Lecture Slides

1. Lecture Slides Elementary StatisticsTwelfth Edition and the Triola Statistics Series by Mario F. Triola

2. Chapter 14Statistical Control Processes 14-1 Review and Preview 14-2 Control Charts for Variation and Mean 14-3 Control Charts for Attributes

3. Key Concept The main objective of this section is to construct run charts, R charts, and charts so that we can monitor important characteristics of data over time. We will use such charts to determine whether some process is statistically stable (or within statistical control).

4. 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, methods, and conditions. Important characteristics of process data can change over time.

5. 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.

6. Example Treating the 100 weights of quarters as a string of consecutive measurements, construct a run chart using a vertical axis for the weights and a horizontal axis to identify the chronological order of the weights.

7. Example - Continued Interpretation: As time progresses left to right, the points appear to exhibit greater variation. This pattern of increasing variation is a classic issue in quality control, and failure to recognize it has caused companies to go out of business.

8. Definition A process is statistically stable (or within statistical control) if it only has natural variation, with no patterns, cycles, or unusual points. Examples of processes that are not statistically stable are given in the next slides.

9. Minitab Increasing Variation Over Time

10. Upward Trend

11. Minitab Downward Trend

12. Minitab Upward Shift

13. Minitab Downward Shift

14. Minitab Exceptional High Value

15. Minitab Exceptional Low Value

16. Minitab Cyclical Pattern

17. 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).

18. A control chart of a process characteristic (such as mean or variation) consists of values plotted sequentially over time, and it includes a centerline 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

19. 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 , 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

20. 1. The data are process data consisting of a sequence of samples all of the same size n. 2. The distribution of the process data is essentially normal. 3. The individual sample data values are independent. Requirements

21. n = size of each sample, or subgroup Notation = mean of the sample ranges (that is, the sum of the sample ranges divided by the number of samples)

22. Points plotted: Sample ranges Graphs Centerline: (mean of sample ranges) Upper Control Limit (UCL): (where is found in Table 14-2) Lower Control Limit (LCL): (where is found in Table 14-2)

23. Control Chart Constants

24. Example - Continued Construct a control chart for R using the weights of the quarters listed in Table 14.1 (see the text, page 691). Use the samples of size n = 5 for each of the 20 days of production. The mean of the 20 sample ranges is: The upper and lower control limits are:

25. Example - Continued

26. Example - Continued Interpretation: Because the points at the end appear to show an upward trend, there is an obvious pattern that is not random. The last two points are lying outside the upper control limit. The first eight points are eight consecutive points all lying below the centerline. We conclude that the variation (not necessarily the mean) of the process is out of control.

27.    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. Caution

28. Interpreting Control Charts When investigating the quality of some process, there are typically two key questions that need to be addressed: • Based on the current behavior of the process, can we conclude that the process is within statistical control? • Do the process goods or services meet design specifications? The methods of this chapter are intended to address the first question, but not the second.

29. 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.

30. 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.

31. Control Chart for Monitoring Means: The Chart The 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 , 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.

32. 1. The data are process data consisting of a sequence of samples all of the same size n. 2. The distribution of the process data is essentially normal. 3. The individual sample data values are independent. Requirements

33. n = size of each sample, or subgroup Notation = mean of the sample means (equal to the mean of all sample values combined)

34. Points plotted: Sample means Center line: = mean of all sample means Upper Control Limit (UCL): where is found in Table 14-2 Lower Control Limit (LCL): where is found in Table 14-2 Control Chart for Monitoring Means: The Chart

35. Example Construct a control chart for using the weights of the quarters listed in Table 14.1 (see the text, page 691). Based on this control chart, determine whether the process mean is within statistical control. The upper and lower control limits are:

36. Example - Continued

37. Example - Continued Interpretation: The process mean is out of statistical control because at least one of the three out-of-control criteria is not satisfied. Specifically, the second criterion is violated because there are points lying beyond the control limits.