1. Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola

2. Slide Slide 2 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. 14-1 Overview 14-2 Control Charts for Variation and Mean 14-3 Control Charts for Attributes Chapter 14 Statistical Process Control

3. Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Section 14-1 and 14-2 Overview and Control Charts for Variation and Mean Created by Erin Hodgess, Houston, Texas Revised to accompany 10th Edition, Jim Zimmer, Chattanooga State, Chattanooga, TN

4. Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept The main objective of this section is to construct run charts, R charts, and charts so that we can monitor important features of data over time. We will use such charts to determine whether some process is statistically stable (or within statistical control).

5. Slide Slide 5 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Control Charts for Variation and Mean Definition. Process data are data arranged according to some time sequence. They are measurements of a characteristic of goods or services that result from some combination of equipment, people, materials, and conditions. Important characteristics of process data can change over time.

6. Slide Slide 6 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Control Charts for Variation and Mean Definition. A run chart is a sequential plot of individual data values over time. One axis (usually vertical) is used for the data values, and the other axis (usually horizontal) is used for the time sequence.

7. Slide Slide 7 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Measuring Aircraft Altimeters For 20 consecutive business days, four altimeters are randomly selected and tested in a pressure chamber that simulates an altitude of 1000 feet. The errors of the readings above or below 1000 feet are shown in Table 14-1.

8. Slide Slide 8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Measuring Aircraft Altimeters Treating the 80 altimeter errors in Table 14-1 as a string of consecutive measurements, construct a run chart by using a vertical axis for the errors and a horizontal axis to identify the order of the sample data. Minitab Figure 14-1 Run Chart of Individual Altimeter Errors in Table 14-1

9. Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Measuring Aircraft Altimeters As time progresses from left to right, the heights of the points appear to show a pattern of increasing variation. The Federal Aviation Administration regulations require errors less than 20 ft, so the altimeters represented by the points at the left are okay, whereas several of the points farther to the right correspond to altimeters not meeting the required specifications. Minitab

10. Slide Slide 10 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition A process is statistically stable (or within statistical control) if it has natural variation, with no patterns, cycles, or any unusual points. Control Charts for Variation and Mean Only when a process is statistically stable can its data be treated as if they came from a population with a constant mean, standard deviation, distribution, and other characteristics.

11. Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(a): There is an obvious upward trend that corresponds to values that are increasing over time. Minitab

12. Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(b): There is an obvious downward trend that corresponds to steadily decreasing values. Minitab

13. Slide Slide 13 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(c): There is an upward shift. A run chart such as this one might result from an adjustment to the filling process, making all subsequent values higher. Minitab

14. Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(d): There is a downward shift—the first few values are relatively stable, and then something happened so that the last several values are relatively stable, but at a much lower level. Minitab

15. Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(e): The process is stable except for one exceptionally high value. Minitab

16. Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(f): There is an exceptionally low value. Minitab

17. Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(g): There is a cyclical pattern (or repeating cycle). This pattern is clearly nonrandom and therefore reveals a statistically unstable process. Minitab

18. Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Figure 14-2 Processes That Are Not Statistically Stable Figure 14-2(h): The variation is increasing over time. This is a common problem in quality control. Minitab

19. Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. A common goal of many different methods of quality control is this: Reduce variation in a product or a service.

20. Slide Slide 20 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definitions Random variation is due to chance; it is the type of variation inherent in any process that is not capable of producing every good or service exactly the same way every time. Assignable variation results from causes that can be identified (such factors as defective machinery, untrained employees, and so on).

21. Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition A control chart of a process characteristic (such as mean or variation) consists of values plotted sequentially over time, and it includes a center line as well as a lower control limit (LCL) and an upper control limit (UCL). The centerline represents a central value of the characteristic measurements, whereas the control limits are boundaries used to separate and identify any points considered to be unusual. Control Chart for Monitoring Variation: The R Chart

22. Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. An R chart (or range chart) is a plot of the sample ranges instead of individual sample values, and it is used to monitor the variation in a process. In addition to plotting the range values, it includes a centerline located at R, which denotes the mean of all sample ranges, as well as another line for the lower control limit and a third line for the upper control limit. Control Chart for Monitoring Variation: The R Chart

23. Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Given: Process data consisting of a sequence of samples all of the same size n, and the distribution of the process data is essentially normal. n = size of each sample, or subgroup R = mean of the sample ranges (that is, the sum of the sample ranges divided by the number of samples) Notation

24. Slide Slide 24 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Points plotted: Sample ranges Centerline: R Upper Control Limit (UCL): D 4 R (where D 4 is found in Table 14-2) Lower Control Limit (LCL): D 3 R (where D 3 is found in Table 14-2) Monitoring Process Variation: Control Chart for R

25. Slide Slide 25 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Table 14-2 Control Chart Constants

26. Slide Slide 26 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Refer to the altimeter errors in Table 14-1. Using the samples of size n = 4 collected each day of manufacturing, construct a control chart for R. Example: Manufacturing Aircraft Altimeters

27. Slide Slide 27 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. MINITAB Display R Chart for Errors Minitab

28. Slide Slide 28 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Upper and lower control limits of a control chart are based on the actual behavior of the process, not the desired behavior. Upper and lower control limits are totally unrelated to any process specifications that may have been decreed by the manufacturer. Interpreting Control Charts

29. Slide Slide 29 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. 1.Based on the current behavior of the process, can we conclude that the process is within statistical control? 2.Do the process goods or services meet design specifications? When investigating the quality of some process, there are typically two key questions that need to be addressed: The methods of this chapter are intended to address the first question, but not the second. Interpreting Control Charts

30. Slide Slide 30 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Criteria for Determining When a Process Is Not Statistically Stable (Out of Statistical Control) 1. There is a pattern, trend, or cycle that is obviously not random. 2. There is a point lying beyond the upper or lower control limits. 3. Run of 8 Rule: There are eight consecutive points all above or all below the center line.

31. Slide Slide 31 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Additional Criteria Used by Some Businesses  There are 6 consecutive points all increasing or all decreasing.  There are 14 consecutive points all alternating between up and down (such as up, down, up, down, and so on).  Two out of three consecutive points are beyond control limits that are 2 standard deviations away from centerline.  Four out of five consecutive points are beyond control limits that are 1 standard deviation away from the centerline.

32. Slide Slide 32 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Statistical Process Control Examine the R chart shown in the Minitab display for the preceding example and determine whether the process variation is within statistical control. 1. There is a pattern, trend, or cycle that is obviously not random: Going from left to right, there is a pattern of upward trend. 2. There is a point (the rightmost point) that lies above the upper control limit.

33. Slide Slide 33 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Statistical Process Control Examine the R chart shown in the Minitab display for the preceding example and determine whether the process variation is within statistical control. Interpretation: We conclude that the variation (not necessarily the mean) of the process is out of statistical control. Because the variation appears to be increasing with time, immediate corrective action must be taken to fix the variation among the altimeter errors.

34. Slide Slide 34 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. The x chart is a plot of the sample means and is used to monitor the center in a process. In addition to plotting the sample means, we include a centerline located at x, which denotes the mean of all sample means, as well as another line for the lower control limit and a third line for the upper control limit. Control Chart for Monitoring Means: The x Chart

35. Slide Slide 35 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Points plotted: Sample means Center line: x = mean of all sample means Upper Control Limit (UCL): x + A 2 R where A 2 is found in Table 14-2 Lower Control Limit (LCL): x – A 2 R where A 2 is found in Table 14-2 Control Chart for Monitoring Means: The x Chart

36. Slide Slide 36 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Manufacturing Aircraft Altimeters Refer to the altimeter errors in Table 14-1. Using samples of size n = 4 collected each working day, construct a control chart for x. Based on the control chart for x only, determine whether the process mean is within statistical control. Before plotting the 20 points corresponding to the 20 values of x, we must first find the value for the centerline and the values for the control limits.

37. Slide Slide 37 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Manufacturing Aircraft Altimeters Upper control limit: Lower control limit:

38. Slide Slide 38 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Manufacturing Aircraft Altimeters Excel

39. Slide Slide 39 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Interpretation Examination of the control chart shows that the process mean is out of statistical control because at least one of the three out-of-control criteria is not satisfied. Specifically, the third criterion is not satisfied because there are eight (or more) consecutive points all below the center line. Also, there does appear to be a pattern of an upward trend. Again, immediate corrective action is required to fix the production process. Example: Manufacturing Aircraft Altimeters

40. Slide Slide 40 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed: Control charts for variation and mean. Run charts determine if characteristics of a process have changed. R charts (or range charts) monitor the variation in a process. x charts monitor the center in a process.

41. Slide Slide 41 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Section 14-3 Control Charts for Attributes Created by Erin Hodgess, Houston, Texas Revised to accompany 10th Edition, Jim Zimmer, Chattanooga State, Chattanooga, TN

42. Slide Slide 42 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept This section presents a method for constructing a control chart to monitor the proportion p for some attribute, such as whether a service or manufactured item is defective or nonconforming. The control chart is interpreted by using the same three criteria from Section 14-2 to determine whether the process is statistically stable.

43. Slide Slide 43 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Control Charts for Attributes  These charts monitor the qualitative attributes of whether an item has some particular characteristic.  In the previous section, the charts monitored the quantitative characteristics.  The control chart for p (or p chart) is used to monitor the proportion p for some attribute.

44. Slide Slide 44 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. p = pooled estimate of proportion of defective items in the process = total number of defects found among all items sampled total number of items sampled q = pooled estimate of the proportion of process items that are not defective = 1 – p n = size of each sample (not the number of samples) Notation

45. Slide Slide 45 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Center line: Upper control limit: Lower control limit: Control Chart for p (If the calculation for the lower control limit results in a negative value, use 0 instead. If the calculation for the upper control limit exceeds 1, use 1 instead.)

46. Slide Slide 46 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. The Altigauge Manufacturing Company produces altimeters in batches of 100, and each altimeter is tested and determined to be acceptable or defective. Listed below are the numbers of defective altimeters in successive batches of 100. Defects: Example: Defective Aircraft Altimeters total number of defects from all samples combined total number of altimeters sampled

47. Slide Slide 47 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Defective Aircraft Altimeters Upper control limit: Lower control limit: Because the lower control limit is less than 0, we use 0 instead.

48. Slide Slide 48 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Defective Aircraft Altimeters

49. Slide Slide 49 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Defective Aircraft Altimeters Interpretation We can interpret the control chart for p by considering the three out-of-control criteria listed in section 14-2. Using those criteria, we conclude that this process is out of statistical control for this reason: There appears to be an upward trend. The company should take immediate action to correct the increasing proportion of defects.

50. Slide Slide 50 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed: A control chart for attributes is a graph of proportions plotted sequentially over time. It includes a centerline, a lower control limit, and an upper control limit. The same three out-of-control criteria listed in Section 14-2 can be used.