Six Sigma Math: A Primer

Six Sigma Math: A Primer

Richard E. Biehl, CSSBB, CSQE Data-Oriented Quality Solutions

Prepared for

American Society for Quality, Section 1105 Knoxville, Tennessee, USA.

Copyright © 2005, Data-Oriented Quality Solutions. All rights reserved.

Six Sigma Math: A Primer ASQ Section 1105

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Six Sigma Math: A Primer

Introduction As a quality movement, Six Sigma is about process capability. It is about reducing the variation in a process, and increasing our control over a process, so that we can predict with considerable accuracy exactly how the process will behave. This level of capability can be used to implement improvements in the process where we set targets for future behaviors, and achieve those targets within the levels of quality control that we choose to design into the improvements.

This Primer provides a very simple introduction to the core math concepts of Six Sigma. It is intended for novice readers that have not yet had any Six Sigma training or experience and want a basic background into the core concepts of Six Sigma centering on variation and control. Individuals with Six Sigma project experience will find this material too superficial, and may even spend more time noting the obvious omissions than concentrating on the materials.

The materials in this Primer are organized into two sections that can be read independently:

1. Conceptual Pass – A discussion of how variation in defects and defectives can be used to understand basic process and product quality, and how the history of the quality movement follows a path of decreasing variation leading up to Six Sigma.

2. Application Pass – A discussion, more technical than the first pass, of how the basic concepts of Statistical Process Control can be used to model the impact that Six Sigma thinking can have on the design of a process, product, or system.

Both sections cover the same concepts, with the second being a little more detailed. Individuals desiring more information beyond this Primer should consult the Bibliography.


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Conceptual Pass

Defects & Defectives

An understanding of Six Sigma Math requires a clarification of a few specific terms that are used by quality professionals, the meanings of which are somewhat narrower than might be expected by a novice:

• Defective – Any product or service instance that fails to meet the requirements of the customer, whether or not those requirements were clearly specified.

• Defect – Any characteristic of a product or service instance that doesn’t conform to its specification, whether such a specification was explicit or implied.

Typically, defective products or services are caused by one or more defects and are referred to as defectives. Not all defects cause defectives. Defects in a product or service that don’t result in the product or service being considered defective are often referred to as latent defects.

The distinction between defect and defective is very important in Six Sigma. All processes exhibit some variability in their outcomes. Inevitably, some portion of the output of a process will contain defects. If the variability of a process can be reduced enough, these inevitable defects will be so close to specification that the customer is unlikely to consider the resulting products or services defective.

Process Variability

Process variability can be seen by selecting one or more important measurable characteristics from a process and then monitoring those characteristics over time. The tool most likely to be used for such monitoring is the Statistical Process Control Chart, or SPC (Appendix A). The basics of SPC can be used to review the history of thinking about defects leading up to Six Sigma.


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1.5 Sigma

The early quality movement began in the face of very high rates of defectives. A process with a very high defect rate is illustrated in Figure 1. The illustrated process has a quality level of 1.5σ, meaning that the characteristic represented by the chart only falls within the customer’s specification limits about 50% of the time.

Figure 1 – Wide variability results in extremely high defect rates

Because the chart’s control limits would be set at 3σ above and below the target, they do not even appear on the scale of Figure 1. Organizations with defect rates this high were unlikely to be using SPC effectively, but the figure illustrates the problems faced by the early quality movement.


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3 Sigma (TQM)

With the advent of quality programs in the 1980s, including Total Quality Management (TQM), the use of SPC to monitor and control processes became more common. TQM programs worked to reduce process variation so that most of the expected variability would fall with the customer’s requirements as represented by the specification limits. Figure 2 illustrates a process that has been brought up to a quality level of 3σ, meaning that the defect rate associated with the charted characteristics could be expected to consistently fall below 10%.

Figure 2 - TQM often aligned the specification limits with the 3 Sigma control limits

TQM thinking aligned defects and defectives. Defectives included any observation outside of the specification limits. Defects included any observation outside of the more restrictive of the control or specification limits. Because many TQM change initiatives designed processes that would achieve customer tolerance, the target control limits often ended up being roughly the same as the customer’s specification limits. The relatively few defect observations that would fall outside of the control limits would be defectives by definition, but the frequency of such observations was dramatically improved compared to pre-TQM programs.


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3 Sigma with 1.5 Sigma shift

One problem with TQM-based alignment of defectives with defects was that processes didn’t remain stable after they were improved. Processes tended to increase their variability over time as a result of everyday deterioration of the conditions under which those processes operated. Such deterioration might include a loss of calibration of sensors, wear and tear on machinery, operator fatigue, supplier quality variation, etc. Process characteristics were seen to wander from their original values by as much as 1.5σ in either direction. This wandering came to be known as the 1.5σ shift, and is illustrated in Figure 3.

Figure 3 - A 1.5 Sigma shift occurs in either direction from the target

This shift resulted in more defects occurring as the process wandered. Under a 1.5σ shift, a process characteristic that had achieved 3σ performance would slip back to 1.5σ performance. Without a proper recognition of the causes of the shift, or corrective actions taken to avoid the shift, it was often perceived that the investment in improving the process had been wasted. TQM programs often fell into disfavor as a result of this fundamental misunderstanding of process behavior.


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4.5 Sigma

For organizations that recognized the problem of shift, the answer became to reduce process variability further in order to allow for the naturally occurring shift. To achieve the benefits of 3σ quality originally achieved by TQM programs, process variability had to be reduced an additional 1.5σ. Figure 4 illustrates such a process characteristic at a 4.5σ quality level.

Figure 4 – New quality levels were achieved by 4.5 Sigma processes

Such low process variability was extremely difficult to achieve, but seemed to provide extremely low defect levels, usually measured in fractions of a percent. In the competitive environment of the late 1980s, such improvements were becoming necessary to survive.


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4.5 Sigma with 1.5 Sigma shift

The reality of the 1.5σ shift gave rise to the concepts of short-term sigma, and long-term sigma. Short-term sigma was the quality level designed into a process and typically achieved on process launch. Long-term sigma was the expected quality level of the process over the time periods in which the 1.5σ shift could be expected to materialize. Figure 5 illustrates the long-term shift of a 4.5σ process back toward 3σ performance.

Figure 5 - 4.5 Sigma processes achieve 3 Sigma quality in the long-term

By taking into account the long-term shift associated with all processes, quality improvement programs could provide the 3σ levels of quality originally targeted by the earlier TQM programs. However, along with this realization came increasing pressure to achieve even higher levels of quality in the late 1980s.


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Six Sigma

The Six Sigma movement pushed these concepts to the extreme of targeting 6σ quality levels in the short-term that would achieve 4.5σ quality levels in the long-term. Defect rates at this quality level fall at about 3.4 defects per million opportunities. The variability of a 6σ process relative to its specification limits is illustrated in Figure 6.

Figure 6 - A short-term 6 Sigma process

The quality movement toward Six Sigma effectively decoupled the definition of defects from defectives. At Six Sigma, defectives are still any observation outside of the customer’s specification limits, although such observations become exceedingly rare. Defects remain however, as roughly 3-7% of all observations will continue to fall outside of the control limits. With the control limits at 3σ, and the specification limits at 6σ, the vast majority of defects do not rise to a level close to resulting in a customer defective. Each defect, though, remains an opportunity to continue to improve the process and bring the expected process shift under control. By managing these defects effectively, the process continuously improves without ever producing a defective for the customer. That distinction – the 3σ to 6σ gap – is the essence and opportunity of Six Sigma Math.


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Application Pass

Brute Force Quality

Historically, quality improvement was carried out as a management-dictated process of applying brute force effort to particular quality problems. For example, management might set a goal of reducing backorders in an order processing environment by 50%, from a current state of 24% to some target of 12% or less. This goal would drive an effort to attack the problem, making changes throughout the problem area and observing the impact of those changes on the targeted measures. After a time, the backorder rate would be seen to have been lowered to some value at or below the targeted 12%, and the improvement program would be declared a success based on that result. (Figure 7) However, the actual process behavior would still vary considerably.

Figure 7 - Wide variability of brute-force outcomes

Problems with the brute force approach were numerous, but centered on the fact that such efforts often focused on incorrect or inappropriate solutions, and the solutions themselves were not usually sustainable. Recognition of these weaknesses caused a shift toward more systematic approaches to quality improvement. Collectively, these approaches came to be known as Total Quality Management (TQM).


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Total Quality Management

TQM involved an expanded use of Statistical Process Control (SPC). The effects of SPC could be seen in two key areas: 1) processes were expected to exhibit variation around an average value, but the variation attributable to the process was expected to remain within certain expected ranges (the control limits), and 2) what a customer wanted from a process (the specification limits) wasn’t necessarily the same thing as what a process would actually be observed to do. When a process was operating outside of its specification limits, it was said to be producing defectives. When a process was operating outside of its control limits, it was said to be out-of-control.

An out-of-control process is a signal that something is wrong with the underlying process, and that it should be addressed using the methods and tools of TQM. In this way, the SPC analysis tells us both where the problems are (producing defectives outside the specification limits) and whether or not we could cost-effectively fix them (out-of-control process behaviors indicating special causes that can be identified and corrected).

Figure 8 - TQM controlled processes to operate within specifications


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Figure 8 illustrates the backorder problem using basic SPC thinking. The original target becomes the upper specification limit (USL) of the desired new process. The objective of the design will be to build a process that doesn’t result in a backorder rate higher than this value, making the design target upper control limit (UCL) also 12%. Presumably the backorder rate should be reduced as much as possible (the lower-the-better, or LTB), and so the lower specification limit (LSL) and lower control limit (LCL) are both set to 0%. The target value for the process redesign is typically the mid-point between the two specification limits, or 6%. The new process is intended to deliver a backorder rate of 6%, with little enough fluctuation that any variation within three standard deviations (or 3σ) from the mean will still be within the 12% USL. The resulting process will exhibit a 3σ quality level.

Six Sigma

What makes the newer Six Sigma movement different from TQM is its emphasis on raising the bar on quality. The processes designed in TQM initiatives became very sensitive to 3σ control exceptions in SPC, with on-going improvement occurring incrementally at these margins. The Six Sigma movement uses all of the tools and techniques of these TQM initiatives, and adds an emphasis on long-term process variability and shift. Processes that were in-control in the short-term (typically operating within 3σ of their mean), would typically appear out-of-control in the long-term as greater variability was seen in human factors and error, equipment wear-and-tear, and gradual deterioration of process conditions.

With this increased variability included, TQM programs failed to deliver adequate quality, even at short-term 3σ levels. The short-term desired defect rate of less than 1% for 3σ processes could be seen to rise above 5% as a result of the long-term shifting of the process. With expectations expanded to 6σ quality, new processes could be defined that provided acceptable levels of quality while taking into account the implications of long-term process shift.


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SPC is still used to monitor and evaluate process performance at 3σ levels. However, the identified exceptions beyond the 3σ control limits are now occurring well within the 6σ specification limits (Appendix B). In TQM, process defects and customer defectives were typically both defined at 3σ, and so process improvement to bring the process back into control was required while also dealing with customer defectives outside the specifications. Six Sigma separates the discussion of process defects (outside 3σ) from the recognition of customer defectives (outside 6σ) to allow processes and systems to self-correct and adjust to results in the 3σ to 6σ range.

Figure 9 - Six Sigma brings even exceptions within specification

Figure 9 illustrates the 6σ viewpoint using the backorder rate example. The specification limits do not change because they represent what the customer wants, which doesn’t change based on how quality is being measured; but, the control limits do change. Design target SPC control limits are still 3σ above and below the target, although the specification limits are now 12σ apart in this new Six Sigma view. This means that the revised UCL is now 9%, or the mid-point between the target value of 6% and the USL of 12%. There is now an improvement zone available between the UCL and the USL. Values above the control limit are process defects that SPC tells us can be economically corrected. If they can be corrected before they rise to the USL, the customer need never see a defective.


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Implications for Practitioners

As processes are redesigned to align with Six Sigma thinking, process engineers have an opportunity to implement controls that take advantage of the improvement zone between 3σ and 6σ process performance. By building critical customer metrics directly into processes they can be made self-correcting by enabling specific actions to be taken when process defects are seen in the improvement zone. These actions need not always involve sophisticated technical solutions to be beneficial. Controls can be as simple as an e-mail notifying support personal of defects above the 3σ level, or a periodic report highlighting activity in the 3σ to 6σ zone. The point isn’t to build processes without defects, but to build business solutions that can be kept from producing defectives in spite of their defects. That is the essence and opportunity of Six Sigma Math.


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Appendix A: SPC Conceptual Highlights

• Statistical Quality Control (SQC)- The application of statistical techniques for measuring and improving the quality of processes. SQC includes SPC, diagnostic tools, sampling plans, and other statistical techniques.

• Statistical Process Control (SPC)- The application of statistical techniques for measuring and analyzing the variation in processes.

• For any particular quality characteristic, variation can be quantified by sampling the output of a process and estimating the parameters of its statistical distribution. Changes in the distribution can be revealed by plotting these parameters over time.

• The variation described by the statistical distribution of a quality characteristic has many separate causes. The Pareto principle illustrates that a few causes will have a major effect upon the total variation. A few more will have a somewhat lesser effect. Most will have a very small effect.

• Shewhart at Bell Labs identified two major components to variation: a steady component that appeared to be inherent in the process, and an intermittent component. Shewhart attributed inherent variation, currently called random variation, to chance and undiscoverable causes, and intermittent variation to assignable causes.

• Shewhart concluded that assignable causes can be economically discovered and removed, but generally random causes can not be removed without making basic changes to the process.

• Each cause can itself vary, and be describable by its own statistical distribution. Each may be causing the observations to move in different directions relative to the others at any point in time. The standard deviation measures the dispersion of each of these distributions.

• Individual standard deviations for the different causes add by the square root of the sum of the squares rule, meaning that the largest causes will have an exponential impact on the process observations.

• Normal distributions are the result of many approximately equal causes of variation. Most measured distributions are not normal, as the Pareto principle predicts that the magnitudes of the causes will not be the same.

• As random causes are eradicated in the improvement process, the causes tend to equalize and the process falls into statistical control.

• Statistical control is not the absence of variation factors, but simply the continuous average counterbalancing of variation factors.

Appendix B: SPC Settings for 6σ Design

Specification limits: LSL = lowest acceptable value USL = highest acceptable value One of these is typically the project goal. The other is typically a natural barrier.

Target value: Mid-point between LSL and USL Design TGT = LSL + [ ( USL – LSL ) / 2 ] Determine if design target is the desired target. If not, indicate desired direction for MTB/LTB.

Target sigma is 1/12th the specification range: σ = ( USL – LSL ) / 12

Design control limits: 3σ above and below target value. Mid-points between target value and spec limits. Design LCL = TGT – 3σ Design UCL = TGT + 3σ


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BIBLIOGRAPHY

These books will provide the basics…

Six Sigma for Managers by Greg Brue. McGraw-Hill, 2002.

The Power of Six Sigma by Subir Chowdhury. Dearborn, 2001.

Design for Six Sigma: The Revolutionary Process for Achieving Extraordinary Profits by Subir Chowdhury. Dearborn, 2002.

Demystifying Six Sigma: A Company-Wide Approach to Continuous Improvement by Alan Larson. AMACON, 2003.

These books are more intermediate, and tend to focus on narrower perspectives…

Six Sigma: SPC and TQM in Manufacturing and Services by Geoff Tennant. Gower, 2001.

Transactional Six Sigma and Lean Servicing: Leveraging Manufacturing Concepts to Achieve World-Class Service by Betsi Harris Ehrlich. St. Lucie Press, 2002.

Six Sigma Team Dynamics: The Elusive Key to Project Success by George Eckes, John Wiley & Sons, 2002.

Juran Institute's Six Sigma Breakthrough and Beyond: Quality Performance Breakthrough Methods by Joseph A. De Feo, & William W. Barnard. McGraw-Hill, 2004.

Design for Six Sigma: Launching New Products and Services Without Failure by Geoff Tennant. Gower, 2002.

Leading Six Sigma: A Step-by-Step Guide Based on Experience with GE and Other Six Sigma Companies by Ronald S. Snee, & Roger W. Hoerl. Prentice Hall, 2002.

Lean Six Sigma: Combining Six Sigma Quality with Lean Speed by Michael L. George. McGraw-Hill, 2002.

AUTHOR

Richard E. Biehl, CSSBB, CSQE

Data-Oriented Quality Solutions 2105 Whitfield Lane Orlando, Florida 32835-5940 USA

[email protected] www.doqs.com

Rick Biehl is a senior information technology quality consultant with twenty-seven years of experience specializing in quality assurance, logical and physical data architectures, application architectures, and strategic planning for the business application of information technology. Rick holds a masters degree in education from Walden University, as well as certifications from the American Society for Quality (ASQ) as a Certified Six Sigma Black Belt (CSSBB) and a Certified Software Quality Engineer (CSQE). He currently serves as the Education Chair of the ASQ Human Development & Leadership Division.

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Six Sigma Math: A Primer