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Statistics for Business and Economics 6th Edition Chapter 18 Introduction to Quality Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-1 Chapter Goals After completing this chapter, you should be able to: Describe the importance of statistical quality control for process improvement Define common and assignable causes of variation Explain process variability and the theory of control charts Construct and interpret control charts for the mean and standard deviation Obtain and explain measures of process capability Construct and interpret control charts for number of occurrences Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-2 The Importance of Quality Primary focus is on process improvement Data is needed to monitor the process and to insure the process is stable with minimum variance Most variation in a process is due to the system, not the individual Focus on prevention of errors, not detection Identify and correct sources of variation Higher quality costs less Increased productivity increased sales higher profit Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-3 Variation A system is a number of components that are logically or physically linked to accomplish some purpose A process is a set of activities operating on a system to transform inputs to outputs From input to output, managers use statistical tools to monitor and improve the process Goal is to reduce process variation Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-4 Sources of Variation Common causes of variation also called random or uncontrollable causes of variation causes that are random in occurrence and are inherent in all processes management, not the workers, are responsible for these causes Assignable causes of variation also called special causes of variation the result of external sources outside the system these causes can and must be detected, and corrective action must be taken to remove them from the process failing to do so will increase variation and lower quality Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-5 Process Variation Total Process Common Assignable = Cause Variation + Cause 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-6 Total Process Variation Total Process Common Assignable = Cause Variation + Cause Variation Variation Variation is often due to differences in: People Machines Materials Methods Measurement Environment Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-7 Common Cause Variation Total Process Common Assignable = Cause Variation + Cause Variation Variation Common cause variation naturally occurring and expected the result of normal variation in materials, tools, machines, operators, and the environment Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-8 Special Cause Variation Total Process Common Assignable = Cause Variation + Cause Variation Variation Special cause variation abnormal or unexpected variation has an assignable cause variation beyond what is considered inherent to the process Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-9 Stable Process A process is stable (in-control) if all assignable causes are removed variation results only from common causes Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-10 Control Charts The behavior of a process can be monitored over time Sampling and statistical analysis are used Control charts are used to monitor variation in a measured value from a process Control charts indicate when changes in data are due to assignable or common causes Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-11 Overview Tools for Quality Improvement Control Charts Process Capability X-chart for the mean s-chart for the standard deviation P-chart for proportions c-chart for number of occurrences Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-12 X-chart and s-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 s-chart and the X-chart must be in control Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-13 Preliminaries Consider K samples of n observations each Data is collected over time from a measurable characteristic of the output of a production process The sample means (denoted xi for i = 1, 2, . . ., K) can be graphed on an X-chart The average of these sample means is the overall mean of the sample observations K x x i /K i1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-14 Preliminaries (continued) The sample standard deviations (denoted si for i = 1, 2, . . . ,K) can be graphed on an s-chart The average sample standard deviation is K s si /K i1 The process standard deviation, σ, is the standard deviation of the population from which the samples were drawn, and it must be estimated from sample data Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-15 Example: Subgroups Sample measurements: Subgroup Individual measurements number (subgroup size = 4) Subgroup measures Mean, x Std. Dev., s 1 15 17 15 11 14.5 2.517 2 12 16 9 15 13.0 3.162 3 17 21 18 20 19.0 1.826 … … … … … … … Average subgroup mean = x Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Average subgroup std. dev. = s Chap 18-16 Estimate of Process Standard Deviation Based on s An estimate of process standard deviation is σˆ s/c 4 Where s is the average sample standard deviation c4 is a control chart factor which depends on the sample size, n Control chart factors are found in Table 18.1 or in Appendix 13 If the population distribution is normal, this estimator is unbiased Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-17 Factors for Control Charts Selected control chart factors (Table 18.1) n c4 A3 B3 B4 2 .789 2.66 0 3.27 3 .886 1.95 0 2.57 4 .921 1.63 0 2.27 5 .940 1.43 0 2.09 6 .952 1.29 0.03 1.97 7 .959 1.18 0.12 1.88 8 .965 1.10 0.18 1.82 9 .969 1.03 0.24 1.76 10 .973 0.98 0.28 1.72 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-18 Control Charts and Control Limits A control chart is a time plot of the sequence of sample outcomes Included is a center line, an upper control limit (UCL) and a lower control limit (LCL) UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations UCL +3σ Process Average - 3σ LCL time Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-19 Control Charts and Control Limits (continued) The 3-standard-deviation control limits are estimated for an X-chart as follows: Process Average 3 Standard Deviations x 3 σˆ / n x 3 s /(c 4 n ) x A3s Where the value of A 3 3 is given in Table 18.1 or in Appendix 13 c4 n Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-20 X-Chart The X-chart is a time plot of the sequence of sample means The center line is CL X x The lower control limit is LCL X x A 3 s The upper control limit is UCL X x A 3 s Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-21 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 mean in control? Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-22 X-Chart Example: Subgroup Data Day 1 2 3 4 5 6 7 Subgroup Subgroup Subgroup Size Mean Std. Dev. 5 5 5 5 5 5 5 These are the xi values for the 7 subgroups Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 5.32 6.59 4.89 5.70 4.07 7.34 6.79 1.85 2.27 1.28 1.99 2.61 2.84 2.22 These are the si values for the 7 subgroups Chap 18-23 X-Chart Control Limits Solution x x i K s s 5.32 6.59 6.79 5.813 7 1.85 2.27 2.22 2.151 K 7 i UCL X x A 3 ( s ) 5.813 (1.43)(2.1 51) 8.889 LCL X x A 3 ( s ) 5.813 (1.43)(2.1 51) 2.737 A3 = 1.43 is from Appendix 13 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-24 X-Chart Control Chart Solution Minutes 8 6 4 2 0 1 UCL = 8.889 _ x = 5.813 LCL = 2.737 2 3 4 Day 5 6 7 Conclusion: Process mean is in statistical control Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-25 s-Chart The s-chart is a time plot of the sequence of sample standard deviations The center line on the s-chart is CL s The lower control limit (for three-standard error limits) is LCLs B3 s The upper control limit is UCL s B4 s Where the control chart constants B3 and B4 are found in Table 18.1 or Appendix 13 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-26 s-Chart Control Limits Solution x x i K s s 5.32 6.59 6.79 5.813 7 1.85 2.27 2.22 2.151 K 7 i UCL s B 4 s (2.09)(2.1 51) 4.496 LCLs B3 s (0)(2.151) 0 B4 and B3 are found in Appendix 13 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-27 s-Chart Control Chart Solution Minutes UCL = 4.496 4 2 _ s = 2.151 0 LCL = 0 1 2 3 4 Day 5 6 7 Conclusion: Variation is in control Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-28 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-29 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-30 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 H1: The process is out of control i.e., assignable cause variation exists If the process is found to be out of control, steps should be taken to find and eliminate the assignable causes of variation Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-31 In-control Process A process is said to be in control when the control chart does not indicate any out-of-control 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 variation due to common causes is too large, you need to alter the process Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-32 Process In Control Process in control: points are randomly distributed around the center line and all points are within the control limits UCL Process Average LCL time Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-33 Process Not in Control Out of control conditions: One or more points outside control limits 6 or more points in a row moving in the same direction either increasing or decreasing 9 or more points in a row on the same side of the center line Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-34 Process Not in Control One or more points outside control limits Nine or more points in a row on one side of the center line UCL UCL Process Average Process Average LCL LCL Six or more points moving in the same direction UCL Process Average LCL Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-35 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-36 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 or process needs, for example) The upper tolerance limit (U) is the largest value that can be obtained and still conform to customers’ expectations The lower tolerance limit (L) is the smallest value that is still conforming Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-37 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-38 Measures of Process Capability Process capability is judged by the extent to which x 3σˆ lies between the tolerance limits L and U Cp Capability Index Appropriate when the sample data are centered between the tolerance limits, i.e. x (L U)/2 The index is UL Cp 6σˆ A satisfactory value of this index is usually taken to be one that is at least 1.33 (i.e., the natural rate of tolerance of the process should be no more than 75% of (U – L), the width of the range of acceptable values) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-39 Measures of Process Capability (continued) Cpk Index Used when the sample data are not centered between the tolerance limits Allows for the fact that the process is operating closer to one tolerance limit than the other The Cpk index is U x x L Cpk Min , ˆ ˆ 3σ 3σ A satisfactory value is at least 1.33 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-40 Process Capability Example You are the manager of a 500-room hotel. You have instituted tolerance limits that luggage deliveries should be completed within ten minutes or less (U = 10, L = 0). 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-41 Process Capability: Hotel Data Day Subgroup Size 1 2 3 4 5 6 7 5 5 5 5 5 5 5 Subgroup Mean 5.32 6.59 4.89 5.70 4.07 7.34 6.79 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Subgroup Std. Dev. 1.85 2.27 1.28 1.99 2.61 2.84 2.22 Chap 18-42 Process Capability: Hotel Example Solution n5 X 5.813 s 2.151 c 4 0.940 s 2.151 Estimate σˆ 2.288 c 4 0.940 U x x L 10 5.813 5.813 0 Cpk Min , Min , ˆ ˆ 3 σ 3 σ 3(2.228) 3(2.228) Min0.610 , 0.847 0.610 The capability index for the luggage delivery process is less than 1. The upper specification limit is less than 3 standard deviations above the mean. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-43 p-Chart Control chart for proportions Is an attribute chart Shows proportion of defective or 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-44 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 large sample size so that the average number of nonconforming items per sample is at least five or six Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-45 Creating a p-Chart Calculate subgroup proportions Graph subgroup proportions Compute average of subgroup proportions Compute the upper and lower control limits Add centerline and control limits to graph Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-46 p-Chart Example Sample Subgroup number, i Sample size Number of successes Proportion, pi 1 150 15 .1000 2 150 12 .0800 3 150 17 .1133 … … … Average sample proportions = p Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-47 Average of Sample Proportions The average of sample proportions = p If equal sample sizes: K p p i1 i K where: pi = sample proportion for subgroup i K = number of subgroups of size n Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-48 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 ) σˆ p n Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-49 Computing Control Limits (continued) The upper and lower control limits for the p-chart are p(1 p) LCLp p 3 n p(1 p) UCLp p 3 n Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0 Chap 18-50 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-51 p Chart Example: Hotel Data Day 1 2 3 4 5 6 7 # Rooms 200 200 200 200 200 200 200 Statistics for Business and Economics, 6e © 2007 Pearson Education, 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 18-52 p Chart Control Limits Solution K p p i1 K i .080 .035 .080 .0864 7 LCLp p 3 p(1 p) .0864(1 .0864) .0864 3 .0268 n 200 UCLp p 3 p(1 p) .0864(1 .0864) .0864 3 .1460 n 200 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-53 p Chart Control Chart Solution P 0.15 UCL = .1460 _ p = .0864 0.10 0.05 0.00 LCL = .0268 1 2 3 4 5 Day 6 7 _ Individual points are distributed around p without any pattern. Any improvement in the process must come from reduction of common-cause variation, which is the responsibility of management. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-54 c-Chart Control chart for number of defects per item Also a type of attribute chart Shows total number of nonconforming items per unit 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-55 Mean and Standard Deviation for a c-Chart The sample mean number of occurrences is c c i K The standard deviation for a c-chart is σˆ c c where: ci = number of successes per item K = number of items sampled Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-56 c-Chart Center and Control Limits The center line for a c-chart is CLc c The control limits for a c-chart are LCLc c 3 c UCL c c 3 c Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. The number of occurrences can never be negative, so if the calculated lower control limit is negative, set LCL = 0 Chap 18-57 Process Control Determine process control for p-chars and c-charts using the same rules as for X and s-charts Out of control conditions: One or more points outside control limits Six or more points moving in the same direction Nine or more points in a row on one side of the center line Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-58 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-59 Constructing the c-Chart The mean and standard deviation are: c c K i 2 1 3 0 5 1 0 1.7143 7 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-60 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-61 Chapter Summary Reviewed the concept of statistical quality control Discussed the theory of control charts Common cause variation vs. special cause variation Constructed and interpreted X and s-charts Obtained and interpreted process capability measures Constructed and interpreted p-charts Constructed and interpreted c-charts Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-62