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2. Measurement Process Characterization
2.2. Statistical control of a measurement process
2.2.3. How is short-term variability controlled?

Control chart for standard deviations

Degradation of instrument or anomalous behavior on one occasion Changes in the precision of the instrument, particularly anomalies and degradation, must be addressed. Changes in precision can be detected by a statistical control procedure based on the F-distribution where the short-term standard deviations are plotted on the control chart.

The base line for this type of control chart is the pooled standard deviation, s1, as defined in Data collection and analysis.

Example of control chart for a mass balance Only the upper control limit, UCL, is of interest for detecting degradation in the instrument. As long as the short-term standard deviations fall within the upper control limit established from historical data, there is reason for confidence that the precision of the instrument has not degraded (i.e., common cause variations).
The control limit is based on the F-distribution The control limit is $$UCL = {\large s}_1 \sqrt{F_{\alpha, \, J-1, \, K(J-1)}}$$ where the quantity under the radical is the upper α critical value from the F table with degrees of freedom (J - 1) and K(J - 1). The numerator degrees of freedom, v1 = (J -1), are associated with the standard deviation computed from the current measurements, and the denominator degrees of freedom, v2 = K(J - 1), correspond to the pooled standard deviation of the historical data. The probability α is chosen to be small, say 0.05.

The justification for this control limit, as opposed to the more conventional standard deviation control limit, is that we are essentially performing the following hypothesis test: $$H_0: \,\, \sigma_1 = \sigma_2$$ $$H_a: \,\, \sigma_1 < \sigma_2$$ where σ1 is the population value for the s1 defined above and σ2 is the value against which the standard deviation of the current value is being tested. Generally, s1 is based on sufficient historical data that it is reasonable to make the assumption that σ1 is a "known" value.

The upper control limit above is then derived based on the standard F test for equal standard deviations. Justification and details of this derivation are given in Cameron and Hailes (1974).

Sample Code Sample code for computing the F value for the case where
α = 0.05, J = 6, and K = 6, is available for both Dataplot and R.
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