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Unformatted text preview: Shewhart’s Theory of Chance Cause Systems of Variation 1 Shewhart’s Chance Cause System h All empirical data are generated by some type of process. Walter Shewhart referred to these processes as chance cause systems of variation. l 2 Shewhart’s Chance Cause System The Shewhart Model of a Chance Cause System of Variation Group 1 M Units from the Process Produced at Time Point 1 Group 2 M Units from the Process Produced at Time Point 2 Group K M Units from the Process Produced at Time Point K Data Production Process ... ... Explainable Causes of Variation Production Mules Figure 4 UE W E E M E E E Fi g u r M 3 Production M E E E UE UE UE UE UE W W Explainable Causes of Variation Sampling Mules UE UE E UE UE UE UE E Figure 1 E E E Sampling Cause System Creating the Sample of N Units from the Batch E UE W Unexplainable Causes of Variation Production Woodpeckers Unexplainable Causes of Variation Sampling Woodpeckers Explainable Causes of Variation Measurement Mules UE UE UE Unexplainable Causes of Variation Measurement Woodpeckers W E Figure 2 E E E UE Y Cause System Creating the Fundamental Variation Across and Within Groups of Widgets Sample of Size N Units from the Batch Measurement Cause System Creating the Observed Data Y Shewhart’s Chance Cause System 3 Shewhart’s Chance Cause System h All chance cause systems are made up of two types of chance causes which are referred to as explainable causes and unexplainable causes. Therefore, chance cause systems can be thought of as satisfying the pseudo equation r chance cause systems = explainable causes + unexplainable causes. 4 Explainable Causes h Explainable causes . . . lie outside the process, and they contribute significantly to the total variation observed in performance measures. performance l The variation created by explainable causes is usually The unpredictable, but it is explainable after it has been observed. unpredictable, 5 The HD-2 Filler Case Study Control Chart for Lane 1 245 Control Chart for Lane 4 245 Gross Fill Weight - Grams Gross Fill Weight - Grams 3.0SL=241.0 240 3.0SL=241.0 240 X=238.0 235 X=238.0 235 -3.0SL=235.0 Evidence of an assignable cause - periodic low weights 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -3.0SL=235.0 Evidence of an assignable cause - a saw tooth pattern 230 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 230 Cup Cup 6 The HD-2 Filler Case Study Control Chart for Lane 4 - Phase 1 245 Control Chart for Lane 4 - Phase 2 245 Gross Fill Weight - Grams 3.0SL=241.0 240 X=238.0 Gross Fill Weight - Grams 3.0SL=241.0 240 X=238.0 235 -3.0SL=235.0 235 -3.0SL=235.0 230 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 230 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Cup Cup 7 Unexplainable Causes Unexplainable causes . . . l May be unidentified explainable causes or they may be random causes that belong to, or are inherent in, the process. they If they are common, unexplained causes then they produce random variation If in the behavior of the performance measurement, but the variation is consistent and predictable. The variation associated with unexplained random causes of variation has a statistical identity. causes The random variation produced by unexplainable causes is often referred to The as noise, because there is no real change in process performance. noise because Noise cannot be traced to a specific cause, and it is therefore, although Noise predictable, it is unexplainable. l l 8 Unexplainable Causes h The random nature of unexplainable cause variation lends itself to The the application of the statistical methods, while the chaotic or pattened variation produced by explainable causes may not. Therefore, the statistical methods can be used to characterize Therefore, the inherent process variation and to develop tools to identify the presence of explainable causes. presence l 9 Unexplainable Causes h It may seem counter intuitive at first to claim that the objective of process It control is to achieve a state where the process behaves in a random fashion. There are numerous instances where we depend upon random behavior to There predict results. l Card games like bridge or poker Card Games of chance in Las Vegas Games 10 10 Shewhart’s Chance Cause System The Woodpeckers and the Mules! 11 11 Shewhart’s General Chance Cause System h The variation due to the woodpeckers in constant causes systems is The unexplainable, but predictable! unexplainable The variation due to mules in a general chance cause system is The explainable, but generally unpredictable! explainable l 12 12 Shewhart’s Theory of Chance Cause Systems of Variation Unexplainable Explainable Random, Predictable Constant Causes Unpredictable Assignable Causes 13 13 Basic Statistical Basic Concepts for Constant Cause Systems Systems Woodpeckers Only ! 14 14 Constant Cause Systems h Chance cause systems that are made up only of unexplainable random causes are referred to as constant cause systems. That is, constant cause systems = unexplainable random causes. 15 15 Constant Cause Systems h Constant cause systems are equivalent to the assumption of independent, identically distributed (IID) random variables. 16 16 Constant Cause Systems Although both unexplainable and explainable causes create variation in the performance measure of interest, unexplainable causes create controlled variation while explainable causes create uncontrolled variation. 17 17 Constant Cause Systems Shewhart defined controlled variation in the following way. “A phenomenon will be said to be controlled when, through the use of past experience, we can predict, at least within limits, how the phenomenon may be expected to vary in the future. Here it is understood that prediction within limits means that we can state, at least approximately, the probability that the observed phenomenon will fall within given limits.” h Constant cause systems are therefore controlled cause systems. 18 18 The Empirical Rule h It a remarkable fact that the following relationships are approximately true for It almost any process distribution associated with a constant cause system. 60% to 75% of the process output lies between µ - σ and µ + σ 90% to 98% of the process output lies between µ - 2σ and µ + 2 σ 99% to 100% of the process output lies between µ - 3σ and µ + 3σ . 19 19 The Empirical Rule 99%-100% 90%-98% 60%- 75% µ − 3σ µ − 2σ µ −σ µ µ+σ µ + 2σ µ + 3σ The Empirical Rule is the Foundation for Shewhart Control Charts 20 20 Understanding and Analyzing a Chance Cause System of Variation 21 21 Rational Subgrouping h One of the most important concepts to understand and master in One order to use use the data from analytic studies to their full potential is the notion of rational subgroups. The key to extraction of information from data is asking and The answering the right questions. answering This can only be achieved by fully understanding and exploiting This the structure of the data obtained from the process. the h l 22 22 Rational Subgrouping Shewhart made the following important observation regarding rational subgroups. “Obviously, the ultimate object is not only to detect trouble but also to find it, and such discovery naturally involves classification. The engineer who is successful in dividing his data initially into rational subgroups based on rational hypotheses is therefore inherently better off in the long run than the one who is not thus successful.” 23 23 Understanding a Chance Cause System of Variation h Many chance cause systems can be rationalized by hypothesizing what factors Many are potentially creating the observed variation in the data. are Factor 1 Factor 2 …. Factor K Time Unknown factors Random, Unexplained variation 24 24 Understanding a Chance Cause System of Variation h For example in the aseptic filler case study: Factor 1 - Lane (1-4) Factor 2 - Phase (1-2) Time = Order of production Unknown factors - pump effect Random, Unexplained variation 25 25 The Concept of Rational Subgrouping The General Structure of Rational Subgroups Rational Subgroups 1 Observations w ithin Subgroups 1 2 … n X11 X12 … X1N X1 R1 S1 2 X21 X22 … X2N X2 R2 S2 … … … … … … … … K XK1 XK2 … XKN XK RK SK Average within Subgroups Range within Subgroups Standard Deviation w ithin Subgroups 26 26 Rational Subgrouping The fundamental concept of rational subgrouping is to study the variation observed across subgroups that are defined in a meaningful way relative to the variation observed within the subgroups, in order to answer important questions. l Rational subgroups represent samples from the process organized in some meaningful way relative to a region of space, time, subprocess or product. 27 27 Rational Subgrouping In general, the statistical analysis methods that are constructed from the data contained in the rational subgroups are designed to answer the following question “Is the variation in the performance measure observed across subgroups greater than predicted based on the variation observed within the subgroups?” 28 28 Rational Subgrouping For a constant cause system the variation within a subgroup is the For same as the variation across subgroups. l Therefore, if the assumption of a constant cause system is correct, Therefore, it should be possible to predict the behavior of summary statistics, like sample averages, ranges, and standard deviations, across subgroups based on the homogeneous variation observed within subgroups. subgroups. 29 29 Evidence of Explainable Causes h Data from a constant cause system of variation will display random, Data unexplainable variation both within and across rational subgroups. unexplainable The range of variation due to constant causes will be within predictable The statistical limits. statistical Nonrandom patterns of variation appearing within or across the rational Nonrandom subgroups, that can be meaningfully interpreted within the context of the cause system of variation, provide evidence that explainable causes are affecting the data. data. u u 30 30 The Attendance Management Case Study The effective management of employee attendance is an important The management responsibility. Within a data entry process, it is important that all 10 data entry specialists scheduled for work are present or the system becomes backlogged, and important deadlines are missed. The supervisors had raised important concerns about the level of absenteeism among the employees within the department. employees Each employee’s attendance rate, defined as the percent of scheduled hours Each actually worked, is recorded each pay period. The employees are paid on a bi-weekly basis, and the payroll department maintains the employee attendance data. Attendance data were available for 22 consecutive pay periods. The actual data for the ten employees are presented in Table 8.1. periods. 31 31 The Attendance Management Case Study PAY PERIOD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 1 100.00 100.00 100.00 100.00 100.00 74.94 100.00 100.00 74.91 100.00 100.00 87.50 100.00 100.00 100.00 60.00 100.00 100.00 100.00 100.00 100.00 100.00 2 100.00 100.00 100.00 100.00 100.00 100.00 20.00 100.00 100.00 100.00 100.00 73.73 100.00 85.71 88.89 100.00 75.00 12.50 54.55 71.53 75.00 62.50 3 96.39 90.00 100.00 100.00 100.00 80.00 100.00 83.13 100.00 100.00 100.00 80.00 100.00 100.00 63.64 90.00 100.00 95.00 100.00 100.00 100.00 100.00 4 70.00 100.00 100.00 100.00 80.00 100.00 100.00 100.00 100.00 100.00 77.78 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 90.00 100.00 100.00 EMPLOYEE 5 10.00 77.21 44.44 60.00 89.74 100.00 100.00 50.00 100.00 100.00 88.89 80.00 50.00 60.00 87.50 90.00 60.00 55.00 89.97 88.89 53.62 62.50 6 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 68.42 100.00 100.00 88.32 88.89 100.00 100.00 100.00 100.00 88.69 100.00 97.81 88.65 7 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 70.00 96.38 80.00 95.00 82.92 8 100.00 100.00 100.00 100.00 97.22 100.00 100.00 78.95 71.43 100.00 100.00 100.00 80.00 86.88 100.00 100.00 100.00 75.00 67.50 100.00 0.00 0.00 9 100.00 100.00 100.00 100.00 100.00 100.00 100.00 94.12 100.00 87.80 100.00 90.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 10 0.00 0.00 100.00 98.10 69.03 100.00 100.00 60.00 82.98 84.81 87.50 77.78 59.75 33.05 80.00 100.00 43.25 70.00 100.00 100.00 82.22 100.00 32 32 The Attendance Management Case Study There are two organizations or structures of the data that were exploited to There answer important questions concerning employee attendance using control charts. Table 8.2 presents the first structure which uses the 22 pay periods as the rational subgroups. The table entries Pij denote the recorded as attendance rate for the ith employee for the jth pay period. attendance The first question that was asked by management was whether or not the The overall department attendance rate was changing over time; i.e., from pay period to pay period. To answer this question, the attendance rates were organized into 22 rational subgroups by pay period. The data within the subgroups were the attendance rates for the 10 employees for the pay period. An average chart was constructed using the pay period as the rational subgroup and the individual employee attendance rate as the basic data within the subgroup. within 33 33 The Attendance Management Case Study Table 8.2. The First Organization of the Employee Attendance Data Rational Subgroup - Pay Period Employee 1 2 1 P1 1 P2 1 2 P1 2 P2 2 … 22 P1 22 P2 22 10 P10 1 P10 2 P10 22 34 34 The Attendance Management Case Study Figure 8.7 presents the control chart for the department attendance rate by pay period. Figure Based on Figure 8.7, there is no evidence that the department attendance rate is changing over time. It appears to be in a reasonable state of statistical control around the average of 89.15%. the Control Chart for Department Attendance Rate 10 0 90 A TT NDA NC R T E E AE 80 70 60 50 40 30 20 1 0 0 PA Y PERI OD 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 20 2122 0 1234567 89 LCL=69.49 X=89.1 5 Figure 8.7. Average Chart for the Department Attendance Rate 35 35 The Attendance Management Case Study The next question that was asked was whether there were differences in the attendance rates across the employees. To answer this question the data were reorganized using the employee as the rational subgroup. Table 8.3 presents the second organization of this data. Table 8.3. The Second Organization of the Employee Attendance Data Rational Subgroup - Employee Pay Period 1 2 1 P1 P1 1 2 P 21 P2 2 3 P 31 P3 2 … 10 P 10 P 10 1 2 2 22 P1 22 P2 22 P3 22 P 10 22 36 36 The Attendance Management Case Study Figure 8.8 presents the average chart produced using the employee as the rational subgroup. This control chart compares the variation in attendance rates across employees to the variation observed over the 22 pay periods within an employee. Comparison of Employee Attendance Rates 10 0 90 ATT ND ANC RAT E E E 80 70 60 50 40 30 20 1 0 0 EMPLOYEE 1 2 3 4 5 6 7 8 9 1 0 X=89.1 5 L L=77.24 C Figure 8.8. The Average Chart for Comparing Employee Attendance Rates 37 37 The Attendance Management Case Study This chart indicates that the variation in the averages across the employees This is larger than expected compared to the variation in attendance within employees. The chart provides a clear signal that the attendance rates for employee #5 and employee #10 fall outside of the expected range due to unexplainable cause variation, and they should be investigated. u Management should work with these two employees on a localized basis in Management an attempt to discover the reasons for their low attendance in order to help them get back into the normal system. them 38 38 The Plastic Cup Flange Width Example The first generation HD-2 filler process was designed to simultaneously fill and The seal preformed cups at four filling and sealing stations, so the original machine filled four cups at a time. filled The second generation machine was designed to simultaneously form cups and The then fill the cups using 24 cup forming cavities, and 24 filling and sealing stations. This new design eliminated the need for an outside cup vendor, and increased the production capacity by a factor of 6. increased After the machine forms and fills the cup, the cup is sealed with a heat treated After foil seal. The integrity of the product in the cup is dependent upon a good seal. The integrity of the seal is very dependent on the width of the cup flange because the heat treatment melts the flange and seats the foil seal into the melted flange. melted 39 39 The Plastic Cup Flange Width Example The functional specification limits for the flange width are 4.5 mm ± 0.5 mm. The cup flange is created by the 24 cavities in the cup forming process. The geometry of the 24 cup forming cavities is presented in Figure 8.9. 40 40 The Plastic Cup Flange Width Example Column Column Column Column Column Column 3 4 5 6 2 1 Row 1 1 2 3 4 5 6 Row 2 7 8 9 10 11 12 Row 3 13 14 15 16 17 18 Row 4 19 20 21 22 23 24 Machine Direction Figure 8.9. Cup Forming Cavity Geometry for the 24 Cavities 41 41 The Plastic Cup Flange Width Example An acceptance test was conducted in which numerous performance characteristics An of the machine were analyzed, including flange width. Two of the questions of interest were whether or not the flange width could be maintained in a state of statistical control during the production run, and whether or not the 24 cavities significantly affect flange width. In order to answer these questions, the acceptance test was designed as follows. acceptance A nine-hour production run was scheduled under normal working conditions. At nine-hour the beginning of each hour, n=4 successive cups were sampled from each of the 24 cavities. This resulted in a total of N = 9x24x4 = 864 flange width measurements. Three different organizations of the data were considered in order to answer the questions of interest. questions 42 42 The Plastic Cup Flange Width Example The first structure for the data is presented in Table 8.4. Using this structure there The are 216 rational subgroups of size n=4. The subgroups have been arranged so that the data for 8:00 A.M. are presented first for all 24 cavities, followed by the data for 9:00 A.M. for all 24 cavities, etc. for The variation within the subgroups is the variation across four consecutive cups The formed by the same cavity at the same point in time. This variation should reflect the inherent or unexplained variation in the process (i.e., the process noise). the The variation across the subgroups is affected not only by the noise in the process, The but also possibly by explainable causes due to cavity differences within a time period and explainable causes across time. period 43 43 The Plastic Cup Flange Width Example Table 8.4. The First Structure of the Acceptance Test Data 216 Rational Subgroups with N = 4 Time Cavity 1 Replicates 2 3 4 Average Range 8:00 A.M. 1 X X X X X1 R1 2 X X X X X2 R2 ... 24 X X X X 1 X X X X 9:00 A.M. 2 X X X X ... 24 X X X X … 1 X X X X 4:00 P.M. 2 X X X X ... 24 X X X X X216 R216 44 44 The Plastic Cup Flange Width Example Figure 8.10 is the average chart and Figure 8.11 is the range chart produced Figure from this organization of the data. It is clear from Figure 8.10 that the process was not in a state of statistical control during the production run. For example, the flange width increased significantly from subgroup 25 to 48 which represents the 9:00 a.m. and 10:00 a.m. time frame. u The flange width then decreased for subgroups 49 through 96 which The represents 11:00 a.m. and 12:00 p.m. time frame. The range chart in Figure 8.11 also indicates that the inherent process variation increased beginning with the 10:00 a.m. subgroup. with 45 45 The Plastic Cup Flange Width Example 15 . 5.0 Flange Width UCL = 4.849 4.5 10 . UCL = 10.55 X = 4.512 LCL = 4.175 Range 0.5 R = 0.4624 4.0 1 0 30 50 70 90 10 1 0 1 0 1 0 1 0 21 13579 0 0.0 1 0 30 50 70 90 10 1 0 1 0 1 0 1 0 21 13579 0 LCL = 0.000 Subgroup Subgroup Figure 8.10. Average Chart for Flange Width Data Structure #1 Figure 8.11. Range Chart for Flange Width Data Structure #1 46 46 The Plastic Cup Flange Width Example Figure 8.12 is the same range chart except only the first 48 subgroups (the 8:00 a.m. and 9:00 a.m. data) were used to set the control limits. That chart clearly shows the process variation to be out of control during the production run. 15 . 10 . RA NGE UCL=0.6321 0.5 R=0.2770 0.0 1 0 30 50 70 90 1 0 1 0 1 0 1 0 1 0 21 13 5 7 9 0 LCL=0.000 SUB GROUP Figure 8.12. Range Chart for Flange Width - Data Structure #1 and Control Limits Set with 8:00 A.M. and 9:00 A.M. Data 47 47 The Plastic Cup Flange Width Example These charts indicate that there are explainable causes of variation in the These cause system affecting both the average flange width and the inherent variation in flange width. These explainable causes should be investigated and removed from the process if possible. and u Figure 8.13 is a histogram of the 864 flange width measurements with the Figure upper and lower functional specification limits superimposed on the graph. Clearly, this process is not capable of meeting the functional specification limits. limits. 48 48 The Plastic Cup Flange Width Example 10 0 USL LSL Frequency 50 0 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 Flange Width Figure 8.13. Comparison of the Histogram and Functional Specification Limits for Flange Width 49 49 The Plastic Cup Flange Width Example The second structure for the data is presented in Table 8.5. Here The the rational subgroups presented in the first organization have simply been rearranged so that the nine time periods for cavity #1 are presented first, followed the nine time periods for cavity #2, etc. are 50 50 The Plastic Cup Flange Width Example Table 8.5. The Second Structure of the Acceptance Test Data 216 Rational Subgroups with N = 4 1 Time 1 Replicates 2 3 4 Average Range 8 am 9 am X X X X X1 R1 X X X X X2 R2 ... 4 pm 8 am 9 am X X X X X X X X X X X X 2 ... 4 pm X X X X … 24 8 am 9 a m X X X X X X X X ... 4 pm X X X X X216 R216 51 51 The Plastic Cup Flange Width Example This organization allows an easy analysis of the performance, and This comparison of, the individual cavities across time. The average chart for the first four cavities is presented in Figure 8.14. chart l This control chart is reproduced in Figure 8.15 which shows where This each of the four cavities begin and end. Only four cavities are presented on the chart in order to see the patterns more clearly. In the actual analysis, all 24 cavities were studied. the 52 52 The Plastic Cup Flange Width Example CA VI TY 1 5.0 5.0 A C VI T 2 C VI TY 3 AY CA VI TY 4 Flange Width Flange Width UCL = 4.849 4.5 UCL = 4.849 X = 4.512 LCL - 4.175 4.0 X = 4.512 LCL - 4.175 4.5 4.0 2 6 1 0 1 4 1 8 22 26 30 34 2 6 1 0 1 4 1 8 22 26 30 34 Subgroup Figure 8.14. Control Chart for the First Four Cavities Subgroup Figure 8.15. Control Chart for the First Four Cavities - Beginning and End Points Identified 53 53 The Plastic Cup Flange Width Example It is clear from Figure 8.15 that the flange width went out of control for each of the It four cavities at 9:00 A.M. which indicates an explainable cause associated with the process that systematically affected all four cavities. (In fact, the complete control chart indicated that it affected all 24 cavities.) The cups are formed from plastic sheets which come in large rolls sheets This shift in flange width was traced to a sheet splice (i.e., the plastic roll was This changed over to a new roll). Similar shifts in the flange width occurred throughout the production run when new sheet rolls were spliced into the process. The explainable cause was traced to a change in sheet thickness. The thickness of the sheet rolls purchased from an outside vendor was not consistent from roll to roll, and the sheet roll vendor was contacted to discuss ways to improve the consistency of the sheet thickness. of 54 54 The Plastic Cup Flange Width Example The third organization of the same data, presented in Table 8.6, was designed to The answer the question about the effects of the 24 cavities. In this case the data were placed into 24 subgroups defined by the 24 cavities. Since a sample of size n = 4 cups was selected from each cavity for each of the 9 time periods, there are n = 36 measurements per subgroup in this case. measurements The variation within the subgroup includes the process noise plus the differences The across time. The variation across subgroups includes the effects of cavities. Since the subgroup sample size is larger than 10, the average and standard deviation charts presented in Figures 8.16, 8.17, and 8.18 were used to analyze the data. charts 55 55 The Plastic Cup Flange Width Example Table 8.6. The Third Structure of the Acceptance Test Data 24 Rational Subgroups with N = 36 Time Replicates 1 Cavity 1 X X X X X X X X X X X X X1 S1 2 X X X X X X X X X X X X X2 S2 ... 24 X X X X X X X X X X X X X24 S24 56 56 8:00 AM 2 3 4 1 2 3 4 9:00 AM ... 1 4:00 PM 2 3 4 Average Range The Plastic Cup Flange Width Example 4 .8 4.8 4.7 R W1 O R W2 O R W3 O R W4 O Flange Width Flange Width 4. 7 UCL = 4.652 4. 6 4. 5 4.4 4.3 2 4 6 8 10 12 14 16 1 20 22 24 8 UCL = 4.652 4.6 X = 4.512 LCL = 4.372 = = 4.5 4.4 4.3 2 4 6 8 10 1 14 16 18 20 22 2 4 2 X = 4.512 LCL = 4.372 Cavity Figure 8.16. Analysis of Cavity Effect on Flange Width Cavity Figure 8.17. Analysis of Cavity Effect on Flange Width - Cavity Row Geometry Identified on the Chart 57 57 The Plastic Cup Flange Width Example Figure 8.16 is the initial average chart, and Figure 8.17 is the same Figure chart with the information on the cavity row geometry described in Figure 8.9 included on the chart. There is a clear signal from these charts that the cavities are having a significant effect on the flange width. There is an obvious nonrandom pattern in the flange width across cavities in each row. cavities 58 58 The Plastic Cup Flange Width Example The flange width generally decreases across the 6 cavities within each of The the four rows. The decrease is associated with a column effect. The explainable cause was traced to an uneven distribution of heat in the forming plates that fit across the rows. The heat distribution system associated with the forming plates was redesigned to obtain a constant heat gradient across each of the four rows. heat 59 59 The Plastic Cup Flange Width Example h The standard deviation chart is presented in Figure 8.18. Since this organization The of the data placed both the process noise and time effects into the rational subgroups, it was decided that no action should be taken based on this chart at this time. this 0.4 UCL = 0.3791 Standard Deviation 0.3 S = 0.2788 0.2 LCL = 0.1784 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 22 24 Cavity Figure 8.18. Analysis of Standard Deviation of Flange Width within Cavities 60 60 ...
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This note was uploaded on 02/19/2012 for the course STATISTICS 641 taught by Professor Staff during the Fall '11 term at Ohio State.

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