
Combining uncertainty components Calculation of combined standard uncertainty (6) Equation (6) is based on a firstorder Taylor series approximation of the measurement equation Y = f(X_{1}, X_{2}, . . . , X_{N}) given in equation (1) and is conveniently referred to as the law of propagation of uncertainty. The partial derivatives of f with respect to the X_{i }(often referred to as sensitivity coefficients) are equal to the partial derivatives of f with respect to the 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}. Simplified forms 
Measurement result: y = a_{1}x_{1 }+ a_{2}x_{2 }+ . . . a_{N}x_{N} Combined standard uncertainty: u_{c}^{2}(y) = a_{1}^{2}u^{2}(x_{1}) + a_{2}^{2}u^{2}(x_{2}) + . . . a_{N}^{2}u^{2}(x_{N})
Measurement result: y = Ax_{1}^{a }x_{2}^{b}. . . x_{N}^{p} Combined standard uncertainty:
Here u_{r}(x_{i}) is the relative standard uncertainty of x_{i }and is defined by u_{r}(x_{i}) = u(x_{i})/x_{i}, where x_{i} is the absolute value of x_{i }and x_{i }is not equal to zero; and u_{c,r}(y) is the relative combined standard uncertainty of y and is defined by u_{c,r}(y) = u_{c}(y)/y, where y is the absolute value of y and y is not equal to zero. Meaning of uncertainty



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