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

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

3. 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.

4. 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)

5. 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.

6. Figure 4.1

7. Basic Control Chart Principles Procedures for setting up new control charts 1.Obtain at least 20 subgroups or at least 100 individual observations (from past or current data) 2.Calculate the trial control limits 3.Data points outside the trial limits should be investigated. Those points can only be removed if the assignable causes can be detected and removed 4.Repeat steps 2 and 3 until no further action can be taken

8. 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)

9. 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

10. Table 4.1 Probabilities of Points Plotting Outside Control Limits n.0027nActual Prob. 10.0027 20.0054 50.01350.0134 100.02700.0267 150.04050.0397 200.05400.0526 250.06750.0654 500.13500.1264 1000.27000.2369 3500.94500.6118

11. 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)

12. 4.3 Control Charts: When to Use, Where to Use, How many to Use Where: 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

13. 4.4 Benefits from the Use of Control Charts Good record keeping Control charts work as an aid in identifying special causes of variation

14. 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

15. 4.6 Basic Statistical Aspects of Control Charts

16. 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.

17. 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

18. Table 4.2 Data in Subgroups Obtained at Regular Intervals SubgroupX1X2X3X4 172847949 256873342 355732260 444805474 597264858 683899162 747665358 888508469 957474146 1013103032 1126395248 1246276334 1349627887 1471638255 1571586970 1667697094 1755637249 1849515576 1972806159 2061746257 X-barRS 71.003515.47 54.505423.64 52.505121.70 63.003616.85 57.257129.68 81.252913.28 56.00198.04 72.753817.23 47.75166.70 21.252211.35 41.252611.53 42.503615.76 69.003816.87 67.752711.53 67.00136.06 75.002712.73 59.75239.98 57.752712.42 68.00219.83 63.50177.33

19. 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 S 2 )

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

21. 4.7.1 R-Chart

22. Figure 4.2 R-Chart

23. 4.7.2 R-Chart with Probability Limits

24. 4.7.3 S-Chart

25. S-Chart

26. 4.7.4 S-Chart with Probability Limits

27. Table 4.3 The.001 and.999 Percentage Points of  2 Distribution n 21.5708E-0610.8276 30.002013.8155 40.024316.2662 50.090818.4668 60.210220.5150 70.381122.4577 80.598524.3219 90.857126.1245 101.151927.8772 111.478729.5883 121.833931.2641 132.214232.9095 142.617234.5282 153.040736.1233

28. S-Chart with Probability Limits

29. 4.7.5 S 2 -Chart

32. 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?

34. 4.7.7 Recomputing Control Limits

35. 4.7.8 Applying Control Limits to Future Production