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The problem is one of pattern recognition, that is, recognizing systematic or nonrandom patterns on the control chart and identifying the reason for this behavior The ability to interpret.

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Figure 16-6 (a) Variability with the cyclic pattern. (b) Variability with the cyclic pattern eliminated.

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a particular pattern in terms of assignable causes requires experience and knowledge of the process. That is, we must not only know the statistical principles of control charts, but we must also have a good understanding of the process. The Western Electric Handbook (1956) suggests a set of decision rules for detecting nonrandom patterns on control charts. Speci cally, the Western Electric rules would conclude that the process is out of control if either 1. 2. 3. 4. One point plots outside 3-sigma control limits. Two out of three consecutive points plot beyond a 2-sigma limit. Four out of ve consecutive points plot at a distance of 1-sigma or beyond from the center line. Eight consecutive points plot on one side of the center line.


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We have found these rules very effective in practice for enhancing the sensitivity of control charts. Rules 2 and 3 apply to one side of the center line at a time. That is, a point above the upper 2-sigma limit followed immediately by a point below the lower 2-sigma limit would not signal an out-of-control alarm. Figure 16-7 shows an X control chart for the piston ring process with the 1-sigma, 2-sigma, and 3-sigma limits used in the Western Electric procedure. Notice that these inner

In this mode space, Maxwell's equations then become much simpler. In fact, they reduce to just two coupled ordinary differential equations:

74.0180 74.0135 74.0090 Zone B 74.0045 Zone C 74.0000 Zone C 73.9955 Zone B 73.9910 Zone A 73.9865 73.9820 LCL = 73.9865 1 X 2 X 3 X UCL = 74.0135 Zone A 3 X 2 X 1 X

10 11 12

= -cknen,s, = cknbn,s,

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limits (sometimes called warning limits) partition the control chart into three zones A, B, and C on each side of the center line. Consequently, the Western Electric rules are sometimes called the run rules for control charts. Notice that the last four points fall in zone B or beyond. Thus, since four of ve consecutive points exceed the 1-sigma limit, the Western Electric procedure will conclude that the pattern is nonrandom and the process is out of control.


(A40) (A41)

When dealing with a quality characteristic that can be expressed as a measurement, it is customary to monitor both the mean value of the quality characteristic and its variability. Control over the average quality is exercised by the control chart for averages, usually called the X chart. Process variability can be controlled by either a range chart (R chart) or a standard deviation chart (S chart), depending on how the population standard deviation is estimated. Suppose that the process mean and standard deviation and are known and that we can assume that the quality characteristic has a normal distribution. Consider the X chart. As discussed previously, we can use as the center line for the control chart, and we can place the upper and lower 3-sigma limits at 1n 1n (16-2)

as can be seen by substituting (A.38) and (A.39) in (AI) to (A.4) and using the orthogonality properties of the mode functions. Together, these equations yield harmonic oscillator equations for the electric and magnetic fields, like (2.51) and (2.52): (A.42) (A43)

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