Estimating Residual Error by Repeated Measurements
Keywords:
Repeatability, residual error, repeated measurementsAbstract
Repeated measurements can be used to estimate the residual error of a measurement process. Residual error, defined as the error remaining after all known sources of error have been accounted for, is what causes differences in the measurement outcome when everything about the measurement process is seemingly identical. Four estimators for the amount of residual error are suggested. These estimators are functions of the outcomes from m repeated measurements on each of n items. Assuming a normal distribution for the residual error, two of the estimators are unbiased estimators for the standard deviation of the residual error, and the third is the maximum likelihood estimator for the standard deviation of the residual error. The fourth is not an estimator for standard deviation, but rather it uses the distance between measurement order statistics as an indicator of the amount of residual error. The efficiencies of the first two estimators and the bias of the maximum likelihood estimator are evaluated. Computations use standard statistical methods and are included in appendices.
This work, motivated by the study of machines used to measure the modulus of elasticity of dimension lumber, has been used to assess the performance of this machinery. An example using data from more than 10 years ago and some recent data show that the residual error then was about double that of today for well-tuned, high-speed production-line machinery.
References
Frank, E. 1959. Electrical measurement analysis. McGraw-Hill Book Company Inc., New York, NY.nHogg, R. V., and A. T. Craig. 1965. Introduction to mathematical statistics. The Macmillan Company, New York, NY.nLogan, J. D. 1991. Introduction: Repeatability measurements in the CLT continuous lumber tester. Metriguard Inc., Pullman, WA.nMiller, K. S. 1964. Multidimensional Gaussian distributions. John Wiley and Sons, Inc., New York, NY.nRao, C. R. 1965. Linear statistical inference and its applications. John Wiley and Sons, Inc., New York, NY.nWilks, S. S. 1962. Mathematical statistics. John Wiley and Sons, Inc. New York, NY.n
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