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

2.2.3.1.

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

where the quantity under the radical is the upper critical value from the F-table with degrees of freedom (J - 1) and K(J - 1). The numerator degrees of freedom, v1 = (J -1), refers to the standard deviation computed from the current measurements, and the denominator degrees of freedom, v2 = K(J -1), refers to the pooled standard deviation of the historical data. The probability 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:

H₀: ₁ = ₂
H_a: ₂ > ₁

where ₁ is the population value for the s₁ defined above and ₂ is the population value for the standard deviation of the current values being tested. Generally, s₁ is based on sufficient historical data that it is reasonable to make the assumption that ₁ 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).

Run software macro for computing the F factor

Dataplot can compute the value of the F-statistic. For the case where alpha = 0.05; J = 6; K = 6, the commands

let alpha = 0.05
let alphau = 1 - alpha
let j = 6
let k = 6
let v1 = j-1
let v2 = k*(v1)
let F = fppf(alphau, v1, v2)

return the following value:

THE COMPUTED VALUE OF THE CONSTANT F = 0.2533555E+01