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The overall mean, range, and standard deviation are 19.48, 2.80 ana 1.18, respectively. The mean and range charts are developed by using the overall mean and range values in Eqs. 6.10 and 6.11. The resulting Shewhart charts are displayed in Figure 6.4. The mean of sample 22 is out of control while the range chart is in control, indicating a significant shift in level. Both the mean and range are out of control at sample 23, indicating significant change in both level and spread of the sample.

Observation Number

Figure 6.4. Shewhart chart for mean (CL = x) and range (CL — R).

Observation Number

Figure 6.4. Shewhart chart for mean (CL = x) and range (CL — R).

The Mean and Standard Deviation Charts

The S chart is preferable for monitoring variation when the sample size is large or varying from sample to sample. Although S2 is an unbiased estimate of a2, the sample standard deviation 5 is not an unbiased estimator of a. For a variable with a normal distribution, S estimates C4CT, where C4 is a parameter that depends on the sample size n. The standard deviation of S is cta/1 — c\. When a is to be estimated from past data,

and S/C4 is an unbiased estimator of a. The exact values for C4 are given in the Table of Control Chart Constants in the Appendix. An approximate relation based on sample size n is

The S Chart

The control limits of the S chart are

C4 v

Defining the constants

the limits of the S chart are expressed as

The x Chart

When a = S/c4, the control limits for the x chart are

c41/n

Defining the constant A3 = the limits of the x chart become

Example The mean and standard deviation charts are developed by using the overall mean and standard deviation values in Eqs. 6.16 and 6.18. The resulting Shewhart charts are displayed in Figure 6.5. The means of samples 22 and 23 are out-of-control, while the standard deviation chart is out-of-control for sample 23, providing similar results as x and R charts.

Interpretation of x Charts

The x charts must be used along with a spread chart. The process spread must be in-control for proper interpretation of the x chart.

The x chart considers only the current data value in assessing the status of the process. In order to include historical information such as trends in data, run rules have been developed. The rim rules sensitize the chart, but they also increase the false alarm probability. If k run rules are used simultaneously and rule i has a Type I error probability of at, the overall Type I error probability Q.totai is k

If 3 rules are used simultaneously and cm = 0.05, then a = 0.143. For on = 0.01, a = 0.0297.

Observation Number

Figure 6.5. Shewhart chart for mean (CL — x) and standard deviation

Observation Number

Figure 6.5. Shewhart chart for mean (CL — x) and standard deviation

The Run Rules

Run rules, also known as Western Electric Rules [111], enable decision making based on trends in data. A process is declared out of control if any one or more of the run rules are met. Some of the criteria used as run rules are:

• One point outside the control limits.

• Two of three consecutive points outside the 2-sigma warning limits but still inside the control limits.

• Four of five consecutive points outside the 1-sigma limits.

• Eight consecutive points on one side of the center line.

• Eight consecutive points forming a run up or a run down.

• A nonrandom or unusual pattern in the data.

Patterns in data could be any systematic behavior such as shifts in process level, cyclical (periodic) behavior, stratification (points clustering around the centerline), trends, or drifts.

Average Run Length (ARL)

The ARL is average number of samples (or sample averages) plotted in order to get an indication that the process is out-of-control. ARL can be used to compare the efficacy of various SPC charts and methods. ARL(O) is the in-control ARL, the ARL to generate an out-of-control signal even though in reality the process remains in control. The ARL to detect a shift in the mean of magnitude ka is represented by ARL(cr) where k is a constant and a is the standard deviation of the variable. A good chart must have a high ARL(O) (for example ARL(0)=400 indicates that there is one false alarm on the average out of 400 successive samples plotted) and a low ARL(ct) (bad news is displayed as soon as possible). For a Shewhart chart, the ARL is calculated from

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