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

:

.

[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.