Non-Normal Data Not Transforming the Data Can Be Fatal to Your Analysis - Breyfogle

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NOT Transforming the Data Can Be Fatal toYour Analysis

By: Forrest Breyfogle III

A case study, with real data, describes the need for data transformation.

Not surprisingly, there was controversy over Forrest Breyfogle's article, "Non-normal Data: ToTransform or Not to Transform[1]," written in response to Donald Wheelers article "Do YouHave Leptokurtophobia?[2]" Wheeler continued the debate with "Transforming the Data CanBe Fatal to Your Analysis[3]." This article is Breyfogles response to Wheelers latest column.

--Editor

Donald Wheeler stated in his second article "Transforming the Data Can Be Fatal to Your

Analysis [3]," "out of respect for those who are interested in learning how to better analyzedata, I feel the need to further explain why the transformation of data can be fatal to youranalysis."

My motivation for writing this article is not only improved data analysis but formulatinganalyses so that a more significant business performance reporting question is alsoaddressed within the assessment; i.e., a paradigm shift from the objectives of the traditional

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Page 1 of 16NOT Transforming the Data Can Be Fatal to Your Analysis

9/26/2009http://www.qualitydigest.com/print/8970


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four-step Shewhart system, which Wheeler referenced in his article.

The described statistical business performance charting methodology can, for example,reduce firefighting when the approach replaces organizational goal-setting red-yellow-greenscorecards, which often have no structured plan for making improvements. This article willshow, using real data, why an appropriate data transformation can be essential to determinethe best action or non-action to take when applying this overall system in both manufacturing

and transactional processes.

Should Traditional Control Charting Procedures be Enhanced?

I appreciate the work of the many statistical-analysis icons. For example, more than 70 yearsago Walter Shewhart introduced statistical control charting. W. Edwards Deming extendedthis work to the business system with his profound knowledge philosophy and introduced theterminology common and special cause variability. I also respect and appreciate the work thatWheeler has done over the years. For example, his book Understanding Variationhasprovided insight to many.

However, Wheeler and I have a difference of opinion about the need to transform data whena transformation makes physical sense. The reason for writing this article is to provideadditional information on the reasoning for my position. I hope that this supplementalexplanation will provide readers with enough insight so that they can make the best logicaldecision relative to considering data transformations or not.

In his last article, Wheeler commented: "However, rather than offering a critique of the pointsraised in my original article, he [Breyfogle] chose to ignore the arguments againsttransforming the data and to simply repeat his mantra of transform, transform, transform.' "

In fact, I had agreed with Wheelers stated position that, "If a transformation makes senseboth in terms of the original data and the objectives of the analysis, then it will be okay to usethat transformation."

What I did take issue with was his statement: " Therefore, we do not have to pre-qualify ourdata before we place them on a process behavior chart. We do not need to check the data fornormality, nor do we need to define a reference distribution prior to computing limits. Anyonewho tells you anything to the contrary is simply trying to complicate your life unnecessarily."

In my article, I stated "I too do not want to complicate peoples lives unnecessarily; however, itis important that someones over-simplification does not cause inappropriate behavior."

Wheeler kept criticizing my random data set parameter selection and the fact that I did notuse real data. However, Wheeler failed to comment on the additional points I made relative toaddressing a fundamental issue that was lacking in his original article, one that goes beyondthe transformation question. This important issue is the reporting of process performancerelative to customer requirement needs, i.e., a goal or specification limits.

This article will elaborate more on the topic using real data which will lead to the sameconclusion as my previous article: For some processes an appropriate transformation is avery important step in leading to the most suitable action or non-action. In addition, this articlewill describe how this metric reporting system can create performance statements, offering aprediction statement of how the process is performing relative to customer needs. It isimportant to reiterate that appropriate transformation selection, when necessary, needs to be




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part of this overall system.

In his second article, Wheeler quoted analysis steps from Shewharts Economic Control ofQuality of Manufactured Product.These steps, which were initiated over seven decades ago,focus on an approach to identify out-of-control conditions. However, the step sequence didnot address whether the process output was capable of achieving a desired performanceoutput level, i.e., expected process non-conformance rate from a stable process.

Wheeler referenced a study " encompassing more than 1,100 probability models where97.3 percent of these models had better than 97.5-percent coverage at three-sigma limits."From this statement, we could infer that Wheeler believes that industry should be satisfiedwith a greater than 2-percent false-signal rate. For processes that do not follow a normaldistribution, this could translate into huge business costs and equipment downtime searchingfor special-cause conditions that are in fact common-cause. After an organization chasesphantom special-cause-occurrences over some period of time, it is not surprising that theywould abandon control charting all together.

Wheeler also points out how traditional control charting has been around for more than 70

years and how the limits have been thoroughly proven. I don't disagree with the threesampling standard deviation limits; however, how often is control charting really usedthroughout businesses? In the 1980s, there was a proliferation of control charts; however,look at us now. Has the usage of these charts continued and, if they are used, are they reallyapplied with the intent originally envisioned by Shewhart? I suggest there is not thewidespread usage of control charts that quality-tools training classes would lead students tobelieve. But why is there not more frequent usage of control charts in both manufacturing andtransactional processes?

To address this underutilization, I am suggesting that while the four-step Shewhartmethodology has its applicability, we now need to revisit how these concepts can better be

used to address the needs of today's businesses, not only in manufacturing but transactionalprocesses as well.

To assess this tool-enhancement need, consider that the output of a process (Y) is a functionof its inputs (Xs) and its step sequence, which can be expressed as Y=f(X). The primarypurpose of Wheeler's described Shewhart four-step sequence is to identify when a specialcause condition occurs so that an appropriate action can be immediately taken. Thisprocedure is most applicable to the tracking of key process input variables (Xs) that arerequired to maintain a desired output process response for situations where the process hasdemonstrated that it provides a satisfactory level of performance for the customer-drivenoutput, i.e., Y.

However, from a business point of view, we need to go beyond what is currently academicallyprovided and determine what the enterprise needs most as a whole. One business-management need is an improved performance reporting system for both manufacturing andtransactional processes. This enhanced reporting system needs to structurally evaluate andreport the Youtput of a process for the purpose of leading to the most appropriate action ornon-action.

An example application transactional process need for such a charting system is telephonehold time in a call center. For situations like this, there is a natural boundary: hold time cannotget below zero. For this type of situation, a log-normal transformation can often be used to

describe adequately the distribution of call-center hold time. Is this distribution a perfect fit?




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No, but it makes physical sense and is often an adequate representation of what physicallyhappens; that is, a general common-cause distribution consolidation bounded by zero with atail of hold times that could get long.

From a business view point, what is desired at a high level is a reporting methodology thatdescribes what the customer experiences relative to hold time, the Youtput for this process.This is an important business requirement need that goes beyond the four-step Shewhart

process.

I will now describe how to address this need through an enhancement to the Shewhart four-step control charting system. This statistical business performance charting (SBPC)methodology provides a high-level view of how the process is performing. With SBPC, we arenot attempting to manage the process in real time. With this performance measurementsystem, we consider assignable-cause differences between sites, working shifts, hours of theday, and days of the week to be a source of common-cause input variability to the overallprocessin other words, Deming's responsibility of management variability.

With the SBPC system, we first evaluate the process for stability. This is accomplished using

an individuals chart where there is an infrequent subgrouping time interval so that inputvariability occurs between subgroups. For example, if we think that Monday's hold time couldbe larger than the other days of the week because of increased demand, we should considerselecting a weekly subgrouping frequency.

With SBPC, we are not attempting to adjust the number of operators available to respond tophone calls in real time since the company would have other systems to do that. What SBPCdoes is assess how well these process-management systems are addressing the overallneeds of its customers and the business as a whole. The reason for doing this is to determinewhich of the following actions or non-actions are most appropriate, as described in Table 1.

The four-step Shewhart model that Wheeler referenced focuses only on step number one.

In my previous article, I used randomly generated data to describe the importance ofconsidering a data transformation when there is a physical reason for such a consideration. Ithought that it would be best to use random data since we knew the answer; however,Wheeler repeatedly criticized me for selecting a too-skewed distribution and not using realdata.

I will now use real data, which will lead to the same conclusion as my previous article.

Real-data Example

Table 1:Statistical Business Performance Charting (SBPC) ActionOptions

1. Is the process unstable or did something out of the ordinaryoccur, which requires action or no action?

2. Is the process stable and meeting internal and externalcustomer needs? If so, no action is required.

3. Is the process stable but does not meet internal and externalcustomer needs? If so, process improvement efforts areneeded.




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A process needs to periodically change from producing one product to producing another. It isimportant for the changeover time to be as small as possible since the production line will beidle during changeover. The example data used in this discussion is a true enterprise view ofa business process. This reports the time to change from one product to another on a processline. It includes six months of data from 14 process lines that involved three different types ofchangeouts, all from a single factory. The factory is consistently managed to rotate throughthe complete product line as needed to replenish stock as it is purchased by the customers.

The corporate leadership considers this process to be predictable enough, as it is run today,to manage a relatively small finished goods inventory. With this process knowledge, what isthe optimal method to report the process behavior with a chart?

Figure 1 is an individuals chart of changeover time. From this control chart, which has notransformation as Wheeler suggests, nine incidents are noted that should have beeninvestigated in real time. In addition, one would conclude that this process is not stable or isout of control. But, is it?

Figure 1: Individuals Chart of Changeover Time (Untransformed Data)

One should note that in Figure 1 the lower control limit is a negative number, which makes nophysical sense since changeover time cannot be less than zero.

Wheeler makes the statement, "Whenever we have skewed data there will be a boundaryvalue on one side that will fall inside the computed three-sigma limits. When this happens, theboundary value takes precedence over the computed limit and we end up with a one-sidedchart." He also says, "The important fact about nonlinear transformations is not that theyreduce the false-alarm rate, but rather that they obscure the signals of process change."

It seems to me that these statements can be contradictory. Consider that the response thatwe are monitoring is time, which has a zero boundary, and where a lower value is better. Formany situations, our lower-control limit will be zero with Wheelers guidelines (e.g., Figure 1).Consider that the purpose of an improvement effort is to reduce changeover time. Animproved reduction in changeover time can be difficult to detect using this one-sided control

chart, when the lower-control limit is at the boundary conditionzero in this case.




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Wheeler's article makes no mention of what the process customer requirements are or thereporting of its capability relative to specifications, which is an important aspect of lean SixSigma programs. Let's address that point now.

The current process has engineering evaluating any change that takes longer than twelvehours. This extra step is expensive and distracts engineering from its core responsibility. Theorganization's current reporting system does not address how frequently this engineering

intervention occurs.

In his article, Wheeler made no mention of making such a computation for either normal ornon-normal distributed situations. This need occurs frequently in industry when processes areto be assessed on how they are performing relative to specification requirements.

Let's consider making this estimate from a normal probability plot of the data, as shown inFigure 2. This estimate would be similar to a practitioner manually calculating the value usinga tabular z-value with a calculated sample mean and standard deviation.

Figure 2:Probability Plot of the Untransformed Data

We note from this plot how the data do not follow a straight line; hence, the normal distributiondoes not appear to be a good model for this data set. Because of this lack-of-fit, thepercentage-of-time estimate for exceeding 12 hours is not accurate; i.e., 46% (100-54 = 46).

We need to highlight that technically we should not be making an assessment such as thisbecause the process is not considered to be in control when plotting untransformed data. Forsome, if not most, processes that have a long tail, we will probably never appear to have anin-control process, no matter what improvements are made; however, does that make sense?

The output of a process is a function of its steps and input variables. Doesnt it seem logical toexpect some level of natural variability from input variables and the execution of process

steps? If we agree to this assumption, shouldnt we expect a large percentage of process




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output variability to have a natural state of fluctuation; i.e., be stable?

To me this statement is true for most transactional and manufacturing processes, with theexception of things like naturally auto-correlated data situations such as the stock market.However, with traditional control charting methods, it is often concluded that the process isnot stable even when logic tells us that we should expect stability.

Why is there this disconnection between our belief and what traditional control charts tell us?The reason is that underlying control-chart-creation assumptions and practices are often notconsistent with what occurs naturally in the real world. One of these practices is not usingsuitable transformations when they are needed to improve the description of processperformance, for instance, when a boundary condition exists.

It is important to keep in mind that the reason for process tracking is to determine whichactions or non-actions are most appropriate, as described in Table 1. Lets now return to ourreal-data example analysis.

For this type of bounded situation, often a log-normal distribution will fit the data well, since

changeover time cannot physically go below a lower limit of zero, such as the previouslydescribed call-center situation. With the SBPC approach, we want first to assess processstability. If a process has a current stable region, we can consider that this process ispredictable. Data from the latest region of stability can be considered a random sample of thefuture, given that the process will continue to operate as it has in the recent past's region ofstability.

When these continuous data are plotted on an appropriate probability plot coordinate system,a prediction statement can be made: What percentage of time will the changeover take longerthan 12 hours? Figure 3 shows a control chart in conjunction with a probability plot. A netting-out of the process analysis results is described below the graphics: The process is predictable

where about 38 percent of the time it takes longer than 12 hours.

Unlike the previous non-transformed analysis, we would now conclude that the process isstable, i.e., in control. Unlike the non-transformed analysis, this analysis considers that theskewed tails, which we expect from this process, to be the result of common-cause processvariability and not a source for special cause investigation. Because of this, we conclude, forthis situation, that the transformation provides an improved process discovery foundationmodel to build upon, when compared to a non-transformed analysis approach.




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Lets now compare the report-outs of both the untransformed (Figure 1) and transformed data

(Figure 3). Consider what actions your organization might take if presented each of thesereport-outs separately.

The Figure 1 report can be attempting to explain common cause events as though eachoccurrence has an assignable cause that needs to be addressed. Actions resulting from thisline of thinking can lead to much frustration and unnecessary process-procedural tamperingthat result in increased process-response variability, as Deming illustrated in his funnelexperiment.

When Figure 3's report-out format is used in our decision making process, we would assessthe options in Table 1 to determine which action is most appropriate. With this data-

transformed analysis, number three in Table 1 would be the most appropriate action,

The process is predictable where about 38% percent of the time ittakes longer than 12 hours.

Figure 3:Report-out.




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assuming that we consider the 38 percent frequency of occurrence estimate above 12 hoursexcessive.

Wheeler stated, "If you are interested in looking for assignable causes you need to use theprocess behavior chart (control chart) in real time. In a retrospective use of the chart you areunlikely to ever look for any assignable causes "

This statement is contrary to what is taught and applied in the analyze phase of lean SixSigmas define-measure-analyze-improve-control (DMAIC) process improvement projectexecution roadmap. Within the DMAIC analyze phase, the practitioner evaluates historicaldata statistically for the purpose of gaining insight into what might be done to improve hisprocess improvement projects process.

It can often be very difficult to determine assignable causes with a real-time-search-for-signals approach that Wheeler suggests, especially when the false signal rate can beamplified in situations where an appropriate transformation was not made. Also, with thisapproach, we often do not have enough data to test that the hypothesis of a particularassignable cause is true; hence, we might think that we have identified an assignable cause

from a signal, but this could have been a chance occurrence that did not, in fact, negativelyimpact our process. In addition, when we consider how organizations can have thousands ofprocesses, the amount of resources to support this search-for-signal effort can be huge.

Deming in Out of the Crisisstated, "I should estimate that in my experience most troubles andmost possibilities for improvement add up to proportions something like this: 94 percentbelong to the system (responsibility of management), 6 percent [are] special."

With a search-for-signal strategy it seems as if we are trying to resolve the 6 percent of issuesthat Deming estimates. It would seem to me that it would be better to focus our efforts on howwe can better address the 94 percent common-cause issues that Deming describes.

To address this matter from a different point of view, consider extending a classicalassignable cause investigation from special cause occurrences to determining what needs tobe done to improve a process common-cause variability response if the process does notmeet the needs of the customer or the business.

I have found with the SBPC reporting approach that assignable causes that negatively impactprocess performance from a common-cause point of view can best be determined bycollecting data over some period of time to test hypotheses that assess differences betweensuch factors as machines, operators, day of the week, raw material lots, and so forth. Whenundergoing process improvement efforts, the team can use collected data within the most

recent region of stability to test out compiled hypothesis statements that it thinks could affectthe process output level. These analyses can provide guiding light insight to process-improvement opportunities. This is a more efficient analytical discovery approach than asearch-for-signals strategy where the customer needs are not defined or addressed relative tocurrent process performance.

For the example SBPC plot in Figure 3, the team discovered through hypotheses tests thatthere was a significant difference in the output as a function of the type of change, the shiftthat made the change, and the number of performed tests made during the change.

This type of information helps the team determine where to focus its efforts in determining

what should be done differently to improve the process. Improvement to the system would be




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demonstrated by a statistically significant shift of the SPBC report-out to a new-improved levelof stability. This system of analysis and discovery would apply for both processes that needdata transformation and those that don't, that is, the SPBC system highlights and does notobscure the signals of process change.

Detection of an Enterprise Process Change

Wheeler stated, "However, in practice, it is not the false-alarm rate that we are concernedwith, but rather the ability to detect signals of process changes. And that is why I used realdata in my article. There we saw that a nonlinear transformation may make the histogram lookmore bell-shaped, but in addition to distorting the original data, it also tends to hide all of thesignals contained within those data."

Let's now examine how well a shift can be detected with our real-data example, using bothnon-transformed and transformed data control charts. A traditional test for a process controlchart is the average run length until an out-of-control indication is detected, typically for a onestandard deviation shift in the mean. This does not translate well to cycle-time-based datawhere there are already values near the natural limit of zero. For our analysis, we will assume

a 30-percent reduction in the average cycle time to be somewhat analogous to a shift of onestandard deviation in the mean of a normally distributed process.

In our sample data, there were 590 changeovers in the six-month period. The 30-percentreduction in cycle time was introduced after point 300. A comparison of the transformed anduntransformed process behavior charting provides a clear example of the benefits of thetransformation, noting that, for the non-transformed report-out, a lower control limit of zero,per Wheeler's suggestion, was included as a lower bound reference line.

In Figure 4a, the two charts show the untransformed data analysis, the upper chart appearingto have become stable after the simulated process change. When a staging has been created(lower chart in Figure 4a), the new chart stage identifies four special cause incidents thatwere considered common-cause events in the transformed data set, as shown in the lowerchart in Figure 4b.




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Figure 4a: Non-transformed Analysis of a Process Shift




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Figure 4b:Transformed Analysis of a Process Shift

The two transformed charts in Figure 4b show a change in the process with the introduction ofthe special-cause indications after the simulated change was implemented.

When staging is introduced into this charting, the special-cause indications are eliminated;i.e., the process is considered stable after the change.

Using Wheelers guidelines, a simple reduction in process cycle time would appear to be theremoval of special causes. This is fine in the short term, but as we collect more data on thenew process and introduce staging into the data charting we would surely, for skewedprocess performance, return to a situation were special-cause signals would reoccur, asshown in the lower chart in Figure 4a. We should highlight that these special cause eventsappear as common-cause variability in the transformed-data-set analysis shown in the lowerchart in Figure 4b.

If you follow a guideline of examining a behavioral chart that has an appropriatetransformation, you would have noted a special cause occurring when the simulated process




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change was interjected. From this analysis, the success of the process improvement effortwould have been recognized and the process behavior chart limits would be re-set to newlimits that reflect the new process response level.

Since the transformed data set process is stable, we can report-out a best estimate for howthe process is performing relative to the 12 hour cycle time criteria. In Figure 5, the SBPCdescription provides an estimate of how the process performed before and after the change,

where 100-63 = 37% and 100-87 = 13%. Since the process has a recent region of stability,the data from this recent region can provide a predictive estimate of future performanceunless something changes within the process and/or its inputs; i.e., our estimate is that 13percent of the cycle times will be above 12 hours.

The process has been predictable since observation 300 wherenow about 13% of the time it takes longer than 12 hours for achangeover. This 13% value is a reduction from about 37% before

the new process change was made.




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I wonder if much of the dissatisfaction and lack of use of business control charting derive fromthe use of non-transformed data. Immediately after the change, the process looks to be good,and the improvement effort is recognized as a success. However, in a few weeks or monthsafter the control chart has been staged, the process will show to be out of control againbecause the original data is no longer the primary driver for the control limits. Theorganization will assume the problem has returned and possibly consider the earlier effort tonow be a failure.

In addition, Wheelers four-step Shewhart process made no mention of how to assess how

well a process is performing relative to customer needs, a very important aspect in the realbusiness world.

Conclusions

The purpose of traditional individuals charting is to identify in a timely fashion when special-cause conditions occur so that corrective actions can be taken. The application of thistechnique is most beneficial when tracking the inputs to a process that has an output levelwhich is capable of meeting customer needs.

False signals can occur if the process measurement by nature is not normally distributed, for

example, in processes that cannot be below zero. Investigation into these false signals can beexpensive and lead to much frustration when no reason is found for out-of-control conditions.

The Wheeler suggested four-step Shewhart process has its application; however, a morepressing issue for businesses is in the area of high-level predictive performancemeasurements. SBPC provides an enhanced control charting system that addresses theseneeds; e.g., an individuals chart in conjunction with an appropriate probability plot forcontinuous data. Appropriate data transformation considerations need to be part of the overallSBPC implementation process.

With SBPC, we are not limited to identifying out-of-control conditions but also are able to

report the capability of the process in regions of stability in terms that everyone canunderstand. With this form of reporting, when there is a recent region of stability, we canconsider data from this region to be a random sample of the future. With statistical businessperforming charting approach, we might be able to report that our process has been stable forthe last 14 weeks with a prediction that 10 percent of our incoming calls will take longer than agoal of one minute.

I expect that the one real issue behind this entire discussion is the idea of "what is goodenough?" Wheeler shared a belief that a control charting method that allows up to 2.5 percentof process measures to trigger a cause and corrective action effort that will not find a truecause in a business as "good enough." Wheeler goes so far as to relate the 95 percent

confidence concept from hypothesis testing to imply that up to 5 percent false special cause

Figure 5:SBPC Report-out Describing Impact of ProcessChange.




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detections are acceptable. Using the above-described concept of transforming process datawhere the transformation is appropriate for the process-data type will lead to processbehavior charting that matches the sensitivity and false-cause detection that we have alllearned to expect when tracking normally distributed data in a typical manufacturingenvironment.

Why would anyone want to have a process behavior chart that will be interpreted differently

for each use in an organization? The answer should be clear: use transformations when theyare appropriate and then your organization can interpret all control charts in the samemanner. Why be "good enough" when you have the ability to be correct?

The "to transform or not transform" issue addressed in this paper led to SBPC reporting andits advantages over the classical control-charting approach described by Wheeler. However,the potential for SBPC predictive reporting has much larger implications than reporting awidget manufacturing process output.

Traditional organizational performance measurement reporting systems have a table ofnumbers, stacked bar charts, pie charts, and red-yellow-green goal-based scorecards that

provide only historical data and make no predictive statements. Using this form of metricreporting to run a business is not unlike driving a car by only looking at the rear view mirror, adangerous practice.

When predictive SBPC system reporting is used to track interconnected business processmap functions, an alternative forward-looking dashboard performance reporting systembecomes available. With this metric system, organizations can systematically evaluate futureexpected performance and make appropriate adjustments if they don't like what they see, notunlike looking out a car's windshield and turning the steering wheel or applying the brake ifthey don't like where they are headed.

How SBPC can be integrated within a business system that analytically/innovativelydetermines strategies with the alignment of improvement projects that positively impact theoverall business will be described in a later article.

Quality Insider Column Six Sigma Column Twitter Ed

Source URL (retrieved on 09/26/2009):http://www.qualitydigest.com/inside/six-sigma-column/not-transforming-data-can-be-fatal-your-analysis.html

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