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where
ni = the i th repeated COP sample size
ȳi = mean value for the i th repeated COP sample
si = standard deviation of i th repeated COP sample
yi j = the j datum of the i th repeated COP sample.
The overall mean value for the k = 10 repeated samples at each point is computed from equation
k
ni
k
1 XX
1X
ȳ =
ni ȳi
yi j =
n i=1 j=1
n i=1
where n =
Pk
i=1 ni
is the total number of measurements.
Uncertainty of Uniformity
The uniformity error is reported as the estimated changes in the
bias over the normal variation process (the same as linearity in R&R studies). Process variation
was estimated as 6× the repeatability standard uncertainty. As this error assumed to follow a
rectangular probability distribution, the uncertainty on uniformity, uun f , is estimated as
uunf =
Uncertainty of Nonlinearity
|slope| × process variation
p
3
(B.5)
The nonlinearity error ("lnr ) is assumed to follow a rectangular
probability distribution. Therefore, the uncertainty for nonlinearity, ulnr , is the square root of the
variance of the rectangular distribution with boundaries the maximum absolute residual of the
linear model and is computed as
ulnr =
Uncertainty of Hysteresis
max|Y − Ŷ |
p
3
(B.6)
The hysteresis error ("h ys ) is assumed to follow a rectangular prob-
ability distribution. Therefore, the uncertainty due to hysteresis (uh ys ) is the square root of the
variance of the rectangular distribution with boundaries the maximum difference between the
upscale and downscale readings among the points P1→ξ and is computed as
uhys =
B.3.5.1.
max|Yupscale − Ydownscale |
p
3
(B.7)
Combine Uncertainties
Using the variance addition rule to combine statistically independent uncertainties from different
sources the uncertainty in the measurement error, uCOP , is
Ç
uCOP = u2ran + u2res + u2hys + u2lnr + u2unf
The expanded uncertainty of measurement (U) is reported as the combined uncertainty of measurement multiplied by the coverage factor k = 2 which for a normal distribution corresponds to a
coverage probability of 95%
U = k × uCOP
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