Statistics Summary of Quality and Statistical Process Control Chapter Goals After completing this chapter, you should be able to: ▪ Use the seven basic tools of quality ▪ Construct and interpret x-charts and R-charts ▪ Construct and interpret p-charts ▪ Construct and interpret c-charts Chap 18-2 Chapter Overview Quality Management and Tools for Improvement Philosophy of Quality Deming’s 14 Points Juran’s 10 Steps to Quality Improvement Tools for Quality Improvement The Basic 7 Tools Control Charts x-bar/R-charts p-charts c-charts Chap 18-3 Themes of Quality Management ◼ Primary focus is on process improvement ◼ Most variations in process are due to systems ◼ Teamwork is integral to quality management ◼ Customer satisfaction is a primary goal ◼ Organization transformation is necessary ◼ It is important to remove fear ◼ Higher quality costs less Chap 18-4 Deming’s 14 Points ◼ 1. Create a constancy of purpose toward improvement ◼ ◼ 2. Adopt the new philosophy ◼ ◼ Better to improve now than to react to problems late 3. Stop depending on inspection to achieve quality -- build in quality from the start ◼ ◼ become more competitive, stay in business, and provide jobs Inspection to find defects at the end of production is too late 4. Stop awarding contracts on the basis of low bids ◼ Better to build long-run purchaser/supplier relationships Chap 18-5 Deming’s 14 Points (continued) ◼ ◼ 5. Improve the system continuously to improve quality and thus constantly reduce costs 6. Institute training on the job ◼ ◼ 7. Institute leadership ◼ ◼ ◼ Workers and managers must know the difference between common cause and special cause variation Know the difference between leadership and supervision 8. Drive out fear so that everyone may work effectively. 9. Break down barriers between departments so that people can work as a team. Chap 18-6 Deming’s 14 Points (continued) ◼ 10. Eliminate slogans and targets for the workforce ◼ ◼ ◼ ◼ ◼ They can create adversarial relationships 11. Eliminate quotas and management by objectives 12. Remove barriers to pride of workmanship 13. Institute a vigorous program of education and self-improvement 14. Make the transformation everyone’s job Chap 18-7 Juran’s 10 Steps to Quality Improvement ◼ 1. Build awareness of both the need for improvement and the opportunity for improvement ◼ 2. Set goals for improvement 3. Organize to meet the goals that have been set ◼ ◼ ◼ 4. Provide training 5. Implement projects aimed at solving problems Chap 18-8 Juran’s 10 Steps to Quality Improvement (continued) ◼ ◼ ◼ ◼ ◼ 6. Report progress 7. Give recognition 8. Communicate the results 9. Keep score 10. Maintain momentum by building improvement into the company’s regular systems Chap 18-9 The Deming Cycle Plan Act The Deming Cycle Study Do The key is a continuous cycle of improvement Chap 18-10 The Basic 7 Tools 1. Process Flowcharts 2. Brainstorming 3. Fishbone Diagram 4. Histogram 5. Trend Charts 6. Scatter Plots 7. Statistical Process Control Charts Chap 18-11 The Basic 7 Tools (continued) 1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts Map out the process to better visualize and understand opportunities for improvement Chap 18-12 The Basic 7 Tools (continued) 1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts Show patterns of variation Fishbone (cause-and-effect) diagram: Major Cause 1 Major Cause 2 Causes Problem Causes Major Cause 3 Major Cause 4 Chap 18-13 The Basic 7 Tools (continued) 1. 2. 3. 4. 5. 6. 7. Identify trend Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts y y time Examine relationships x Chap 18-14 The Basic 7 Tools (continued) 1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts Examine the performance of a process over time X time Chap 18-15 Multiple Meanings Of Six Sigma 1. Six Sigma = Management Philosophy View processes  Measures completely from customer point of view  Continual improvement  Integration of quality and daily work  Completely satisfying customer needs profitably  Chap 18-16 Multiple Meanings Of Six Sigma 2. Sigma = Process Capability  A statistical measure of a process’s ability to meet customer requirements (CTQs – Critical To Quality)  Process Sigma ZST = 6; equates to 3.4 Defects Per Million Opportunities Chap 18-17 Multiple Meanings Of Six Sigma 3. Sigma = Standard Deviation The Greek symbol “sigma” which means standard deviation. It is a measure of variation Chap 18-18 GE Six Sigma Appendix B – Six Sigma Overview Chap 18-19 Six Sigma’s Methodologies Design NEW products and processes that meet customer needs Improve EXISTING processes so that their outputs meet customer requirements Six Sigma Process Management Control and manage cross-function processes to meet business goals Chap 18-20 Common Six Sigma Tools • Project management • Voice of the Customer • Process mapping • Data collection • Data graphs • Gage R&R • Operational definitions • Process Capability Assessment • Hypothesis testing • Regression analysis • Designed experiments • Statistical process control • FMEA • Stakeholder analysis • Implementation planning • Tollgate reviews Chap 18-21 DMAIC Methodology 1. 2. 3. 4. 5. Define Measure Analyze Improve Control Chap 18-22 Define ◼ ◼ ◼ Describe the problem in operational terms Drill down to a specific problem statement (project scoping) Identify customers and CTQs, performance metrics, and cost/revenue implications Chap 18-23 Define Business Case Voice of the Customer VOC Delighters Key Issue CTQ More Is Better Must Be Initial Process Mapping Inputs Process Outputs Yield: 60% CUSTOMERS Problem Statement: Goal: Business Case: Scope: Cost Benefit Projection: Milestones: SUPPLIERS Project Charter Yield: 90% Yield: 45% Yield: 98% Chap 18-24 Measure ◼ Key data collection questions ◼ ◼ ◼ ◼ ◼ What questions are we trying to answer? What type of data will we need to answer the question? Where can we find the data? Who can provide the data? How can we collect the data with minimum effort and with minimum chance of error? Chap 18-25 Measure Identify the Metrics Identify Process Capability LSL I P O USL Display Data UCL 1000 Cp = 0.4 s = 2.7 X 0 LCL -1000 10 Input Measures Process Measures 20 30 D B F A C E Other Output Measures Measure the process Prioritize the Metrics Data Collection Plan Data Collection Plan O1 O2 O3 O4 I1 I2 I3 I4 FMEA What questions do you want to answer? Operational Definition and Procedures Data What Measure type/ How Related Sampling How/ Data type measured conditions notes where How will you ensure consistency and stability? What is your plan for starting data collection? How will the data be displayed? Validate Measurement Systems Col # Inspector Sample # 1 2 3 4 5 Totals Averages 1 2 3 4 5 6 1st Trial 2.0 2.0 1.5 3.0 2.0 10.5 2.1 A 2nd Trial 1.0 3.0 1.0 3.0 1.5 9.5 1.9 Diff 1.0 1.0 0.5 0.0 0.5 3.0 0.6 1st Trial 1.5 2.5 2.0 2.0 1.5 9.5 1.9 B 2nd Trial 1.5 2.5 1.5 2.5 0.5 8.5 1.7 Diff 0.0 0.0 0.5 0.5 1.0 2.0 0.4 Sum XA 4.0 2.0 R A Sum XB 3.6 1.8 R B Chap 18-26 Analyze ◼ ◼ ◼ ◼ Focus on why defects, errors, or excessive variation occur Seek the root cause 5-Why technique Experimentation and verification Chap 18-27 Analyze Process Door VA Data Door NVA 22 O O 21 Cause & Effect O 20 O 19 n 18 X 17 n O n X n X O O O n X n 15 n O 16 X X O X n 14 n X n 13 X 12 . Hypothesis-Testing Chi-Square 1 2 3 4 5 6 7 8 9 10 Design of Experiments t-test ANOVA c² Regression Analysis Y Regression X1 . . Chap 18-28 Improve ◼ ◼ ◼ ◼ Idea generation Brainstorming Evaluation and selection Implementation planning Chap 18-29 Improve Plan Implementation Generate Solutions A B C D 4 1 3 2 1 2 3 4 5 6 7 8 9 10 A B C D G E F G H Perform CostBenefit Analysis I J Run Pilot Full scale Original Test Assess Risks Select the Solution FMEA Chap 18-30 Control ◼ ◼ ◼ ◼ ◼ Maintain improvements Standard operating procedures Training Checklist or reviews Statistical process control charts Chap 18-31 Control Document & Standardize Key Learnings Closure Results Training Curriculum Training Manual Learnings • • • Fill to here Recommendations next QC Process Chart Date of Issue: Revision Date Product Name Process Name Process Code # Flowchart Issued by: Reason Approved by: Signature Process Owner Control/Check Points Response to Abnormality Work Control Immediate Permanent Instructions Code # CharacW ho Fix Who Notes teristicsLimitsMethod Fix . Evaluate Project Results 1 2 12 Ownership & Monitoring Process Change Management } A1 A2 A3 A4 . LCL 3.7 1.4 LSL s = 2.7 Cp = 0.4 USL After Before UCL Before s = Cp = Improvement A2 A1 A3 A4 After Step 4 changes implemented Good } Improvement } Remaining Gap Target Chap 18-32 Introduction to Control Charts ◼ Control Charts are used to monitor variation in a measured value from a process ◼ Exhibits trend ◼ Can make correction before process is out of control ◼ A process is a repeatable series of steps leading to a specific goal ◼ Inherent variation refers to process variation that exists naturally. This variation can be reduced but not eliminated Chap 18-33 Process Variation Total Process Common Cause Special Cause = + Variation Variation Variation ◼ ◼ ◼ Variation is natural; inherent in the world around us No two products or service experiences are exactly the same With a fine enough gauge, all things can be seen to differ Chap 18-34 Sources of Variation Total Process Common Cause Special Cause = + Variation Variation Variation Variation is often due to differences in: ◼ People ◼ Machines ◼ Materials ◼ Methods ◼ Measurement ◼ Environment Chap 18-35 Common Cause Variation Total Process Common Cause Special Cause = + Variation Variation Variation Common cause variation ◼ naturally occurring and expected ◼ the result of normal variation in materials, tools, machines, operators, and the environment Chap 18-36 Special Cause Variation Total Process Common Cause Special Cause = + Variation Variation Variation Special cause variation ◼ abnormal or unexpected variation ◼ has an assignable cause ◼ variation beyond what is considered inherent to the process Chap 18-37 Statistical Process Control Charts ◼ Show when changes in data are due to: ◼ ◼ Special or assignable causes ◼ Fluctuations not inherent to a process ◼ Represents problems to be corrected ◼ Data outside control limits or trend Common causes or chance ◼ Inherent random variations ◼ Consist of numerous small causes of random variability Chap 18-38 Control Chart Basics Special Cause Variation: Range of unexpected variability UCL Common Cause Variation: range of expected variability +3σ Process Average - 3σ LCL time UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations Chap 18-39 Process Variability Special Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present UCL ±3σ → 99.7% of process values should be in this range Process Average LCL time UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations Chap 18-40 Statistical Process Control Charts Statistical Process Control Charts x-chart and R-chart p-chart c-chart Used for measured numeric data Used for proportions (attribute data) Used for number of attributes per sampling unit Chap 18-41 x-chart and R-chart ◼ ◼ ◼ Used for measured numeric data from a process Start with at least 20 subgroups of observed values Subgroups usually contain 3 to 6 observations each Chap 18-42 Steps to create an x-chart and an R-chart ◼ Calculate subgroup means and ranges ◼ Compute the average of the subgroup means and the average range value ◼ Prepare graphs of the subgroup means and ranges as a line chart Chap 18-43 Steps to create an x-chart and an R-chart (continued) ◼ Compute the upper and lower control limits for the x-chart ◼ Compute the upper and lower control limits for the R-chart ◼ Use lines to show the control limits on the x-chart and R-chart Chap 18-44 Example: x-chart ◼ Process measurements: Subgroup measures Subgroup Individual measurements number Mean, x Range, R 1 15 17 15 11 14.5 6 2 12 16 9 15 13.0 7 3 17 21 18 20 19.0 4 … … … … … … … Average subgroup mean = x Average subgroup range = R Chap 18-45 Average of Subgroup Means and Ranges Average of subgroup means: x  x= i k where: Average of subgroup ranges: R  R= i k where: xi = ith subgroup average k = number of subgroups Ri = ith subgroup range k = number of subgroups Chap 18-46 Computing Control Limits ◼ The upper and lower control limits for an x-chart are generally defined as UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations ◼ or UCL = x + 3σ LCL = x − 3σ Chap 18-47 Computing Control Limits (continued) ◼ Since control charts were developed before it was easy to calculate σ, the interval was formed using R instead ◼ The value A2R is used to estimate 3σ , where A2 is from Appendix Q ◼ The upper and lower control limits are UCL = x + A 2 ( R ) LCL = x − A 2 ( R ) where A2 = Shewhart factor for subgroup size n from appendix Q Chap 18-48 Example: R-chart ◼ The upper and lower control limits for an R-chart are UCL = D 4 ( R ) LCL = D3 ( R ) where: D4 and D3 are taken from the Shewhart table (appendix Q) for subgroup size = n Chap 18-49 x-chart and R-chart UCL x-chart x LCL time UCL R-chart R LCL time Chap 18-50 Using Control Charts ◼ Control Charts are used to check for process control H0: The process is in control i.e., variation is only due to common causes HA: The process is out of control i.e., special cause variation exists ◼ If the process is found to be out of control, steps should be taken to find and eliminate the special causes of variation Chap 18-51 Process In Control ◼ Process in control: points are randomly distributed around the center line and all points are within the control limits x UCL x LCL time Chap 18-52 Process Not in Control Out of control conditions: ◼ One or more points outside control limits ◼ Nine or more points in a row on one side of the center line ◼ Six or more points moving in the same direction ◼ 14 or more points alternating above and below the center line Chap 18-53 Process Not in Control ◼ ◼ One or more points outside control limits UCL ◼ Nine or more points in a row on one side of the center line UCL x x LCL LCL Six or more points moving in the same direction UCL ◼ 14 or more points alternating above and below the center line UCL x x LCL LCL Chap 18-54 Out-of-Control Processes ◼ When the control chart indicates an out-ofcontrol condition (a point outside the control limits or exhibiting trend, for example) ◼ ◼ Contains both common causes of variation and assignable causes of variation The assignable causes of variation must be identified ◼ If detrimental to the quality, assignable causes of variation must be removed ◼ If increases quality, assignable causes must be incorporated into the process design Chap 18-55 p-Chart ◼ Control chart for proportions ◼ ◼ Is an attribute chart Shows proportion of nonconforming items ◼ Example -- Computer chips: Count the number of defective chips and divide by total chips inspected ◼ ◼ Chip is either defective or not defective Finding a defective chip can be classified a “success” Chap 18-56 p-Chart (continued) ◼ ◼ Used with equal or unequal sample sizes (subgroups) over time ◼ Unequal sizes should not differ by more than ±25% from average sample sizes ◼ Easier to develop with equal sample sizes Should have np ≥ 5 and n(1-p) ≥ 5 Chap 18-57 Creating a p-Chart ◼ Calculate subgroup proportions ◼ Compute the average of the subgroup proportions ◼ Prepare graphs of the subgroup proportions as a line chart ◼ Compute the upper and lower control limits ◼ Use lines to show the control limits on the p-chart Chap 18-58 p-Chart Example Subgroup number Sample size Number of successes Proportion of successes, p 1 150 15 .1000 2 150 12 .0800 3 150 17 .1133 … … … Average subgroup proportion = p Chap 18-59 Average of Subgroup Proportions The average of subgroup proportions = p If equal sample sizes: p  p= i k where: pi = sample proportion for subgroup i k = number of subgroups of size n If unequal sample sizes: np  p= n i i i where: ni = number of items in sample i ni = total number of items sampled in k samples Chap 18-60 Computing Control Limits ◼ The upper and lower control limits for an p-chart are UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion – 3 Standard Deviations ◼ or UCL = p + 3σ LCL = p − 3σ Chap 18-61 Standard Deviation of Subgroup Proportions ◼ The estimate of the standard deviation for the subgroup proportions is If equal sample sizes: (p)(1 − p) sp = n where: If unequal sample sizes: Generally, s p is computed separately for each different sample size p = mean subgroup proportion n = common sample size Chap 18-62 Computing Control Limits (continued) ◼ The upper and lower control limits for the p-chart are UCL = p + 3(s p ) LCL = p − 3(s p ) ◼ If sample sizes are equal, this becomes (p)(1 − p) UCL = p + 3 n Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0 (p)(1 − p) LCL = p − 3 n Chap 18-63 p-Chart Examples ◼ For equal sample sizes ◼ For unequal sample sizes UCL UCL p p LCL LCL s p is constant since s p varies for each n is the same for all subgroups subgroup since ni varies Chap 18-64 c-Chart ◼ Control chart for number of nonconformities (occurrences) per sampling unit (an area of opportunity) ◼ ◼ Shows total number of nonconforming items per unit ◼ ◼ Also a type of attribute chart examples: number of flaws per pane of glass number of errors per page of code Assume that the size of each sampling unit remains constant Chap 18-65 Mean and Standard Deviation for a c-Chart ◼ The mean for a c-chart is x  c= ◼ The standard deviation for a c-chart is i k s= c where: xi = number of successes per sampling unit k = number of sampling units Chap 18-66 c-Chart Control Limits The control limits for a c-chart are UCL = c + 3 c LCL = c − 3 c Chap 18-67 Process Control Determine process control for p-chars and c-charts using the same rules as for x-bar and R-charts Out of control conditions: ◼ One or more points outside control limits ◼ Nine or more points in a row on one side of the center line ◼ Six or more points moving in the same direction ◼ 14 or more points alternating above and below the center line Chap 18-68 c-Chart Example ◼ A weaving machine makes cloth in a standard width. Random samples of 10 meters of cloth are examined for flaws. Is the process in control? Sample number 1 2 3 4 5 6 7 Flaws found 2 1 3 0 5 1 0 Chap 18-69 Constructing the c-Chart ◼ The mean and standard deviation are: x  c= k i 2 + 1+ 3 + 0 + 5 + 1+ 0 = = 1.7143 7 s = c = 1.7143 = 1.3093 ◼ The control limits are: UCL = c + 3 c = 1.7143 + 3(1.3093) = 5.642 LCL = c − 3 c = 1.7143 − 3(1.3093) = −2.214 Note: LCL < 0 so set LCL = 0 Chap 18-70 The completed c-Chart 6 UCL = 5.642 5 4 3 2 c = 1.714 1 0 LCL = 0 1 2 3 4 5 6 7 Sample number The process is in control. Individual points are distributed around the center line without any pattern. Any improvement in the process must come from reduction in common-cause variation Chap 18-71 Chapter Summary ◼ Reviewed the philosophy of quality management ◼ ◼ ◼ ◼ Demings 14 points Juran’s 10 steps Described the seven basic tools of quality Discussed the theory of control charts ◼ Common cause variation vs. special cause variation ▪ Constructed and interpreted x-charts and Rcharts ▪ Constructed and interpreted p-charts ▪ Constructed and interpreted c-charts Chap 18-72 Project Description and Rubric Select one of the two options and use the Template in the second worksheet of this Excel file: #1. Suppose that you were developing a small restaurant with a dining area and local delivery. #2 - Select a topic of your choice to conduct a House of Quality Project. Requirements for the House of Quality: 1-Develop a list of 10 Customer Requirements, 2-Develop a list of 10 Technical Requirements (that provide the foundation for the product or service design). Summarize the results of Steps 1-4. 3-Develop a relationship matrix between the Customer Requirements and the Technical Requirements. 4-Develop the relationships among the Technical Requirements. 5- Summarize your results. Step 1 Step 2 Step 3 Step 4 20-Exceeds 10-Meets 5-Partially Meets Expectations Expectations Expectations > 10 10 5-9 > 10 10 5-9 > 10 10 5-9 > 10 10 5-9 1. Paper is logically organized Summary 2. Easily followed 3. Effective, smooth and logical transitions 4. Professional format 1. Paper has a clear organizational structure with some digressions, ambiguities or irrelevances. 2. Easily followed 0-Inadequate
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