Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 18
Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 18 Statistical Applications in Quality Management Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-1 Learning Objectives In this chapter, you learn: The basic themes of quality management and Deming’s 14 points The basic aspects of Six Sigma management How to construct various control charts Which control chart to use for a particular type of data How to measure the capability of a process Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-2 Chapter Overview Quality Management and Tools for Improvement Philosophy of Quality Deming’s 14 Points Sigma® Six Management Tools for Quality Improvement Control Charts Process Capability p chart R chart X chart Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-3 Total Quality Management Primary focus is on process improvement Most variation in a process is due to the system, not the individual 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-4 Deming’s 14 Points 1. Create a constancy of purpose toward improvement become more competitive, stay in business, and provide jobs 2. Adopt the new philosophy Better to improve now than to react to problems later 3. Stop depending on inspection to achieve quality -- build in quality from the start 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-5 Deming’s 14 Points 5. Improve the system continuously to improve quality and thus constantly reduce costs 6. Institute training on the job Workers and managers must know the difference between common cause and special cause variation 7. Institute leadership 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. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-6 Deming’s 14 Points 10. Eliminate slogans and targets for the workforce They can create adversarial relationships 11. Eliminate quotas and management by numerical goals 12. Remove barriers to pride of workmanship 13. Institute a vigorous program of education and selfimprovement 14. Make the transformation everyone’s job Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-7 The Shewhart-Deming Cycle Plan Act The ShewhartDeming Cycle Study Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Do The key is a continuous cycle of improvement Chap 18-8 Six Sigma® Management A method for breaking a process into a series of steps: The goal is to reduce defects and produce near perfect results The Six Sigma® approach allows for a shift of as much as 1.5 standard deviations, so is essentially a ±4.5 standard deviation goal The mean of a normal distribution ±4.5 standard deviations includes all but 3.4 out of a million Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-9 The Six Sigma® DMAIC Model DMAIC represents Define -- define the problem to be solved; list costs, benefits, and impact to customer Measure – need consistent measurements for each Critical-to-Quality characteristic Analyze – find the root causes of defects Improve – use experiments to determine importance of each Critical-to-Quality variable Control – maintain gains that have been made Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-10 Theory of Control Charts A process is a repeatable series of steps leading to a specific goal Control Charts are used to monitor variation in a measured value from a process Inherent variation refers to process variation that exists naturally. This variation can be reduced but not eliminated Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-11 Theory of Control Charts Control charts indicate when changes in data are due to: Special or assignable causes Fluctuations not inherent to a process Data outside control limits or trend Represents problems to be corrected or improvements to incorporate into the process Chance or common causes Inherent random variations Consist of numerous small causes of random variability Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-12 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-13 Process Variation Total Process Common Cause Special Cause = + Variation Variation Variation Variation is often due to differences in: People Machines Materials Methods Measurement Environment Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-14 Process 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-15 Process 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-16 Control Limits Forming the Upper control limit (UCL) and the Lower control limit (LCL): UCL = Process Mean + 3 Standard Deviations LCL = Process Mean – 3 Standard Deviations UCL +3σ Process Average - 3σ LCL time Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-17 Control Chart Basics Special Cause Variation: Range of unexpected variability UCL Common Cause Variation: range of expected variability +3σ Process Mean - 3σ LCL time UCL = Process Mean + 3 Standard Deviations LCL = Process Mean – 3 Standard Deviations Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-18 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 Mean LCL time UCL = Process Mean + 3 Standard Deviations LCL = Process Mean – 3 Standard Deviations Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-19 Using Control Charts Control Charts are used to check for process control If the process is found to be out of control, steps should be taken to find and eliminate the special causes of variation Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-20 In-control Process A process is said to be in control when the control chart does not indicate any out-ofcontrol condition Contains only common causes of variation If the common causes of variation is small, then control chart can be used to monitor the process If the common causes of variation is too large, you need to alter the process Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-21 Process In Control Process in control: points are randomly distributed around the center line and all points are within the control limits UCL Process Mean LCL time Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-22 Process Not in Control Out of control conditions: One or more points outside control limits 8 or more points in a row on one side of the center line 8 or more points in a row moving in the same direction Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-23 Process Not in Control One or more points outside control limits Eight or more points in a row on one side of the center line UCL UCL Process Average Process Average LCL LCL Eight or more points in a row moving in the same direction UCL Process Average LCL Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-24 Out-of-control Processes When the control chart indicates an out-of- control 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-25 Control Chart for the Proportion: 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” Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-26 Control Chart for the Proportion: p Chart 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-27 Creating a p Chart Calculate subgroup proportions Graph subgroup proportions Compute mean proportion Compute the upper and lower control limits Add centerline and control limits to graph Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-28 Average of Subgroup Proportions The average of subgroup proportions = p If equal sample sizes: If unequal sample sizes: k k p pi i1 k p X i1 k n i 1 i i where: where: pi = sample proportion for subgroup i Xi = the number of nonconforming k = number of subgroups of size n items in sample i ni = total number of items sampled in k samples Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-29 Computing Control Limits The upper and lower control limits for a p chart are UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion – 3 Standard Deviations The standard deviation for the subgroup proportions is (p)(1 p) n Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. where: n = mean subgroup size Chap 18-30 Computing Control Limits The upper and lower control limits for the p chart are p(1 p) UCL p 3 n p(1 p) LCL p 3 n Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0 Chap 18-31 p Chart Example You are the manager of a 500-room hotel. You want to achieve the highest level of service. For seven days, you collect data on the readiness of 200 rooms. Is the process in control? Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-32 p Chart Example Day 1 2 3 4 5 6 7 # Rooms 200 200 200 200 200 200 200 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. # Not Ready 16 7 21 17 25 19 16 Proportion 0.080 0.035 0.105 0.085 0.125 0.095 0.080 Chap 18-33 p Chart Example k p X i1 k i n i1 16 7 16 121 .0864 200 200 200 1400 i k n n i i1 k 200 200 200 200 7 UCL p 3 p(1 p) .0864(1 .0864) .0864 3 .1460 200 n LCL p 3 p(1 p) .0864(1 .0864) .0864 3 .0268 200 n Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-34 p Chart Example P 0.15 UCL = .1460 _ p = .0864 0.10 0.05 0.00 LCL = .0268 1 2 3 4 5 6 7 Day _ Individual points are distributed around p without any pattern. Any improvement in the process must come from reduction of commoncause variation, which is the responsibility of management. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-35 Understanding Process Variability: Red Bead Experiment The experiment: From a box with 20% red beads and 80% white beads, have “workers” scoop out 50 beads Tell the workers their job is to get white beads Some workers will get better over time, some will get worse Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-36 Morals of the Red Bead Experiment 1. 2. 3. 4. 5. Variation is an inherent part of any process. The system is primarily responsible for worker performance. Only management can change the system. Some workers will always be above average, and some will be below. Setting unrealistic goals is detrimental to a firm’s wellbeing. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-37 R chart and X 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 For the process to be in control, both the R chart and the X-bar chart must be in control Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-38 Example: Subgroups Process measurements: Subgroup measures Subgroup number Individual measurements (subgroup size = 4) 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 Average subgroup mean = X range = R Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-39 The R Chart Monitors variability in a process The characteristic of interest is measured on a numerical scale Is a variables control chart Shows the sample range over time Range = difference between smallest and largest values in the subgroup Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-40 The R Chart Find the mean of the subgroup ranges (the center line of the R chart) 2. Compute the upper and lower control limits for the R chart 3. Use lines to show the center and control limits on the R chart 4. Plot the successive subgroup ranges as a line chart 1. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-41 Average of Subgroup Ranges Average of subgroup ranges: R R i k where: Ri = ith subgroup range k = number of subgroups Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-42 R Chart Control Limits The upper and lower control limits for an R chart are UCL D4 ( R ) LCL D3 ( R ) where: D4 and D3 are taken from the table (Appendix Table E.11) for subgroup size = n Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-43 R Chart Example You are the manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the variation in the process in control? Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-44 R Chart Example Day 1 2 3 4 5 6 7 Subgroup Size 5 5 5 5 5 5 5 Subgroup Average 5.32 6.59 4.89 5.70 4.07 7.34 6.79 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Subgroup Range 3.85 4.27 3.28 2.99 3.61 5.04 4.22 Chap 18-45 R Chart Example R R i k 3.85 4.27 ... 4.22 3.894 7 UCL D4 (R ) (2.114)(3.894) 8.232 LCL D3 (R ) (0)(3.894) 0 D4 and D3 are from Table E.11 (n = 5) Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-46 R Chart Control Chart Solution Minutes UCL = 8.232 8 6 4 2 0 _ R = 3.894 LCL = 0 1 2 3 4 Day 5 6 7 Conclusion: Variation is in control Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-47 The X Chart Shows the means of successive subgroups over time Monitors process average Must be preceded by examination of the R chart to make sure that the variation in the process is in control Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-48 The X Chart Compute the mean of the subgroup means (the center line of the X chart) Compute the upper and lower control limits for the X chart Graph the subgroup means Add the center line and control limits to the graph Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-49 Average of Subgroup Means Average of subgroup means: X X i k where: Xi = ith subgroup average k = number of subgroups Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-50 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 R Use d 2 to estimate the standard deviation of the process average, where d2 is from appendix Table E.11 n Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-51 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 so UCL X 3 LCL X 3 R d2 n R d2 n Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-52 Computing Control Limits Simplify the control limit calculations by using UCL X A 2 (R ) LCL X A 2 (R ) where A2 (from table E.11) = Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. 3 d2 n Chap 18-53 X Chart Example You are the manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For seven days, you collect data on five deliveries per day. Is the process average in control? Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-54 X Chart Example Day 1 2 3 4 5 6 7 Subgroup Size 5 5 5 5 5 5 5 Subgroup Average 5.32 6.59 4.89 5.70 4.07 7.34 6.79 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Subgroup Range 3.85 4.27 3.28 2.99 3.61 5.04 4.22 Chap 18-55 X Chart Control Limits Solution X X k R R k i i 5.32 6.59 6.79 5.814 7 3.85 4.27 4.22 3.894 7 UCL X A2 ( R ) 5.813 (0.577)(3.894) 8.061 LCL X A2 ( R ) 5.813 (0.577)(3.894) 3.567 A2 is from Table E.11 (n = 5) Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-56 X Chart Control Chart Solution Minutes 8 6 4 2 0 1 UCL = 8.061 _ _ X = 5.814 LCL = 3.567 2 3 4 Day 5 6 7 Conclusion: Process average is in statistical control Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-57 Process Capability Process capability is the ability of a process to consistently meet specified customer-driven requirements Specification limits are set by management in response to customers’ expectations The upper specification limit (USL) is the largest value that can be obtained and still conform to customers’ expectations The lower specification limit (LSL) is the smallest value that is still conforming Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-58 Estimating Process Capability Must first have an in-control process Estimate the percentage of product or service within specification Assume the population of X values is approximately normally distributed with mean estimated by X and standard deviation estimated by R / d2 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-59 Estimating Process Capability For a characteristic with a LSL and a USL P(outcome will be within specifications) USL X LSL X P(LSL X USL) P Z R R d2 d2 Where Z is a standardized normal random variable Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-60 Estimating Process Capability For a characteristic with only an USL P(outcome will be within specifications) USL X P( X USL) P Z R d2 Where Z is a standardized normal random variable Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-61 Estimating Process Capability For a characteristic with only a LSL P(outcome will be within specifications) LSL X P(LSL X) P Z R d2 Where Z is a standardized normal random variable Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-62 Process Capability Example You are the manager of a 500-room hotel. You have instituted a policy that 99% of all luggage deliveries must be completed within ten minutes or less. For seven days, you collect data on five deliveries per day. You know from prior analysis that the process is in control. Is the process capable? Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-63 Process Capability Example Day Subgroup Size Subgroup Average Subgroup Range 1 2 3 4 5 6 7 5 5 5 5 5 5 5 5.32 6.59 4.89 5.70 4.07 7.34 6.79 3.85 4.27 3.28 2.99 3.61 5.04 4.22 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-64 Process Capability Example n5 X 5.814 R 3.894 d 2 2.326 P(outcome will be within specifications) 10 5.814 P( X 10) P Z 3.894 2.326 P( Z 2.50) .9938 Therefore, we estimate that 99.38% of the luggage deliveries will be made within the ten minutes or less specification. The process is capable of meeting the 99% goal. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-65 Capability Indices A process capability index is an aggregate measure of a process’s ability to meet specification limits The larger the value, the more capable a process is of meeting requirements Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-66 Cp Index A measure of potential process performance is the Cp index USL LSL specification spread Cp process spread 6( R / d 2 ) Cp > 1 implies a process has the potential of having more than 99.73% of outcomes within specifications Cp > 2 implies a process has the potential of meeting the expectations set forth in six sigma management Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-67 CPL and CPU To measure capability in terms of actual process performance: X LSL CPL 3(R / d2 ) CPU USL X 3(R / d2 ) CPL (CPU) > 1 implies that the process mean is more than 3 standard deviation away from the lower (upper) specification limit Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-68 CPL and CPU Used for one-sided specification limits Use CPU when a characteristic only has a UCL Use CPL when a characteristic only has an LCL Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-69 Cpk Index The most commonly used capability index is the Cpk index Measures actual process performance for characteristics with two-sided specification limits Cpk = min(CPL, CPU) Cpk = 1 indicates that the process average is 3 standard deviation away from the closest specification limit Larger Cpk indicates greater capability of meeting the requirements, e.g., Cpk > 1.5 indicates compliance with six sigma management Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-70 Process Capability Example You are the manager of a 500-room hotel. You have instituted a policy that all luggage deliveries must be completed within ten minutes or less. For seven days, you collect data on five deliveries per day. You know from prior analysis that the process is in control. Compute an appropriate capability index for the delivery process. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-71 Process Capability Example n5 X 5.814 R 3.894 d 2 2.326 USL X 10 5.814 CPU 0.8335 3( R / d 2 ) 3(3.894 / 2.326) Since there is only the upper specification limit, we need to only compute CPU. The capability index for the luggage delivery process is .8337, which is less than 1. The upper specification limit is less than 3 standard deviation above the mean. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-72 Chapter Summary In this chapter, we have Reviewed the philosophy of quality management Deming’s 14 points Discussed Six Sigma® Management Reduce defects to no more than 3.4 per million Uses DMAIC model for process improvement Discussed the theory of control charts Common cause variation vs. special cause variation Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-73 Chapter Summary In this chapter, we have Constructed and interpreted p charts Constructed and interpreted X and R charts Obtained and interpreted process capability measures Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 18-74
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