Chapter 4 Control Charts for Measurements with Subgrouping (for One Variable)

Presumptions Subgroups (samples) of data are formed. Measurements are made and the values are obtained with sufficient speed.

4.1 Basic Control Chart Principles Stage 1: Control charts can be used to determine if a process has been in a state of statistical control by examining past data. (retrospective data analysis) Stage 2: Recent data can be used to determine control limits that would apply to future data obtained from a process.

Basic Control Chart Principles Control charts alone cannot produce statistical control Control charts can indicate whether or not statistical control is being maintained and provide users with other signals from the data To avoid unnecessary and undesired process adjustments Control charts can also be used to study process capability (Chapter 7)

Basic Control Chart Principles Best results will generally be obtained when control charts are applied primarily to process variables than to product variables. In general, it is desirable to monitor all process variables that affect important product variables. Control charts are essentially plots of data over time.

Figure 4.1

Basic Control Chart Principles Procedures for setting up new control charts Obtain at least 20 subgroups or at least 100 individual observations (from past or current data) Calculate the trial control limits Data points outside the trial limits should be investigated. Those points can only be removed if the assignable causes can be detected and removed Repeat steps 2 and 3 until no further action can be taken

Basic Control Chart Principles After a process has been brought into a state of statistical control, a process capability study can be initiated to determine the capability of the process in regard to meeting the specifications. Process performance indices: using long term sigma (when a process is not in a state of statistical control)

4.2 Real-time Control Charting vs Analysis of Past Data When a set of points is plotted all at once, the probability of observing at least one point that is outside the control limit will be much greater than .0027. When points are plotted individually, 3-sigma limits are used, A normal distribution is assumed, and Parameter of the appropriate distribution are assumed to be known

Table 4.1 Probabilities of Points Plotting Outside Control Limits Actual Prob. 1 0.0027 2 0.0054 5 0.0135 0.0134 10 0.0270 0.0267 15 0.0405 0.0397 20 0.0540 0.0526 25 0.0675 0.0654 50 0.1350 0.1264 100 0.2700 0.2369 350 0.9450 0.6118

Real-time Control Charting vs Analysis of Past Data The calculation of probabilities can only be done if the parameters are assumed to be known. In general, the possible use of k-sigma limits in Stage 1 needs to be addressed. Costs of shutting down the process Costs of making units outside specifications Costs of false looking for assignable causes Medical practitioners often tend to favor 2-sigma limits. (assignable causes can be detected as quickly as possible)

4.3 Control Charts: When to Use, Where to Use, How many to Use Not at every work station Nature of the product often preclude measurements No need for control charts in a process that is highly unlikely the process ever go out of control Control charts should be used where trouble is likely to occur They should be used where the potential for cost reduction is substantial Number of charts Computer vs manual charting

4.4 Benefits from the Use of Control Charts Good record keeping Control charts work as an aid in identifying special causes of variation

4.5 Rational Subgroups Important Requirement: data chosen for the subgroups come from the same population (same operator, shift, machine, etc.) If data are mixed, the control limits will correspond to a mixture (aggregated) distribution

4.6 Basic Statistical Aspects of Control Charts  

Basic Statistical Aspects of Control Charts When X is highly asymmetric, data can usually be transformed (log, square root, power, reciprocal, Cox-cox, etc.) to be approximately normal.

4.7 Illustrative Example Important Requirement: data chosen for the subgroups come from the same population (same operator, shift, machine, etc.) If data are mixed, the control limits will correspond to a mixture (aggregated) distribution

Table 4.2 Data in Subgroups Obtained at Regular Intervals X1 X2 X3 X4 1 72 84 79 49 2 56 87 33 42 3 55 73 22 60 4 44 80 54 74 5 97 26 48 58 6 83 89 91 62 7 47 66 53 8 88 50 69 9 57 41 46 10 13 30 32 11 39 52 12 27 63 34 78 14 71 82 15 70 16 67 94 17 18 51 76 19 61 59 20 X-bar R S 71.00 35 15.47 54.50 54 23.64 52.50 51 21.70 63.00 36 16.85 57.25 71 29.68 81.25 29 13.28 56.00 19 8.04 72.75 38 17.23 47.75 16 6.70 21.25 22 11.35 41.25 26 11.53 42.50 15.76 69.00 16.87 67.75 27 67.00 13 6.06 75.00 12.73 59.75 23 9.98 57.75 12.42 68.00 21 9.83 63.50 17 7.33

R vs. S Charts Either R or S charts could be used in controlling the process variability. S-chart is preferable since it uses all the observations in each subgroup. Other statistical methods in quality improvement are generally based on S (or S2)

Estimating of Population Parameters by Sample Statistics Population Statistics:  , usually unknown Using Sample Statistics to estimate population statistics: Point estimates

4.7.1 R-Chart  

Figure 4.2 R-Chart

4.7.2 R-Chart with Probability Limits  

4.7.3 S-Chart  

S-Chart

4.7.4 S-Chart with Probability Limits  

Table 4.3 The .001 and .999 Percentage Points of 2 Distribution 10.8276 3 0.0020 13.8155 4 0.0243 16.2662 5 0.0908 18.4668 6 0.2102 20.5150 7 0.3811 22.4577 8 0.5985 24.3219 9 0.8571 26.1245 10 1.1519 27.8772 11 1.4787 29.5883 12 1.8339 31.2641 13 2.2142 32.9095 14 2.6172 34.5282 15 3.0407 36.1233

S-Chart with Probability Limits  

4.7.5 S2-Chart  

   

 

  Questions to ask: What is the likelihood to have one of 20 sample data fall outside the control limits? What is the probability for an subgroup average as small as 21.25?

   

4.7.7 Recomputing Control Limits  

4.7.8 Applying Control Limits to Future Production