# Tool for Quality Control Design and Evaluation

This Demonstration can be used to estimate various parameters of a measurement process and to design the quality control rule to be applied. You define the parameters of the control measurements that are the assigned mean, the observed mean, and the standard deviation, in arbitrary measurement units. In addition, you define the quality specifications of the measurement process, that is, the total allowable analytical error (as a percentage of the assigned mean), the maximum acceptable fraction of measurements nonconforming to the specifications, and the minimum acceptable probabilities for random and systematic error detection. Finally, you choose the number of control measurements.
is a quality control rule that rejects the analytical run if at least one of the control measurements is less than or greater than , where is a positive real number and and are the mean and the standard deviation of the control measurements. The quantities and are the lower and the upper quality control decision limits. Then the fraction nonconforming , the critical random and systematic errors of the measurement process, as well as the quality control decision limits and the respective probabilities for critical random and systematic error detection and for false rejection are estimated. Finally, by choosing the type of output, you can see the estimated quality control (qc) parameters or plot the probability density function (pdf), the cumulative density function (cdf) of the control measurements, or the power function graphs for the random error (pfg re) and systematic error (pfg se).
The parameters are estimated and the functions are plotted if , where is the fraction nonconforming and is the maximum acceptable fraction nonconforming.

### DETAILS

Let , , and be the assigned mean, the observed mean, and the standard deviation of the control measurements. Let be the total allowable analytical error (expressed as a fraction), the maximum (acceptable) fraction nonconforming, and and the minimum (acceptable) probabilities for critical random and systematic error detection. Then the following equations are used to estimate the respective parameters [1, 2]:
(a) the fraction nonconforming: ,
(b) the critical random error : ,
(c) the critical systematic error : , where if and otherwise,
(d) the factor of the decision limits of the quality control rule is the minimum solution of both of the following two equations for the variable :
(1) ,
(2) , with as before in (c),
(e) the probability for false rejection of the quality control rule : , and
(f) the probability for error detection of the random error and the systematic error of the quality control rule : .
The power function graphs ("pfg") are the plots of the probabilities for error detection versus the size of the error.
This Demonstration can be used as a tool for the design and evaluation of alternative quality control rules for a measurement process.
References
[1] A. T. Hatjimihail, "A Tool for the Design and Evaluation of Alternative Quality Control Procedures," Clinical Chemistry 38, 1992 pp. 204–210.
[2] A. T. Hatjimihail, "Estimation of the Optimal Statistical Quality Control Sampling Time Intervals Using a Residual Risk Measure," PLoS ONE 4(6), 2009 p. e5770.

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