1. Statistics for Quality: Control and Capability - Statistical Process Control and Using Control Charts PBS Chapters 12.1 and 12.2 © 2009 W.H. Freeman and Company

2. Objectives (PBS Chapter 12.1 and 12.2) Statistical process control and Using control charts  Processes  Systematic approach to process improvement  Process improvement toolkit  and s charts for process monitoring  Using and setting up control charts  Comments on statistical control  Don’t confuse control with capability!

3. Processes  Processing an application for admission to a university and deciding whether or not to admit the student.  Reviewing an employee’s expense report for a business trip and issuing a reimbursement check.  Hot forging to shape a billet of titanium into a blank that, after machining, will become part of a medical implant for hip, knee, or shoulder replacement.

4. How processes are like populations Think of a population containing all the outputs that would be produced by the process if it ran forever in its present state. The outputs produced today or this week are a sample from this population.

5. Systematic approach to process improvement  A systematic approach to process improvement is captured in the Plan-Do- Check-Act (PDCA).  Plan the intended work.  Then Do the implementation of the solution or change.  Check to see if improvement efforts have been successful.  Act by implementing the changes.

6. Process improvement toolkit Describing processes graphically: A flowchart is a picture of the stages of a process

7. Describing processes graphically: cause-and-effect diagram Organizes the logical relationships between the inputs and stages of a process and an output.

8. Statistical process control Goal: make a process stable over time and then keep it stable unless planned changes are made. All processes have variation. Statistical stability means the pattern of variation remains stable, not that there is no variation in the variable measured. Statistical Control: A variable that continues to be described by the same distribution when observed over time is said to be in control. Control charts: Statistical tools that monitor a process and alert us when the process has been disturbed so that it is now out of control. This is a signal to find and correct the cause of the disturbance.

9. The idea of statistical process control Common cause variation  A process that is in control has only common cause variation.  Common cause variation is the inherent variability of the system, due to many small causes that are always present. Special cause variation  When the normal functioning of the process is disturbed by some unpredictable event, special cause variation is added to the common cause variation.  We hope to be able to discover what lies behind special cause variation and eliminate that cause to restore the stable functioning of the process.

10. Control charts Control charts distinguish between the common cause variation and the special cause variation. A control chart sounds an alarm when it sees too much variation. The point X indicates a data point for sample number 13 that is “out of control.”

11. charts for process monitoring Procedure for applying control charts to a process: 1.Chart setup stage a)Collect data from the process. b)Establish control by uncovering and removing special causes. c)Set up control charts to maintain control. 1.Process monitoring a)Observe the process operating in control for some time. b)Understand usual process behavior. c)Have a long run of data from the process. d)Keep control charts to monitor the process because a special cause could erupt at any time.

12. charts for process monitoring  Process monitoring conditions:  Measure a quantitative variable x that has a Normal distribution.  The process has been operating in control for a long period, so that we know the process mean  and the process standard deviation  that describe the distribution of x as long as the process remains in control.

13. charts for process monitoring 1.Take samples of size n from the process at regular intervals. Plot the means of these samples against the order in which the samples were taken. 2.We know that the sampling distribution of under the process- monitoring conditions is Normal with a mean  and a standard deviation. Draw a solid center line on the chart at height . 3.The 99.7 part of the 68-95-99.7 rule for Normal distributions says that as long as the process remains in control, 99.7% of the values of will fall within three standard deviations of the mean. Draw dashed control limits on the chart at these heights. The control limits mark off the range of variation in sample means that we expect when the process remains in control.

14. charts for process monitoring  Sample mean that is out of control.

15. Assessing improvement efforts a) shows a case where the control chart demonstrates a successful attempt to decrease the time needed to obtain lab results. b) The control chart indicates no impact from the attempted process improvement.

16. General procedure for control charts Three-sigma (3  ) control charts for any statistic Q: 1.Take samples from the process at regular intervals and plot the values of the statistic Q against the order in which the samples were taken. 2.Draw a center line on the chart at height  Q, the mean of the statistic when the process is in control. 3.Draw upper and lower control limits on the chart 3 standard deviations of Q (  Q ) above and below the mean. 4.The chart produces an out-of-control signal when a plotted point lies outside the control limits.

17. s charts for process monitoring For a Normally distributed process characteristic: 1.The mean of s is a constant times the process standard deviation . This is the center line of an s chart. 2.The standard deviation of s is also a constant times the process standard deviation. The control limits for an s chart are The control chart constants c 4, B 5, and B 6 depend on the sample size n.

18. Control chart constants Use an LCL = 0 for small n

19. s charts for process monitoring In controlOut of control

20. Comparing to s control charts Do both types of control charts show the same information? Here are two control charts for mesh tension:

21. Comparing to s control charts  Lack of control on an s chart is due to special causes that affect the observations within a sample differently. examples: new and non-uniform material, new and poorly trained operator, mixing results from several machines or several operators  Look at the s chart first.  Lack of control on an chart responds to s-type causes as well as to longer-range changes in the process, so it is important to eliminate the s- type causes first. examples of longer-range change: new raw material that differs from that used in the past or a gradual drift in the process level caused by wear in a cutting tool.

22. Process control record sheet

23. Using control charts  and R charts: an R chart is based on the sample range for spread instead of the sample standard deviation. Range = largest observation – smallest observation. Less informative than s charts.  Additional out-of-control signals

24. Setting up control charts

25. Comments on statistical control  Focus on the process rather than on the products.  If the process is kept in control, we know what to expect in the finished product.  We want to do it right the first time, not inspect and fix finished product.  Rational subgroups.  We want the variation within a sample to reflect only the item-to-item chance variation that, when in control, results from many small common causes.  Samples of consecutive items are rational subgroups when we are monitoring the output of a single activity that does the same thing over and over again.  Think about causes of variation in your process and decide which are common causes and do not need to be eliminated.  Why statistical control is desirable.  If the process is kept in control, we know what to expect in the finished product.  Caution: distinguish between natural tolerances and control limits.

26. Don’t confuse control with capability! There is no guarantee that a process in control produces products of satisfactory quality. “Satisfactory quality” is measured by comparing the product to some standard outside the process, set by technical specifications, customer expectations, or the goals of the organization. Statistical quality control only pays attention to the internal state of the process. Capability refers to the ability of a process to meet or exceed the requirements placed on it. Capability has nothing to do with control; except if a process is not in control, it is hard to tell if it is capable or not. If a process is in control but does not have adequate capability, fundamental changes in the process are needed. Better training for workers, new equipment, better raw materials, etc.

27. Statistics for Quality: Control and Capability - Process Capability Indexes and Control Charts for Sample Proportions PBS Chapters 12.3 and 12.4 © 2009 W.H. Freeman and Company

28. Objectives (PBS Chapter 12.3 and 12.4) Process capability indexes and Control charts for sample proportions  The capability indexes C p and C pk  Cautions about capability indexes  Control limits for p charts

29. Process Capability Indexes  Capability relates the actual performance of a process in control, after special causes have been removed, to the desired performance.  Suppose that there are specifications set for some characteristic of the process output.  We can then measure capability by the percent of output that meets specifications.

30. Percent Meeting Specifications  Percentage meeting specifications is a poor measure of capability.  This figure compares the distributions of the diameter of the same part manufactured by two processes.  All of the parts from Process A meet the specifications, but the parts from Process B have a higher proportion of diameters close to the target.

31. The capability indexes  Consider a process with specification limits LSL and USL for some measured characteristic of its output. The process mean for this characteristic is μ and the standard deviation is σ. The capability index C p is C p = (USL - LSL) / 6σ  The capability index C pk is C pk = |μ - nearer spec limit| / 3σ  Set C pk = 0 if the process mean μ lies outside the specification limits. Large values of C p or C pk indicate more capable processes.

32. Interpreting Capability Indexes How capability indexes work: (a) Process centered, process width equal to specification width. (b) Process off-center, process width equal to specification width. (c) Process off-center, process width equal to half the specification width. (d) Process centered, process width equal to half the specification width.

33. Cautions about capability indexes  There are two different ways of estimating σ. The sample standard deviation s will usually be larger than the control chart estimates which is based on averaging the sample standard deviations. A supplier can make C pk a bit larger by using the smaller estimate. That’s cheating.  Capability indexes are strongly affected by non-Normality. Apply capability indexes only when the distribution is at least roughly Normal.  Capability indexes are statistics subject to sampling variation. Estimates based on small samples can differ from the true process C pk in either direction.

34. Control charts for sample proportions A p chart is a control chart based on plotting sample proportions from regular samples from a process against the order in which the samples were taken.

35. Control limits for p charts