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where ni = the i th repeated COP sample size yÌi = 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 yÌ = ni yÌi 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 6Ã 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| Ã 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 â YÌ | 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 P1âÎ¾ and is computed as uhys = B.3.5.1. max|Yupscale â 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 Ã 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 Ã uCOP 154