To be fair, much of this dialogue is healthy. Six Sigma will never solve all problems and a smart leader knows he or she needs a complete set of quality approaches – not just one. Any continuous improvement processes must also be customized to the environment in which it is employed and must evolve as that environment changes. By far, however, the main reason for the clamor calling for new problemsolving methods is that Six Sigma has become more complex than necessary.
What follows in this article is a description of Six Sigma reduced to its fundamental assumptions, or theorems. If these simple concepts are understood, all the tools, all the tollgate deliverables, and all the statistics and jargon are put in their proper supporting roles. Rather than mastering all the tools and attempting to build the program from the bottom up, this approach depends on a topdown, theoretical foundation. Rather than a cookbook approach, Six Sigma should be seen as a mathematical proof.
Customers only pay for value.
It seems so simple. Of course the customer only pays for value, but most businesses define value incorrectly. The products and services being sold are not what confer value to the customer. Products and services are vehicles to deliver value. Value is only created when a specific need the customer has is fulfilled. If a customer need is not met, even if the product or service is perfect, no value is created. Quality is not a measure of perfection, but of effect.
Failing to understand customers and their needs is the biggest driver of cost in most businesses. Quality for many products is defined by whether engineering specifications are met rather than whether the product delivers to actual customer needs. Similarly, in many service environments, service quality is defined by what the customer wants or complains about rather than how the customer uses the service. This distinction is important because it is possible to deliver everything a customer asks for and do so perfectly while not satisfying his or her needs. When this happens, the costs to deliver go up but the revenue from delivering does not.
First Theorem of Six Sigma: Changes in critical to quality (CTQ) parameters – and only changes in CTQ parameters – alter the fiscal relationship an organization has with its customers.
Those product or service qualities that alter the way a customer behaves with regard to purchase decisions are CTQs. Changes in CTQs, be they good or bad, drive customer loyalty. When CTQs are altered, a company’s ability to create customer value is altered. Improving the customer’s perception of value changes ultimately affects the fiscal relationship with that customer either in terms of price or cost to deliver.
The problem with managing to customer CTQs is not a question of intent. No business intentionally fails to deliver to CTQs. Companies fail to deliver to CTQs because of gaps in process knowledge. These gaps manifest themselves in process variation and poor process capabilities.
If a business has a profound and complete process knowledge, the products and services being delivered can be controlled so as to always create customer value. Gaps in process knowledge are the primary causes of failure and defect.
Process control focuses on ensuring that that the process is managed and executed in a consistent manner. If there are gaps in the understanding of how processes work or gaps in the understanding of how the customer ascribes value to the products and services being generated, process control is simply not possible. Six Sigma (and all continuous improvement processes) is fundamentally focused on closing these knowledge gaps.
Second Theorem of Six Sigma: Process outputs are caused by process, system and environmental inputs.
Y = ƒ (X_{1} . . . X_{n})
The “holy grail” of Six Sigma is the process transfer function. Once this is properly defined, managers have all the tools they need to make the process perform in any manner desired. The transfer function is never really perfect, but were a company to ever have complete transfer functions for all their processes, optimizing costs, production and customer value would be a simple matter of arithmetic.
The big myth is that Six Sigma is about quality control and statistics. It is all that – but a helluva lot more! Six Sigma ultimately drives leadership to be better by providing the tools to think through tough issues. —Jack Welch
Good leaders strive to make good decisions all of the time. If, however, there are gaps in understanding, incorrect assumptions or false assumptions, what looks like a good decision will result in bad outcomes. Six Sigma is about reducing the probability of these bad outcomes.
Most processes have subject matter experts and institutional knowledge. In most cases, these subject matter experts and institutional knowledge allow businesses to create transfer functions that are 90 percent to 95 percent correct, which gives the illusion of expert knowledge. It is the illusion of expert knowledge that makes it difficult. Since most of the rules for how to run processes are known and since the knowledge gaps are subtle, it is believed that the issues are executionrelated, not understandingrelated. This is a dangerous situation as these small process knowledge gaps, compounded over a multistep process, can result in significant losses even when people attempt to execute the process to the best of their abilities.
All variation is caused.
Often the only clue to gaps in process knowledge is the degree of variability in the process. Processes always have variation (remember – entropy increases!) but it does not spontaneously occur. It must be caused. Variation is just another output of systems. Transfer functions can be written to describe it; the more complete those functions are the better variation can be controlled. There is no myth or magic. When a system does not perform in exactly the same manner for a static set of process inputs, that simply means there are additional factors that are not yet understood that influence those processes.
Third Theorem of Six Sigma: Variation in process outputs are caused by process, system and environmental inputs.
∂Y / ∂X = ƒ′ (X_{1} . . . X_{n})
If Taguchi’s loss hypothesis (the cost of a system increases as it diverges from the performance expectations of its customer) is accepted, then process variation is the leading cause of customer dissatisfaction and operating costs. The factors that drive the process output and the process variation can be defined and controlled. Processes can then perform at the optimum balance between delivery and stability, thus driving the lowest possible cost and the highest possible customer satisfaction. In other words, if enough “profound knowledge” is added to the system, maximum value can be produced. The goal of Six Sigma, therefore, is always to increase process understanding. The goal is to populate transfer functions to the degree needed, and warranted, in order to best serve customers.
Given the right process knowledge and the ability to deliver products and services that satisfy the customers’ CTQs, management will always make the decision that most benefits the customer and achieves the highest possible return on investment.
The number one assumption when doing any type of continuous improvement is that if leadership is provided with the knowhow, if knowledge gaps are filled in, leadership will make (or allow others to make) decisions that are best for the longterm wellbeing of the business. Some people are afraid to test this assumption; it is a trust issue. If power is held because of the ability to solve a recurring problem, if teams are rewarded for firefighting rather than preventing problems, then this process collapses. While it is tempting to let experience and tradition supersede structured problem solving, in the long term when overall systems understanding is improved, and leadership employs that learning for the advantage of all parties in the supply chain, the business prospers.
Given a choice between longterm sustainable growth and shortterm profit, longterm growth will always outperform the shortterm gain.
This is the final and most critical assumption. It is cornerstone to total quality management, the Toyota Production System (Lean) and Six Sigma. If people are helped with controlling their own destinies, and if they are provided with the wherewithal to achieve selfdetermination, they will naturally do what profits them the most. When educated about the longterm benefits of creating sustainable customer relationships, most people will choose to create value and maximize their payback on the relationship. This is a question of ethics.
There are three foundational theorems and a simple set of postulates or assumptions that these theorems are based upon. All the tools and processes of Six Sigma – both DMAIC (Define, Measure, Analyze, Improve, Control) and DFSS (Design for Six Sigma) – are grounded in these simple foundations. Get the assumptions correct and all else is commentary.
]]>The question was posed to me: “I have five samples to test from my population. From that data, how can I estimate capability against our specifications?” Of course, the brutally honest answer is, “Poorly.”
But Black Belts do not survive by being sarcastic, so a better answer might be: “Any estimate of sigma from a small sample will have very large confidence intervals, giving you little knowledge of the actual population.” The question, however, pointed me toward an interesting avenue of exploration into the various ways that we may estimate the sigma in small sample sizes.
First, let’s review some of the more common methods of estimating sigma (or standard deviation, SD):
A. Using the average difference between an observation and the mean adjusted by sample size (the classic formula for sigma).
B. Using the range of the data divided by a factor, C, where C varies with sample size. Common values of C include 4 and 6, depending on sample size.
SD = Range/C
C. Using the moving range (MR) of time ordered data (where we subgroup on successive data points), divided by a factor. This is the method used in individual moving range control charts.
SD = MR/1.128
D. Using the interquartile range (IQR) of the data divided by a factor (D) where D = 1.35 is the most commonly proposed value.
SD = IQR/D
E. Using the mean of successive differences (MSSD) to estimate variance.^{1}
F. Using the mean absolute deviation (MAD) , absolute deviation (AD) is an estimate for SD.^{2}
G. Also, you can use the data minimum, maximum, and median to calculate an estimate of variance for small sample sizes.
For the purposes of my study I chose to evaluate all of the methods except for F. I chose methods to study that were easy to calculate (B, C, D), were available in Minitab (A, E) and sparked my interest (G).
A successful search for a better estimate of sigma, when the sample size is N ≤ 10, would meet the following two criteria:
In other words, if I repeatedly take random samples from a normally distributed population and calculate sigma, then all my samplings of sigma should begin to form a distribution with the average estimate of sigma equaling the true sigma of the data. Improved estimates of sigma will have a tighter distribution of estimates from this repeated sampling than other methods.
(Note: I will be primarily using a visual analysis of dot plots of the distribution of estimates of sigma to evaluate each method of estimating sigma.)
I also calculated the absolute deviation from true sigma (ADTS) using a formula similar to the mean absolute deviation (Method F) in order to numerically gauge each method’s performance against the criteria using the formula:
where Xi is each individual estimate of sigma generated in the simulation and sigma (σ) is the true population standard deviation from which the random normal data was calculated. In this case, N is the number of total estimates in the simulation. This formula sums the absolute value of the difference between each estimate of sigma and the true sigma of the population and divides by the number of estimates. The higher the ADTS factor, the worse the estimation of sigma. (Note: The distribution of standard deviation is a chisquare distribution and is, therefore, not evenly balanced about the mean. This unbalance can be seen in some of the simulations, but was not always clearly observed.)
Minitab’s Random Data calculator was used to create random normal data for testing of the various methods. I decided to study only normally distributed data sets as calculating sigma for the purposes of this article.
I used Minitab to create thousands of random normal data points. These thousands of pieces of data were then sampled using varying sample sizes, from 5 to 50 samples. In most of my explorations that follow, I used 5,000 pieces of random normal data.
Out of the 5,000 random normal data points, I used the following:
From all these samplings of the random data, I could then calculate the sigma using each of the methods. This gave me a set of data for each method of calculating sigma. A dot plot clearly shows the spread and frequency of each of the estimates of sigma. I could have used other displays of this data such as the classic histogram, but I felt that the dot plot improved the visual clarity of the data.
For example, the dot plot shown below in Figure 1 was created from selecting 1,000 fivepiece samples out of 5,000 pieces of random normal data with mean = 0 and sigma = 1. The sigma was calculated for each of these 1,000 fivepiece samples using the classic formula (Method A).
We observe from the dot plot the range of all 1,000 estimates of sigma. As the population was created using a sigma of 1, we can add that line to the dot plot and observe how all of the 1,000 estimates of sigma gathered about the “true” value. In real life we would never know that true value unless we had some previous knowledge from a larger sample. What this shows us is that if we take five random samples from a population where the actual sigma is 1.0, we might get a sigma anywhere from as low as 0.2 to as high as 2.3.
Calculating the ADTS factor for this set of sigma estimations gives us a value of 0.28. Values of ADTS can be compared to each other as long as the estimates of sigma come from the same population (same population mean and sigma).
A randomly generated set of 5,000 normally distributed data points with mean = 0 and SD = 1 was used to evaluate the various candidates for estimating sigma.
Moving Range Evaluation (Method C)
Using the moving range (average MR/1.128) for an estimate of sigma is problematic as it requires that the data is in time order. If we have a small sample size of data and if the order of this data is either unknown or not relevant, then using MR to estimate sigma is not valid.
N < 10, the MR is a better estimate of sigma than the classic formula. I performed a test of this using a smaller subset of my normal population dataset and calculated sigma using both the classic formula and the MR. Figure 2 below shows that sigma calculated from MR gives no obvious advantage over the classic formula. The calculated ADTS values support this conclusion.
MSSD Evaluation (Method E)
From my 5,000piece random normal data set, I calculated sigma using the classic formula (Method A) and compared that estimate to the SQRT (MSSD) method. Using Minitab’s Store Descriptive Statistics function along with the calculator function, I compared these two methods for sample sizes of 5, 10 and 50.
As can be seen in the dot plot comparison and ADTS values in Figure 3, the two methods have nearly identical centers and spread. Note that the value of ADTS becomes smaller as the sample size increases. This is as expected since the estimate of sigma improves as sample size increases. Given that the MSSD method provides nothing new over the classic formula, I dropped MSSD from the rest of my evaluations.
The “Hozo” Method (Method G)
In their article, “Estimating the mean and variance from the median, range, and the size of the sample,” mathematicians Stela Pudar Hozo, Benjamin Djulbegovic and Iztok Hozo proposed Method G for estimating variance and, thereby, SD involving sample size, minimum value, maximum value and the median.^{3}
The authors used simulations and “determined that for very small samples (up to 15) the best estimator for the variance is the formula” shown here as Method G.
I tested this formula using my 5,000piece population of random normal data and compared to the classic sigma formula as shown in Figures 4 and 5.
When the sample size is five, the Hozo method shifts the range of estimates to the left. Even though the spread of estimates is better for the Hozo method, the ADTS values show the classic standard deviation formula is slightly better. At the sample size of 10, the shift is not as great and the ADTS values indicate little difference between the classic sigma and Hozo methods.
However, in fairness to Hozo et al, their study parameters were different from mine. Instead of a mean of 0 and sigma of 1 (which I used to create normal data), they “drew 200 random samples of sizes ranging from 8 to 100 from a normal distribution with a population mean [of] 50 and sigma [of] 17.” This is a large sigma when compared to the mean (17/50). I am not sure that is practical in engineering.
Therefore, I created 5,000 random normal data points using mean = 50 and SD = 17 to see how the Hozo method compared to the sigma formula for a sample size of five and 10. This simulation still shows that with N = 5 the estimate of sigma is shifted away from the actual population sigma of 17. For N = 10 the shift is less but still present. The ADTS values bear out this conclusion.
In their study, Hozo et al found that “when the sample size increases, range/4 is the best estimator for the sigma until the sample sizes reach about 70. For large samples (size more than 70)[,] range/6 is actually the best estimator for the sigma (and variance).”
Range/C (Method B) and IQR/D (Method D) Evaluation
While all the other formulae are definitive in their variables, the range and IQR methods require some way to decide what to use for the values of C and D.
SD = Range/C
SD = IQR/D
From Hozo et al I found that commonly used values for C are 4 and 6, and that the most commonly used value for D is 1.35.
I decided to test these values against alternate values of C and D in the hopes of finding an improved range or IQR formula for small sample sizes (e.g., <10). After preliminary modeling using a wide selection of values for C and D, I settled on testing the following factors:
Once I determined the values of C and D to be evaluated, I used the 5,000 pieces of random normal data to calculate the spread of estimates of sigma using values of C and D as defined above and compared them to the classically calculated formula for sigma along with their ADTS factor.
In the dot plots that follow I left out the plots of the classic sigma formula as I am focusing on the centering and spread of the various estimates using R/C and IQR/D.
The code of labeling is as follows:
Range Evaluation
Figure 8 shows that R/4 is greatly leftshifted when N < 10 and that R/2.5 is centered – although the spread of the R/2.5 data is large. At values of N > 10, using R/4 centers around the true sigma much better than R/2.5.
Plotting all the ADTS factors by sample size and sigma calculated by R/C and classic standard deviation shows how R/2.5 and R/4 change with sample size (see Figure 9). We also can see how these estimates of sigma compare to the classic standard deviation formula.
IQR Evaluation
In Figure 10, the IQR/D estimates show that when the sample size is 5 or 10, then IQR/1.55 is more centered and has less spread of estimates than when compared to IQR/1.35. With sample sizes greater than 10, however, this pattern shifts and IQR/1.35 slightly improves.
As with the range data, I plotted the ASTS values for each IQR/D method and compared them to the classic formula for standard deviation (Figure 11).
A summary chart of these two methods is shown in Figure 12. This chart shows that all methods are almost equal when N = 5. When N = 10, the IQR/1.55 is better than R/2.5 but not as good as the classic formula for standard deviation.
Given that the best estimates for sigma appear to be IQR/1.55, R/4 or R/6 (depending on sample size), I created a new set of 5,000 pieces of random normal data and reran all of the calculations of ADTS for each combination.
The graph in Figure 13 is interesting in that it shows how IQR/1.55 is actually pretty robust over sample size. The IQR/1.55 method would be a good choice if picking a method for estimating sigma (that was not the classic formula).
The IQR/1.55 method has another advantage. Both the R/C method and the classic sigma method are prone to outliers, especially with small sample sizes. The IQR/1.55 method is not affected by an extreme outlier in a small sample of data.
For example, let’s look at a set of seven data points to see how an outlier affects our estimates of sigma. Below are two sevenpiece random and normal samples of data from a population with a known sigma of 1.0. The first set of data does not have an outlier; the second set of data does have an outlier (2.0) as confirmed by Grubb’s test for outliers.
Table 1: Sample Data  
Data 1  Data 1_1 
0.229762  0.22976 
0.370426  0.37043 
0.402137  0.40214 
0.589118  0.58912 
0.776588  0.77659 
0.845852  0.84585 
0.969874  2.00000 
From this data we can calculate the following estimates of sigma and see how the IQR method is robust to an outlier. The classic method and the R/2.5 method change significantly with the presence of an outlier.
Table 2: Comparing IQR and R to Classic Sigma  
Classic Sigma  IQR/1.55  Range/2.5  
No Outlier  0.76  0.307  0.296 
With Outlier  0.596  0.307  0.708 
For ease of calculations, if you are given a choice, a sample size of N = 7 allows the IQR to be easily solved. For N = 7, the third quartile is the sixth data point in ordered data, and the first quartile is the second data point in the ordered data. When N = 11, then the third quartile is the ninth point and the first quartile is the third point.
A successful search for a better estimate of sigma centers the estimates about the true population sigma and has a tighter spread of estimates than given by the classic formula for standard deviation.
In this study, we examined several candidate formulae for sigma when N ≤ 10. We also hypothesized two new formulae (R/C, where C = 2.5 and IQR/D where D = 1.55).
Other methods of estimating sigma (MSSD, Hozo and MR) do not appear to offer any advantages over the classic formula for sigma. The Hozo method also seems to shift the center of the estimates of sigma to the left of the “true” sigma.
Estimates of sigma using IQR/1.55 appear to be good when the population data is normal – regardless of sample size. Although the classic formula appears to give lower ADTS scores for every sample size.
It is recommended that when the sample size is small, that a test for outliers (e.g., Grubb’s test) be performed. If an outlier exists and if the reason for the outlier cannot be determined, then the IQR/1.55 method is recommended.
The decision by an investigator to use the IQR/1.55 method, Range/C method or the classic standard deviation formula is situational. But it may be argued from this data that the classic formula of standard deviation is the best estimator of sigma – regardless of sample size.
USL, upper specification limit; LSL, lower specification limit.
*Estimated sigma = average range/d2
Common understanding includes the fact that C_{pk} climbs as a process improves – the higher the C_{pk}, the better the product or process. Using the formula above, it’s easy to calculate C_{pk} once the mean, standard deviation, and upper and lower specification limits are known.
But what if you have only one specification or tolerance – for example, an upper, but no lower, tolerance? Is C_{pk} advisable under these circumstances?
When faced with a missing specification, consider one of the following three options:
Examining a specific situation may clarify the outcome of each of these possibilities. A customer of a plastic pellet manufacturer has specified that the pellets should have a low amount of moisture content. The lower the moisture content, the better, but no more than 0.5 units is allowed; too much moisture will create manufacturing problems for the customer. The process is in statistical control.
This customer would undoubtedly not be satisfied with option 1 as C_{pk} has been specifically requested. With option 2, it could be argued that the LSL is 0, since moisture levels below zero are impossible. With a USL at 0.5 and LSL at 0, the C_{pk} calculation would be as follows.
Assume the Xbar = 0.0025 and estimated sigma is 0.15.
The customer is not likely to be satisfied with a C_{pk} of 0.005, and that number does not represent the process capability accurately.
Option 3 assumes that the lower specification is missing. Without an LSL, Z_{lower} is missing or nonexistent. Z_{min} becomes Z_{upper} and C_{pk} becomes Z_{upper} / 3.
Z_{upper} = 3.316 (from above)
C_{pk} = 3.316 / 3 = 1.10
A C_{pk} of 1.10 is more realistic than one of 0.005 for the data given in this example, and is more representative of the process itself.
As the example demonstrates, setting the lower specification to 0 results in a lower, misleading C_{pk}. In fact, as the process improves (here, moisture content decreases), the C_{pk} should have increased. If 0 was used as the LSL, however, the C_{pk} would have decreased. This is one clue that entering an arbitrary specification is not advised.
“What should be done when only one specification exists?” The (only) specification you have should be used, and the other specification should be left out of consideration or treated as missing. In these cases, use only Z_{upper} or Z_{lower}.
C_{pk} can and should be calculated when only one specification exists, provided only the remaining valid specification is used. As the example demonstrates, the missing specification should remain missing and not be artificially inserted into the calculation.
]]>While one of the statistical methods widely used in the Analyze phase is regression analysis, there are situations that warrant the use of other nonparametric methods. Violation of the basic assumptions of normally and independently distributed residuals, and the presence of nonlinear relationships, are the most common situations where using a nonparametric method, such as a classification and regression tree (CART), is more appropriate. In addition, CART can be appropriate in service industries such as banking and healthcare where many potential causes of variation and defects are categorical in nature (e.g., geographical locations, products, channels, partners). The problem with using regression or generalized linear models (GLM) in such cases is that a lot of dummy variables make it difficult to interpret the results. CART is a useful nonparametric technique that can be used to explain a continuous or categorical dependent variable in terms of multiple independent variables. The independent variables can be continuous or categorical. CART employs a partitioning approach generally known as “divide and conquer.”
Assume there is a set of credit card transactions labeled as fraudulent or authentic. There are two attributes of each transaction: amount (of transaction) and age of customer. Figure 1 displays an example map of fraudulent and authentic transactions.
The CART algorithm works to find the independent variable that creates the best homogeneous group when splitting the data. For a classification problem where the response variable is categorical, this is decided by calculating the information gained based upon the entropy resulting from the split. For numeric response, homogeneity is measured by statistics such as standard deviation or variance. (For more information on this please refer to Machine Learning with R by Brett Lantz.)
Two important parameters of the CART technique are the minimum split criterion and the complexity parameter (C_{p}). The minimum split criterion is the minimum number of records that must be present in a node before a split can be attempted. This has to be specified at the outset. C_{p} is a complexity parameter that avoids splitting those nodes that are obviously not worthwhile. Another way to consider these parameters is that the C_{p} value is determined after “growing the tree” and the optimal value is used to “prune the tree.”
In this example, Figure 2 shows that the first rule formed is x2 > 35 → fraudulent transaction. Similarly, other rules are formed as shown in Figures 3 and 4.
In this way, the CART algorithm keeps dividing the data set until each “leaf” node is left with the minimum number of records as specified by minimum split criterion. This results in a treelike structure as shown in Figure 5. The C_{p} value is then plotted against various levels of the tree and the optimum value is used to prune the tree.
The following example contains a hypothetical dataset of 600 dispatch transactions of a bank.
The dependent variable is the attribute “defective,” which is a categorical variable with two classes (yes and no). Each transaction is labeled either “yes” or “no” based on whether there is any printing error in the deliverable. The independent variables are “amount,” “channel,” “service type,” “customer category” and “department involved.” The first step in applying any analytical method is to explore the data using descriptive statistics. Assume that in exploring the data all of the independent variables seem to have a significant relationship with the dependent variable. In order to carry out the CART analysis, the dataset is randomly split into two sets, the training and testing sets. Nonparametric studies are not based upon theoreticalprobability distributions; it is widely accepted practice to build a model on one set of data and test it on another. This helps in ascertaining the accuracy of the model on unknown future records.
The CART model is used to find out the relationship among defective transactions and “amount,” “channel,” “service type,” “customer category” and “department involved.” After building the model, the C_{p} value is checked across the levels of tree to find out the optimum level at which the relative error is minimum. The optimum C_{p} value is then used to prune the tree.
Postpruning, the “final” tree can be created as shown in Figure 8. The model can also be validated against test data to ascertain its accuracy.
As with other nonparametric techniques, CART does not require any assumptions for underlying distributions. It is easy to use and can quickly provide valuable insights into massive amounts of data. These insights can be further used to drill down to a particular cause and find effective, quick solutions. The solution is easily interpretable, intuitive and can be verified with existing data; it is a good way to present solutions to management.
Like any technique, CART also has limitations to take into account before doing the analysis and making any decisions. The biggest limitation is the fact that it is a nonparametric technique; it is not recommended to make any generalization on the underlying phenomenon based upon the results observed. Although the rules obtained through the analysis can be tested on new data, it must be remembered that the model is built based upon the sample without making any inference about the underlying probability distribution. In addition to this, another limitation of CART is that the tree becomes quite complex after seven or eight layers. Interpreting the results in this situation is not intuitive.
Conclusion
CART can be used efficiently to assess massive datasets and can provide quick solutions in the Analyze phase of DMAIC. CART can be one of the quickest and most effective tools in the bag of any process improvement practitioner. CART should not, however, replace corresponding parametric techniques. The latter is always more powerful in terms of explaining any phenomenon owing to the nature of underlying distribution.
]]>Contact
Barbara A. Cleary, PhD
8007773020
Dayton, Ohio, April 14, 2017—An update to GAGEpack 12 from PQ Systems demonstrates the commitment of developers to respond to customer needs with enhanced ease of use and improved user interface.
In this updated version, visual improvements streamline the window view by collapsing the navigation panel and moving it to the ribbon, making filter information more readily available, and altering lessused commands by invoking them through buttons.
Additional improvements to the user experience include:
Customers with maintenance agreements will receive the updated solution automatically, allowing for a smooth transition to the use of new features.
GAGEpack is a powerful gage calibration solution that maintains complete histories of measurement devices, instruments, and gages. To guarantee timely calibration, the software provides a variety of tools, such as:
o Calibration schedules and reports
o Alerts about failed and past due calibrations
o Gage location and status tracking
o Gage repair records
o Audit trail for traceability
o A Task tab with a “To do” list
o Gage event alert system
About PQ Systems: PQ Systems www.pqsystems.com is a privatelyheld company headquartered in Dayton, OH, with representation in Europe, Australia, Central and South America, Asia, and Africa and customers in more than 60 countries. For more than 30 years, the company has been helping businesses drive strategic quality outcomes by providing intuitive solutions to help manufacturers optimize process performance, improve product quality, and mitigate supply chain risk. The company’s scalable solutions include SQCpack® for data analysis and statistical process control and GAGEpack® for measurement intelligence. PQ Systems’ worldclass consulting, training, and support services ensure that clients receive the maximum return on their software implementations.
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]]>There are two types of metrics to consider when selecting KPIs for a project: outcome metrics and process metrics.
Outcome metrics provide insight into the output, or end result, of a process. Outcome metrics typically have an associated datalag due to time passing before the outcome of a process is known. The primary outcome metric for a project is typically identified by project teams early on in their project work. This metric for most projects can be found by answering the question, “What are you trying to accomplish?”
Process metrics provide feedback on the performance of elements of the process as it happens. It is common for process metrics to focus on the identified drivers of process performance. Process metrics can provide a preview of process performance for project teams and allow them to work proactively to address performance concerns.
Consider an example of KPIs for a healthcarefocused improvement project:
In the example above the project has one primary outcome metric and four process metrics that compose the KPIs the team is monitoring. Wellcrafted improvement project KPIs will include both outcome metrics and process metrics. Having a mix of both provides the balance of information that the team needs to successfully monitor performance and progress towards goals.
Teams should develop no more than three to six KPIs for a project. Moving beyond six metrics can dilute the effects of the data and make it more challenging to effectively communicate the progress of a project.
Common questions coaches can use with teams to generate conversation about potential KPIs include:
Coaches should keep the three Ms of crafting KPIs in mind when working with teams.
Remember that successful KPIs:
Crafting KPIs is an important step to guide teams through a continuous improvement process. A coach needs to keep the team focused on what success looks like and how best to measure it.
]]>Depending on the environment, organization’s maturity, people and processes that must be dealt with, LSS practitioners may be in situations that prevent them from following a textbook project. These situations may include:
Assuming one or more of the above elements are true, running a project can be a challenge. Not only can everything one has learned be viewed as a waste of potential and time, but a project may also be seen as too difficult to attempt. A simple way to turn these challenges into an opportunity is by extracting the maximum from the existing situation instead of fighting against it. A couple of examples demonstrate that using individual tools when they fit the purpose can be as rewarding as applying the whole framework – by the book.
Every LSS training or guideline instructs to start a project by analyzing the voice of the customer (VOC). But what if there is no project and the practitioner does not interact with the endproduct customer? There is a trick to adjust the VOC tool to help improve the organization.
Treat the SMEs as the customers. Let’s take the simplest scenario where one LSS expert is assigned to a team of SMEs attempting to improve their own processes. The objective here is threefold:
When it comes to implementing these principles, as with any customer feedback, the key is to establish a structured method for gathering, storing and reviewing the improvement ideas. The following are some tips that can assist in building a simple mechanism to manage a team’s improvement ideas:
With numerous “customers,” this process is more complicated; however, given team discipline and collaboration it can turn into success. The end goal should be for the team to become selfsufficient in improving their processes when the LSS expert is no longer available.
VOC is not the only instrument that teams can use by themselves. Other examples of useful tools that every team can use in everyday work include project management documents that bring structure to every initiative.
Even if LSS is not used in everyday operations, a smart expert can still smuggle a few useful tools into the workplace. This is because every organization runs projects; all projects typically bring change and opportunities for improvement. As these tools are simple and universal, no matter what methodology an organization uses, LSS best practices around project management documentation can often be the first big win. This can apply to any initiative, starting with a local team project and finishing with a global organizational change. Here are some examples of tools that each person running a project should befriend:
What if the LSS expert is assigned to a team that does not run any projects? One might argue that the space for improving their operations is limited. There is a tool, though, that can be applied in any circumstances and implemented by the team independently – 5S (sort, set in order, shine, standardize, sustain) only sounds like one tool, but it is by far one of the most helpful. Apart from the visible improvements 5S offers in each of the five phases of DMAIC (Define, Measure, Analyze, Improve, Control), 5S offers plenty of opportunities to embed the continuous improvement mindset quickly and effectively. (Again, if it is not possible to use all the elements at once, fitforpurpose is the most sensible approach to follow.)
In operations like human resources, finance or outsourcing, some 5S techniques can be applied as successfully as in manufacturing. Good analogies for a service environment relate to virtual workplaces. Some examples include setting up document repositories and shared locations, standardizing service inputs or outputs, and keeping the PC workplace tidy.
The previous examples demonstrate how to improve an existing process with little effort. But what if the process is not there yet? In such instances, the LSS expert might be asked to design and implement an activity that was not previously performed.
When establishing a new function, going through reorganization or simply starting a new activity, a couple of LSS tools can be utilized to help define and document the change taking place.
When the reality is different from what was taught during training, the choices are to give up or adjust one’s approach. By using a fitforpurpose approach and simplifying the tools, it is easier to make the tools easier to remember and, therefore, encourage the staff to use them more often. Many small improvements have a big chance of translating into a continuous improvement culture for the whole organization.
]]>Press Contact
Diane Tilley
(888) 7446295
Kitchener, Ontario February 27, 2017—SigmaXL Inc., a leading provider of user friendly Excel Addins for Statistical and Graphical analysis, announces the release of SigmaXL Version 8.
“SigmaXL was designed from the ground up to be a costeffective, powerful, but easy to use tool that enables users to measure, analyze, improve and control their service, transactional, and manufacturing processes. As an addin to the already familiar Microsoft Excel, SigmaXL is ideal for Lean Six Sigma training or use in a college statistics course. Our slogan for Version 8 is Multiple Comparisons Made Easy,” said John Noguera, CTO, SigmaXL.
New features in Version 8 include:
Dr. Peter Wludyka, coauthor of the book, The Analysis of Means: A Graphical Method for Comparing Means, Rates, and Proportions, “I am happy to endorse the ANOM charts introduced in SigmaXL Version 8. They are easy to use and accurately handle balanced and unbalanced data. We collaborated to extend Multiway Slicing to Binomial and Poisson and these are included in the TwoWay charts, where SigmaXL automatically recommends Slice Charts when the interaction is significant.”A free 30day trial version is available for download from the SigmaXL website at: www.SigmaXL.com.
About SigmaXL Inc.
SigmaXL is a leading provider of user friendly Excel Addins for Lean Six Sigma tools and Monte Carlo Simulation. SigmaXL customers include market leaders like Agilent, Diebold, FedEx, Microsoft, Motorola, and Shell. SigmaXL software is also used by numerous colleges, universities and government agencies.
For more information, visit http://www.SigmaXL.com or call 1888SigmaXL (8887446295).
]]>The business is going through cultural transformation in all of its plants. It is implementing a corporate strategy to support common continuous improvement thinking and language across the enterprise – laying the groundwork required for a sustainable continuous improvement culture. The business is using four phases in its continuous improvement rollout, as shown in Figure 1 below.
I. Foundation and the organizational alignment
II. Expansion and discipline
III. Integration and reinforcement
IV. Sustaining momentum
The Kansas City plant, my plant, has completed Phase II and is working its way through Phase III.
The company sent all employees through a simulated work environment (SWE) where they assembled and disassembled wood cars on a real line using real tools and bolts. (This concept was taken from Caterpillar, which went through the same cultural transformation years ago.) This teaches everyone how to use some Lean tools and shows everyone how the team leads will be used in the new environment. During a twoday training, employees eliminated waste between the different runs, and watched their quality and delivery rates improve after each run. After SWE training, the team leads train the employees on what Lean tools are to be used and how to use them. The Lean tools being taught are: 5S, cyclical and noncyclical standard work, total productive maintenance, quick changeover, inventory management, value stream mapping, error proofing, process problem solving, and Kaizen. The company provides an overview of the same tools to all employees so that they understand what team leads are trying to accomplish.
The plant located in Kansas City committed to more than 30,000 hours of training since starting its LSS journey.
The continuous improvement lead at the Kansas City plant was asked how important Lean Six Sigma is for implementing process improvements in manufacturing. He replied that the company’s employees work for its shareholders as a publicly traded company. LSS fits into the strategic pillar: growth, leadership development, continuous improvement and sustainability. The purpose of the LSS program is to develop a continuous improvement culture and mindset through people, processes and systems that enable bestinclass productivity, quality and time to market – all helping to create a better value for the customer. The following examples are a look at some of the ways in which my company is implementing process improvements.
Example 1: Continuous Improvement Waste Elimination
At my company, a tactical manufacturing engineer’s job is to work with the team leads and assist them with problem solving and implementing improvements. The engineers teach the team leads the tools they should use and what data to collect to make sure improvements are made. One project was the team lead’s Yellow Belt project – the redesign of the fender support cell. For his project he had to complete an A3 (a Lean tool for problem solving), which included a problem statement, future state conditions, current conditions and an implementation plan.
The team’s problem statement was that the fender support subassembly had 27.36 seconds of walk time in each cycle, which added up to 9.12 hours of walk time a week (474.24 hours annually), costing the company approximately $10,883 annually in nonvalueadded tasks. The team’s objective was to reduce walk time by 66 percent (18 seconds a cycle), freeing up more time for valueadded tasks without negatively impacting quality. The team determined what area(s) to focus on by doing a pareto chart of the steps taken per cycle and looking at why the steps were necessary (see Figure 2). The team determined that locations of parts and materials were the main reasons for the high number of steps.
As a team they captured the current state layout and performed a spaghetti diagram to show the process flow of the subassembly area. Then they cut out all the material and equipment in the subarea and played around with a layout for a new placement location for the future state. Improvements that were generated from this process included a smaller work table with a material rack built on the back of it, a new layout of the material rowpacks and a buffer table reduced from 48 pieces to 1 piece. Figure 3 presents the layout of the work station – before and after.
As a result of this new layout, the team reduced the walk time. The buffer reduction resulted in an increase of quality products – from 18 average defects per day down to nine. The work envelope was smaller and the Yamazumi board (another Lean tool) reflected a 18second reduction in walk time. The new layout opened up 2,000 square inches of free space on the work floor. This was a good process improvement for the team and the line because it showed that substantial improvements can be made with just a little time and teamwork.
Example 2: O_{2} Sensors Causing HighScrap Costs
The first Green Belt project that I mentored was focused on a quality issue on some O_{2} sensors causing high scrap costs. From January 2, 2014, to March 31, 2014, the nonconforming material (NCM) for O_{2} sensors and exhaust pipes resulting from O_{2} sensors being stripped or cross threaded was 17,394 parts per million (PPM), which cost the company $29,552 annually. The improvements reduced the NCM for O_{2} sensors and exhaust pipes from 17,394 PPM to 2,253 PPM, an 87 percent improvement in performance, saving $26,689 annually.
Following DMAIC (Define, Measure, Analyze, Improve, Control) helped the project stay on track. The root cause of the problem was that the pipe thread quality was not being met, which caused crossthreading during assembly. There was also better capability if the operators hand started the O_{2} sensor before doing the final torque. The Green Belt candidate worked with the suppler to meet print specification on the threads which yielded the 87 percent improvement. Figure 4 presents a control chart showing the beforeandafter data of O_{2} sensors stripped or crossthreaded.
Example 3: Addressing Ergonomics
Six Sigma can also be followed when trying to reduce ergonomic scores. My company has an ergonomic score on every job performed. From January 24, 2013, to December 12, 2013, the handlebar subassembly stations had ergonomic job measurement system (EJMS) with room for improvement; the stations cost the company $131,492 annually. An unused subassembly carousel was repurposed to replace the work tables used to build the handlebars. This eliminated two major handlebar lifts as well as additional handling of the handlebars. EJMS was reduced an average of 27.5 percent, resulting in a cost avoidance of $108,393. Additionally, operator efficiency improved to support a volume increase of 30.4 percent, resulting in a $47,170 cost avoidance for 2014.
Ergonomics is a big part of assembly; any time improvements can be made to reduce ergonomic scores leadership is always on board. These projects can be difficult to implement because equipment can be expensive; repurposing equipment reduces those costs. This change was a big improvement for this line and helped with the efficiency of the operator. Since this has been implemented, we reduced the head count to two operators from three operators. This was due to the lower volume and the higher operator efficiency. This never would have happened if the conveyor was not implemented (Figure 5).
Example 4: Improving the ErrorProofing System
Another project addressed by a Green Belt candidate was improving the errorproofing system that the company uses on the assembly line. The issue was addressed from January 2, 2014, to January 24, 2014; the average number of assembly verification and information system (AVIS) station bypasses was 166 per day on the lines, which cost the company $77,522 annually. The improvement actions taken were to change and verify the configuration settings for the adjusternut AVIS station. The results reduced the number of AVIS bypasses from 43 percent defective (166/day) to 3.7 percent defective (22/day), saving $77,522 annually.
A pareto chart was performed on all the AVIS stations which helped the team focus on the one problem (Figure 6). After following the DMAIC methodology, the Green Belt candidate determined that the station configuration was incorrect and that the recipe program was incorrect. The candidate fixed these two issues and this problem was eliminated. The problem with this project is the sustainability of AVIS knowledge in the manufacturing engineering group. There are only two engineers who know how to program these AVIS stations and it is time consuming to train others on the process.
As asked in The Toyota Way by Jeffrey Liker:
“What do we know about changing a culture?
1. Start from the top – this may require an executive leadership shakeup.
2. Involve from the bottom up.
3. Use middle managers as change agents.
4. It takes time to develop people who really understand and live the philosophy.
5. On a scale of difficulty, it is ‘extremely’ difficult.”
This is what I see at my current company. Enacting a changeimprovement culture takes time, some leadership changes have occurred, the company lacks middle managers as change agents and change is extremely difficult.
Lean coupled with Six Sigma tools drive decisionmaking by data and metrics and provides a mechanism to quantify the potential for variation, defects and risk – as well as valueadded and resource optimization before implementing actual changes.
Leadership must be involved in order to understand and communicate the importance of LSS and its deployment. Leadership must support a transformation and lead employees in the change. If projects are completed and the results are shared to the “nonbelievers,” a LSS implementation will be successful.
Middle management is focused on making improvements and ensuring bottomline savings, which can lead to a need for lower staffing levels. This is a problem at my company not only with supervisors, but also with manufacturing engineers. They are doing the same amount of, or more, work as before with fewer middlemanagement employees. Middle managers can be overstressed and overworked, leading to increased turnover.
LSS is the future of current company and it will be a long journey to become fully implemented into the continuous improvement culture; however, LSS process improvements will help the company keep costs low, quality high and customer satisfaction high.
]]>Classical statistical process control (SPC) methods, such as individual and moving range, Xbar and R charts, were developed in the era of mass production of identical parts. Production runs lasted for weeks, months and even years. Many SPC rules of thumb were created for this environment (as noted in The Six Sigma Handbook by Thomas Pyzdek). This may not have been a problem in lowmix, highvolume production, but it is impractical or impossible in today’s highmix, lowvolume production. In a lowvolume, highmix situation the entire production run can be fewer parts than are required to start a standard control chart. Standard SPC methods can be modified slightly to work with small runs.
As a rule of thumb, if at least 10 different values occur and repeat values make up no more than 20 percent of the data set, data can be considered variable. Otherwise the data is considered to be discrete and attribute control charts should be used.1 There are several approaches for short runs using variable data, but the ZMR chart is preferred because all the subgroups are used; other methods exclude subgroups. The following explains what the ZMR chart is and how practitioners can use it.
Statisticians and engineers often use normalizing transformations. Sigma level and process capability are two common applications of normalizing transforms. Sigma level is the same thing as Zvalue – this normalization is simply the number of standard deviations from a value of interest and the mean of the data. The Zvalue can be used to create control charts that are independent of the units of measure. Several different characteristics can be plotted on the same control chart as long as they are produced in a similar process. Zcharts are independent of the units of measure and can be thought of as true process control charts. A ZMR chart can be used with shortrun processes when there is not enough data in each run to calculate proper control limits. ZMR charts standardize the measurement data by subtracting the mean to center the data, then dividing by the standard deviation.
Standardizing allows a practitioner to evaluate data from different runs by interpreting a single control chart. The Zchart option is supported by Minitab (and other statistical software products). The standardized data comes from a population with the mean = 0 and the standard deviation = 1. With that, a single plot can be used for the standardized data from different parts or products. The resulting control chart has a center line at 0, an upper limit at +3 and a lower limit at 3.
Example of a Zchart
A specialty manufacturer of pickandplace heads for small parts has a new process for making a vacuum orifice. This process is being used on eight parts with differentsized orifices ranging from 10 microns to 30 microns in diameter. These parts are hard to measure and are run in small batches. There are, thus, few samples to study but whether the process is stable and controlled needs to be understood. (Note that the measurement system has been validated.)
The two parts of Table 1 show the first set of data.
Table 1: Sample Data Set (Part 1)  
Part Numbers  Measurement 
1  10.34 
1  9.23 
1  10.54 
1  9.84 
1  10.30 
2  17.56 
2  19.26 
2  22.72 
2  18.45 
2  21.42 
3  25.08 
3  25.02 
3  24.46 
3  24.80 
3  24.39 
4  20.01 
4  19.93 
4  19.96 
4  19.97 
4  19.89 
5  10.58 
5  9.12 
5  10.67 
5  10.38 
5  10.39 
6  29.37 
6  29.43 
6  30.16 
6  31.56 
6  30.23 
7  29.52 
7  30.56 
7  26.59 
7  27.57 
7  29.66 
“>8  9.57 
8  9.90 
8  10.20 
8  13.50 
8  12.67 
Table 1: Sample Data Set (Part 2)  
Part Number  Mean  Standard Deviation  Range 
1  10.049  0.523  1.303 
2  19.88  2.136  5.16 
3  27.749  0.315  0.686 
4  19.953  0.0438  0.116 
5  10.229  0.632  1.552 
6  30.149  0.883  2.185 
7  28.782  1.639  3.969 
8  11.168  1.788  3.932 
If this data is put in an individual and moving range (IMR) chart, the result has little meaning as there is not enough data to calculate statistically correct control limits. A Zchart (Figure 1) can overcome this limitation.
To build the Zchart using Minitab: choose Stat > Control Charts > Variables Charts for Individuals > ZMR Chart. In Variables, select Measurement and Part Numbers for part indicator.
Minitab provides four methods for estimating , process standard deviations. Choose an estimation method based on the properties of the particular process or product at hand. Or enter a historical value. The data plotted in the Zchart is Z_{i} where . Make assumptions about the process variation, but note that this should not be taken lightly as results will differ between assumptions. Based on the assumptions made, the estimate of standard deviation changes.
Use Table 2 to help select a method of estimation.
Table 2: How to Select a Method of Estimation  
Method Type  Use When  Does This 
Constant (pool all data)  All the output from the process has the same variance – regardless of the size of the measurement  Pools all the data across runs and parts to obtain a common estimate of s 
Relative to size (pool all data, use log [data])  The variance increases in a fairly constant manner as the size of the measurement increases 

By parts (pool all runs of same part/batch)  All runs of a particular part or product have the same variance  Combines all runs of the same part or product to estimate s 
By runs (no pooling) * default option  It cannot be assumed that all runs of a particular part or product have the same variance  Estimates s for each run independently 
Under ZMR Options, select estimates and pick “by Runs” (default) as equal variance cannot be assumed, as shown in Figure 3.
Click OK and OK again. The resulting control chart is shown in Figure 4.
The process is stable and in control.
There are two issues with plotting attribute data from shortrun processes.
Because of these difficulties, many people believe that SPC is practical only for long, highvolume runs. This is not necessarily true. In many cases, stabilized attribute charts can eliminate both of these problems. The downsides to stabilized charts are that they are more complicated to develop and there are not standard options in most common statistical software. These charts must be made manually or a macro must be created. They may require more effort but they can be useful.
Stabilized attribute charts may be used if a process is producing parts or features that are similar. Stabilized (Z) attribute control charts also solve the issue of varying control limits and central lines due to varying sample sizes, making the chart easy to visibly interpret.
Here is a typical scenario: A jobshop welding operation produces small quantities of custom items. The operation, however, always involves joining parts of similar material and similar size. The process control statistic is weld imperfections per 100 inches of weld.
Methods used to create stabilized (Z) attribute control charts are all based on their corresponding classical longrun attribute control chart methods. There are four basic types of control charts involved:
All of these charts are based on the following transformation:
Stabilized (Z) attribute charts can be used for longrun u and p charts with varying sample sizes. This can be used to eliminate the varying and confusing control limits.
For example, 10 part numbers run in different small runs. The parts are similar but different. The number of defective units has been recorded and it is desired to determine if the process is in control. Table 4 displays the application of the formulae above. The calculated Z scores can then be plotted and compared to ±3 standard deviations. As all our values fall within ±3, our process is in statistical control for defective units.
Table 4: Results of Small Run Results  
Part Number  Sample Size  Defectives (np)  p  Z  UCL  LCL  
1  10  1  0.1  1  0.942809  0.942809  0  3  3 
2  15  2  0.133333  1.06066  3  3  
3  20  2  0.1  1.06066  3  3  
4  15  1  0.066667  0  3  3  
5  8  1  0.125  0  3  3  
6  10  1  0.1  0  3  3  
7  12  1  0.083333  0  3  3  
8  15  0  0  1.06066  3  3  
9  10  0  0  1.06066  3  3 
SPC can be used for short production runs and may be helpful in any operation. At a minimum, these charts are more tools to include in the continuous improvement toolbox.
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