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2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.5. Propagation of error considerations

2.5.5.1.

Formulas for functions of one variable

Case: Y=f(X,Z) Standard deviations of reported values that are functions of a single variable are reproduced from a paper by H. Ku (Ku).

The reported value, Y, is a function of the average of N measurements on a single variable.

Notes
Function Y of Xbar

Xbaris an average of N measurements

Standard deviation of Y

s(x) = standard deviation of X.

 
Y = Xbar
(1/SQRT(N))*s(x)
 
Y = Xbar/(1 + Xbar)
s(x)/{SQRT(N)*(1 + Xbar)**2}
 
Y = (Xbar)**2
(2*Xbar/SQRT(N))*s(x)
 
Y = SQRT(Xbar)
s(x)/{2*SQRT(N*Xbar)}
 
Y = LN(Xbar)
s(x)/{SQRT(N)*Xbar}
Approximation could be seriously in error if n is small--
Y = EXP(Xbar)
(EXP(Xbar)/SQRT(N))*s(x)
Not directly derived from the formulas
Y = 100*s(x)/Xbar
Y/SQRT(2*(N-1))

Note: we need to assume that the original data follow an approximately normal distribution.

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