A.1 In many cases a measurand Y is not measured directly, but is determined from N other quantities X_{1}, X_{2}, ... , X_{N} through a functional relation f:
Included among the quantities X_{i } are corrections (or correction factors) as described in subsection 5.2, as well as quantities that take into account other sources of variability, such as different observers, instruments, samples, laboratories, and times at which observations are made (e.g., different days). Thus the function f of Eq. (A-1) should express not simply a physical law but a measurement process, and in particular, it should contain all quantities that can contribute a significant uncertainty to the measurement result.
A.2 An estimate of the measurand or output quantity Y, denoted by y, is obtained from Eq. (A-1) using input estimates x_{1}, x_{2} , ... , x_{N} for the values of the N input quantities X_{1}, X_{2}, ... , X_{N}. Thus the output estimate y, which is the result of the measurement, is given by
A.3 The combined standard uncertainty of the measurement result y, designated by u_{c}(y) and taken to represent the estimated standard deviation of the result, is the positive square root of the estimated variance u_{c}^{2}(y) obtained from
Equation (A-3) is based on a first-order Taylor series approximation of Y = f (X_{1}, X_{2}, ... , X_{N}) and is conveniently referred to as the law of propagation of uncertainty. The partial derivatives ∂f/∂x_{i } (often referred to as sensitivity coefficients) are equal to ∂f/∂X_{i } evaluated at X_{i } = x_{i }; u(x_{i }) is the standard uncertainty associated with the input estimate x_{i }; and u(x_{i }, x_{j }) is the estimated covariance associated with x_{i } and x_{j }.
A.4 As an example of a Type A evaluation, consider an input quantity X_{i } whose value is estimated from n independent observations X_{i,k} of X_{i } obtained under the same conditions of measurement. In this case the input estimate x_{i } is usually the sample mean
and the standard uncertainty u(x_{i }) to be associated with x_{i } is the estimated standard deviation of the mean
A.5 As an example of a Type B evaluation, consider an input quantity X_{i } whose value is estimated from an assumed rectangular probability distribution of lower limit a_{-} and upper limit a_{+}. In this case the input estimate is usually the expectation of the distribution
and the standard uncertainty u(x_{i }) to be associated with x_{i } is the positive square root of the variance of the distribution
where a = (a_{+} - a_{-})/2 (see subsection 4.6).
NOTE -- When x_{i } is obtained from an assumed distribution, the associated variance is appropriately written as u^{2}(X_{i }) and the associated standard uncertainty as u(X_{i }), but for simplicity, u^{2}(x_{i}) and u(x_{i }) are used. Similar considerations apply to the symbols u_{c}^{2}(y) and u_{c}(y).