So how can the joint effects of many sources of uncertainty be understood? First, apply all corrections (e.g., from calibration) to compensate for the known systematic errors.The relevant uncertainty to associate with each correction is the standard uncertainty of the applied correction.
Note: The uncertainty of a correction applied to a measurement result in order to compensate for the systematic effect is NOT the systematic error in the result for that effect. Rather, it is a measure of the uncertainty of the result due to incomplete knowledge of the required value of the correction. The difference between error and uncertainty should always be kept in mind. For example, the result of a measurement after correction can unknowably be very close to the unknown value of the measurand, and thus have a negligible error, even though it may have a large uncertainty.
(From NIST Guidelines).
After compensating for the systematic errors, the combined standard uncertainty can be estimated using the law of propagation of uncertainty, since in many cases, the measurand Y (using NIST notation) is not measured directly, but is determined from N other quantities X1, X2, ...XN such that
Included among the quantities needed to determine Y are the corrections for systematic errors and the correlations among the basic measurements used to determine Y.
In keeping with the NIST notation, the estimate of the measurand (output quantity) Y, denoted by y, is obtained from input quantity estimates x1, x, ...xN using the same functional relationship as above: y = f(x1, x2, ...xN). The combined standard uncertainty uc(y) of the measurement result y is taken as the positive square root of the estimated variance obtained from
Where u(xi) is the standard uncertainty associated with the estimate xiand u(xi, xj) is the covariance associated with xi and xj. The partial derivatives are the sensitivity coefficients (static sensitivities) of the relationship between y and xi. The equation above is based on a first order Taylor series approximation of Y = f(X1, X2, ...XN) and is referred to as the Law of Propagation of Uncertainty.
Note: It is recommended that the number of degrees of freedom associated with uncertainty estimate (e.g., N-1 where N is the number of measurements entering a standard deviation) be reported where applicable. This is often omitted when large, but when small it can affect how the uncertainty estimate is used. Some recommendations regarding how to determine degrees of freedom for uncertainty estimates arising from many components are contained in the NIST document referenced earlier; see the Welch-Satterthwaite formula.