Chap 18-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 18 Introduction to Quality Statistics for Business and Economics 6 th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-2 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 18-3 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 18-4 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 18-5 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 18-6 Process 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 Total Process Variation Common Cause Variation Assignable Cause Variation =+

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-7 Total Process Variation Common Cause Variation Assignable Cause Variation =+ People Machines Materials Methods Measurement Environment Variation is often due to differences in:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-8 Common Cause Variation Common cause variation naturally occurring and expected the result of normal variation in materials, tools, machines, operators, and the environment Total Process Variation Common Cause Variation Assignable Cause Variation =+

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-9 Special Cause Variation Special cause variation abnormal or unexpected variation has an assignable cause variation beyond what is considered inherent to the process Total Process Variation Common Cause Variation Assignable Cause Variation =+

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 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 Overview Process Capability Tools for Quality Improvement Control Charts 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 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 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 x i 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Preliminaries The sample standard deviations (denoted s i for i = 1, 2,...,K) can be graphed on an s-chart The average sample standard deviation is 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 (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Example: Subgroups Sample measurements: Subgroup measures Subgroup number Individual measurements (subgroup size = 4) Mean, x Std. Dev., s 123…123… … … … … … … Average subgroup mean = Average subgroup std. dev. = s

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Estimate of Process Standard Deviation Based on s An estimate of process standard deviation is Where s is the average sample standard deviation c 4 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 Factors for Control Charts nc4c4 A3A3 B3B3 B4B Selected control chart factors (Table 18.1)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Process Average Control Charts and Control Limits UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations UCL LCL +3σ - 3σ- 3σ time 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)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Control Charts and Control Limits The 3-standard-deviation control limits are estimated for an X-chart as follows: (continued) Where the value of is given in Table 18.1 or in Appendix 13

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap X-Chart The X-chart is a time plot of the sequence of sample means The center line is The lower control limit is The upper control limit is

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 X-Chart Example: Subgroup Data DaySubgroup Size Subgroup Mean Subgroup Std. Dev These are the x i values for the 7 subgroups These are the s i values for the 7 subgroups

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap X-Chart Control Limits Solution A 3 = 1.43 is from Appendix 13

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap X-Chart Control Chart Solution UCL = LCL = Minutes Day x = _ _ Conclusion: Process mean is in statistical control

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap s-Chart The s-chart is a time plot of the sequence of sample standard deviations The center line on the s-chart is The lower control limit (for three-standard error limits) is The upper control limit is Where the control chart constants B 3 and B 4 are found in Table 18.1 or Appendix 13

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap s-Chart Control Limits Solution B 4 and B 3 are found in Appendix 13

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap s-Chart Control Chart Solution UCL = Minutes Day LCL = 0 s = _ Conclusion: Variation is in control

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Process Average Control Chart Basics UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations UCL LCL +3σ - 3σ- 3σ Common Cause Variation: range of expected variability Special Cause Variation: Range of unexpected variability time

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Process Average Process Variability UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations UCL LCL ±3σ 99.7% of process values should be in this range time Special Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Using Control Charts Control Charts are used to check for process control H 0 : The process is in control i.e., variation is only due to common causes H 1 : 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 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 Process In Control Process in control: points are randomly distributed around the center line and all points are within the control limits UCL LCL time Process Average

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 Process Not in Control One or more points outside control limits UCL LCL Nine or more points in a row on one side of the center line UCL LCL Six or more points moving in the same direction UCL LCL Process Average

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 Capability Indices A process capability index is an aggregate measure of a processs 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 Measures of Process Capability Process capability is judged by the extent to which lies between the tolerance limits L and U C p Capability Index Appropriate when the sample data are centered between the tolerance limits, i.e. The index is 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 Measures of Process Capability C pk 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 C pk index is A satisfactory value is at least 1.33 (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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? Process Capability Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Process Capability: Hotel Data DaySubgroup Size Subgroup Mean Subgroup Std. Dev

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Process Capability: Hotel Example Solution 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 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 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 Should have large sample size so that the average number of nonconforming items per sample is at least five or six (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 p-Chart Example Subgroup number, i Sample size Number of successes Sample Proportion, p i 123…123… … … Average sample proportions = p

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Average of Sample Proportions The average of sample proportions = p where: p i = sample proportion for subgroup i K = number of subgroups of size n If equal sample sizes:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Computing Control Limits The upper and lower control limits for a p-chart are The standard deviation for the subgroup proportions is UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion – 3 Standard Deviations

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Computing Control Limits The upper and lower control limits for the p-chart are (continued) Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 p Chart Example: Hotel Data # Not Day# RoomsReady Proportion

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap p Chart Control Limits Solution

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap p =.0864 p Chart Control Chart Solution UCL =.1460 LCL = P Day 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 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 Mean and Standard Deviation for a c-Chart The sample mean number of occurrences is The standard deviation for a c-chart is where: c i = number of successes per item K = number of items sampled

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap c-Chart Center and Control Limits The control limits for a c-chart are The center line for a c-chart is The number of occurrences can never be negative, so if the calculated lower control limit is negative, set LCL = 0

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 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 Flaws found

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Constructing the c-Chart The mean and standard deviation are: The control limits are: Note: LCL < 0 so set LCL = 0

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The completed c-Chart 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 UCL = LCL = 0 Sample number c =

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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