: for the uncorrelated case it is
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^Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". Journal of Sound and Vibrations. 332 (11): 2750–2776. doi:10.1016/j.jsv.2012.12.009.