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					                                           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                                                      Tools for Quality
                   Quality                                                          Improvement

                      Deming’s 14                                              Control           Process
                        Points                                                 Charts           Capability
                                                                               p chart
                      Six       Sigma®
                      Management                                               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 self-
           improvement
        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

                                                          The
                                                        Shewhart-
                   Act                                   Deming                   Do
                                                          Cycle
                                                                               The key is a
                                                                               continuous cycle
                                                             Study             of improvement

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.               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                                                +3σ
        Variation: range of                                                    Process Mean
        expected variability                                        - 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
                                                                               Process Mean
        be in this range

                                                                                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-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 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                                                Eight or more points in a row on one
  control limits                                                             side of the center line
                                                UCL                                                        UCL
                                                Process                                                    Process
                                                Average                                                    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

                             pi                                                          X     i
                                                                                    p     i1
              p             i1                                                            k

                                  k                                                       n
                                                                                           i1
                                                                                                 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)
                                                                               where:
                                                                  n             n = mean subgroup size


Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.                    Chap 18-30
                       Computing Control Limits

            The upper and lower control limits for the p
                 chart are

                                                     p(1  p)
                                         UCL  p  3
                                                        n
                                                                               Proportions are never
                                                     p(1  p)                  negative, so if the
                                         LCL  p  3                           calculated lower
                                                        n                      control limit is
                                                                               negative, set LCL = 0


Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.                  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
                                                                               # Not
                    Day                   # Rooms                              Ready   Proportion
                     1                      200                                  16      0.080
                     2                      200                                   7      0.035
                     3                      200                                  21      0.105
                     4                      200                                  17      0.085
                     5                      200                                  25      0.125
                     6                      200                                  19      0.095
                     7                      200                                  16      0.080

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.                        Chap 18-33
                       p Chart Example
                                  k

                                X      i
                                                    16  7    16    121
                        p       i1
                                                                          .0864
                                  k
                                                  200  200    200 1400
                                n
                                 i1
                                        i

                                              k

                                             n       i
                                                            200  200    200
                                      n     i1
                                                                                200
                                                  k                  7


                                            p(1  p)             .0864(1  .0864)
              UCL  p  3                             .0864  3                   .1460
                                               n                       200

                                            p(1  p)             .0864(1  .0864)
              LCL  p  3                             .0864  3                   .0268
                                               n                       200
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.            Chap 18-34
                       p Chart Example
                   P
                 0.15                                                                          UCL = .1460
                                                                                               _
                 0.10                                                                           p = .0864
                 0.05
                                                                                               LCL = .0268
                 0.00
                             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 common-
  cause 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.       Variation is an inherent part of any process.
       2.       The system is primarily responsible for worker
                performance.
       3.       Only management can change the system.
       4.       Some workers will always be above average, and some will
                be below.
       5.       Setting unrealistic goals is detrimental to a firm’s well-
                being.



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                Individual measurements
    number                                                                       Mean, X     Range, R
                              (subgroup size = 4)
          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      Average
                                                                               subgroup     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

                       1. 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

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  D 4 ( 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                 Subgroup                  Subgroup   Subgroup
                                                 Size                     Average    Range
                              1                   5                        5.32       3.85
                              2                   5                        6.59       4.27
                              3                   5                        4.89       3.28
                              4                   5                        5.70       2.99
                              5                   5                        4.07       3.61
                              6                   5                        7.34       5.04
                              7                   5                        6.79       4.22

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.                   Chap 18-45
                       R Chart Example

                  R
                     R                  i
                                             
                                               3.85  4.27  ...  4.22
                                                                         3.894
                                  k                      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
                       8                                                                   UCL = 8.232
                       6                                                                   _
                       4                                                                   R = 3.894
                       2
                       0                                                                   LCL = 0
                             1             2             3         4           5   6   7
                                                                  Day

                                  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
                                   n
             average, where d2 is from appendix Table E.11



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                                                        R
                            UCL  X  3
                                                              d2 n
                                                                  R
                            LCL  X  3
                                                             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 )
                                                                                3
                                      where A2 (from table E.11) =
                                                                               d2 n



Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.          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                 Subgroup                  Subgroup   Subgroup
                                                 Size                     Average    Range
                              1                   5                        5.32       3.85
                              2                   5                        6.59       4.27
                              3                   5                        4.89       3.28
                              4                   5                        5.70       2.99
                              5                   5                        4.07       3.61
                              6                   5                        7.34       5.04
                              7                   5                        6.79       4.22

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.                   Chap 18-55
                       X Chart
                       Control Limits Solution
                    X
                       X                   i
                                                
                                                  5.32  6.59    6.79
                                                                          5.814
                                    k                       7

                 R
                    R                  i
                                            
                                              3.85  4.27    4.22
                                                                      3.894
                                k                       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                                                                              UCL = 8.061
                                                                                           _
                                                                                           _
            6                                                                              X = 5.814
            4
                                                                                           LCL = 3.567
            2
            0
                       1             2            3            4               5   6   7
                                                               Day

              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)
                                                     
                                                     
                                 LSL  X     USL  X 
           P(LSL  X  USL)  P         Z         
                                    R           R
                                
                                 d                   
                                                      
                                    2          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
                                         
                                          d          
                                                      
                                             2       

                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               Subgroup   Subgroup
                                                          Size                  Average    Range
                                   1                            5                5.32       3.85
                                   2                            5                6.59       4.27
                                   3                            5                4.89       3.28
                                   4                            5                5.70       2.99
                                   5                            5                4.07       3.61
                                   6                            5                7.34       5.04
                                   7                            5                6.79       4.22

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.                       Chap 18-64
                       Process Capability
                       Example
                   n5                   X  5.814                       R  3.894   d 2  2.326
                                   P(outcome will be within specificat ions)
                                                                    
                                                         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                
                                      6( R / d 2 )   process spread

                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 )

                                                                 USL  X
                                                 CPU 
                                                                 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
                   n5                   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

				
DOCUMENT INFO
Description: Materi ini merupakan bahan ajar sebagai pelengkap e-materi mata kuliah statistika bisnis. Levine, D. M., Stephan, D. F., Krehbiel, T. C. & Berenson, M. L. (2008). Statistics for Managers Using Microsoft Excel. Pearson.