1 2002 North Haven Group Lean Six Sigma DMAIC: Measure Phase.

1 2002 North Haven Group

Lean Six Sigma

DMAIC: Measure Phase


Six Sigma Improvement Process Descriptive Statistics: Nominal Data

• Introduction

• Pareto Charts

• Summary of Methods for Displaying Data


Objectives

After completing this section, participants should be able to:

Construct and interpret Pareto charts;

Realize the value of Pareto charts for “drilling down,” thereby stratifying causes and focusing process improvement efforts;

Evaluate ways of “weighting” the categories in constructing a Pareto chart;

Interpret a mosaic plot;

Understand how to use Frequency and Weight variables in constructing charts for nominal data.


Introduction

In the Measure and Analyze phase, teams often begin to identify which factors are likely to have an effect on their KPI.

In this section, we address graphical output for nominal data that is especially relevant to this aspect of a project team’s work.

For nominal factors, the main tool is the Pareto chart.

Pareto charts can be constructed in terms of frequency of occurrence or using weights (such as costs).

Relationships between nominal factors are also often of interest.

Mosaic plots are useful in studying such relationships.


Pareto Charts

The distribution of nominal data can be illustrated using bar charts.

However, in improvement efforts one is almost always interested in which categories of nominal data occur most frequently:

What are the most frequent causes of delays?

What are the most frequent causes of defective items?

What are the most costly sources of errors?

A Pareto chart arranges categories in order of decreasing frequency so that the most frequently occurring categories are easily identified.


Pareto ChartsIn constructing a Pareto chart, the following steps are taken:

1. Decide what to investigate.

2. Decide the time period to be studied.

3. Collect the data.

4. Calculate the frequency of each category.

5. Arrange the categories from largest to smallest frequency.

6. Graph the data.

At least 50 data points are usually required for a meaningful diagram.

Note: The time period should not be so short that not enough data is available. The time period should also not be so long that the process has changed substantially.


Pareto Charts

Missing Parts and Damaged are the two most frequent reasons for returned shipments, accounting for about 56% of returns.

Note the “Other” category. If such a category is large, it needs to be investigated.


Pareto Charts

We now construct a Pareto by Product.

Products C and I account for the largest numbers of returned shipments.

Note that the cumulative percent curve is not shown on this chart.


Pareto Charts

Pareto charts provide an effective means for evaluating progress towards team goals.

Suppose a team decided to focus its efforts on reducing returns for products C and I. Below, we see pareto plots for the three month period prior to and the three month period following implementation of process changes.

Evidence of improvement


Pareto Charts

Pareto charts can be used in levels to continue to break down, or stratify, problems or causes.

The first level of a Pareto diagram may include types of problems, the second level may break down the primary type of problem into reasons or causes. These causes or reasons can be broken down until the underlying cause is identified.

10

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Types of reject

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Reasons for Main Type of Reject


Pareto Charts

We look for the following types of patterns when interpreting a Pareto chart:

0

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50

60

0

10

20

30

40

50

60

One primary problem or cause.

Easy to identify.

Stratify this one factor to breakdown further.

2 or 3 primary problems or causes.

Priorities may depend on cost or ease of improvement.

May need to focus on more than one problem or cause.


Pareto Charts

Interpreting Pareto charts (continued):

No primary factors - not useful to prioritize.

Need to summarize data differently.

Miscellaneous or ‘other’ category is major.

Look more closely at breaking down miscellaneous into specific problems or causes.

0

5

10

15

20

0

5

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25

30


Pareto ChartsPareto charts have many benefits:

They can be used to help establish priorities and force you to look at the process from different points of view.

They can help re-define the critical outputs, or ‘Y’s.

They can uncover clues to potential problems or causes.

They can illustrate the effects of changes to a process.

There are also some major limitations and difficulties:

Interpretations may be incorrect if the process is not stable.

Since data is summarized over time, Pareto charts give no indication as to whether a process is stable or not.


Pareto Charts

Limitations and other comments:

The ‘biggest’ problem isn’t necessarily the problem that should be addressed first.

For example:

The costs associated with smaller categories may be greater than those for larger categories (data should be also stratified by value or associated cost),

Smaller problems may be more important or troublesome to the customer, or

Some problems may be much easier to tackle than others due to financial, political, or other reasons.


Summary of Methods for Displaying DataThe following table summarizes the graphical methods that

we’ve covered thus far.

X (Factor) Y (Response) Display Method

None Continuous Histogram

None Nominal Pareto Chart

Continuous Continuous Scatter Plot

Time Order Continuous Run Chart (Or Control Chart)

Note: If data is summarized by counts, put counts in “Freq” field. If there is a weighting variable, enter it in the “Weight” field.


Problem Solving Tools

Team Techniques: Measure and Analyze Phases


Six Sigma Improvement Process Basic Problem Solving Tools

• Introduction• Brainstorming• Multi-Voting• Affinity Diagrams• Cause-and-Effect Diagrams• Cause-and-Effect Matrices• Communication Plan• Skills Practice


Introduction

• The Six Sigma Improvement Process utilizes various problem solving tools for generating information and exploring potential sources of variability.

• Brainstorming is probably the best known method for generating a list of potential causes, factors, or solutions.

• Multi-Voting, Nominal Group Technique and Affinity Diagrams are common methods for generating lists ideas and for narrowing down the list to a few ideas or groupings of ideas.

• Cause-and-Effect Diagrams and Cause-and-Effect Matrices are most commonly used to identify potential causes of problems.


Brainstorming• Brainstorming is an effective way of generating ideas within a

team. It can be used to identify potential problems, experimental factors, or possible solutions.

• A successful brainstorming session allows team members to be creative, allowing them to generate a wide range of ideas.

• General ground rules for conducting a brainstorming session are:

– Solicit quantity– Encourage participation– Welcome exaggeration– Build on ideas– Don’t criticize or evaluate– Record ideas


Brainstorming

• Brainstorming sessions are most effective when guided by a facilitator (or a team member acting as facilitator). In general, brainstorming follows a given sequence of events:

1. Review the topic: Define the subject.

2. Give everyone a minute or two to think about the problem and to generate a list of ideas (in silence). Notify members in advance of the meeting to give them a chance to come prepared - this helps generate participation.

3. Either in a round robin format (moving around the table) or using free-form, invite everyone to offer their ideas (one participant at a time, one idea at a time).

4. Record all ideas on a flipchart. No idea is a bad idea!


Multi-Voting

• Multi-Voting is a simple way to select the most important idea from a list, and is an effective technique for teams that find it difficult to reach a consensus.

• Multi-Voting starts with a brainstorming session, and ends in a series of votes to narrow down the list of items to a manageable number.

• To conduct a multi-vote:

1. Generate a list of items.

2. Review the items, and combine them if the group agrees that they are the same (don’t go overboard).

3. Number the items.


Multi-Voting

• Multi-Voting Steps (Continued):

4. Have each member write down the numbers of the items they think are important (around 1/3 of the total number of items).

5. Review the individual lists to tally votes.

6. Reduce the original list by eliminating items with no or few votes.

7. Repeat steps 5 through 7 on the remaining list until only a few items remain.

If needed, the group can discuss which item should be the top priority, or the group can have one final vote.


Affinity Diagrams• Using an affinity diagram is an alternative method for generating a

list of ideas and for narrowing the list to a few general groupings of ideas.

• Creating an affinity diagram consists of two major steps: Idea generation and idea consolidation.

• 1. Idea Generation. This is essentially a brainstorming session.

Each member first generates a list of ideas;

Using a round-robin format, members post ideas on a flipchart (using sticky notes).

Ideas are briefly introduced as they are posted, but they are not evaluated.


Affinity Diagrams


Affinity Diagrams

2. Idea Consolidation: Ideas are reviewed and consolidated into common ‘themes’.

The team reviews the ideas posted on the flip chart;

Ideas are clarified;

Ideas are consolidated into ‘like’ themes. Sticky notes are physically moved into groupings;

For example, ideas may be related to a specific area of the business, or the ideas may express similar content.

Finally, the themes are reviewed and named.

The end result is a short list, containing 3-6 general categories of ideas.


Affinity Diagrams

Theme 1 Theme 2 Theme 3

Theme 4 Theme 5


Affinity DiagramsComments on affinity diagrams:

The team may encounter ideas that do not fit into any of the categories.

The group will need to make a decision whether to discard the ideas, force them into an existing category, or to create a new category.

During the consolidation step, an alternative approach is to have each team member form general categories of ideas.

These categories are then reviewed by the team to make final decisions on groupings.

This approach allows for greater involvement and buy-in from individual members.


Affinity Diagram

• Example: A design team is considering a new type of child’s safety seat for automobiles (from “Quality Function Deployment”, A Practitioners Approach by James L. Bossert).

• During the early stages of a QFD, exercise they decided to brainstorm the design requirements for the new safety seat.

• The next slide contains the affinity diagram generated from their brainstorming activities.

• The large rectangular areas represent larger categories, while the interior rectangles indicate subcategories.

• In this example, several iterations were required in order to achieve the final groupings.


Affinity Diagram


Cause-and-Effect Diagrams

• A variation of brainstorming that provides grouping of ideas into logical categories and drills down to root causes is a Cause-and-Effect Diagram.

• This type of diagram was initially used by Kaoru Ishikawa, and is thus often referred to as an “Ishikawa Diagram”.

• The diagram is used as a tool to identify potential root causes of a problem, and is best used after the process or problem has been well-defined.

• Since a cause and effect diagram organizes potential causes ‘visually’, the technique allows people to easily see the relationship between factors.


Cause-and-Effect Diagrams• Cause-and-Effect Analysis Steps:

• 1. Define the problem: Make sure the process has been described and the problem is well defined.

• 2. Define Categories: List categories of potential root causes.

• 3. Brainstorm Possible Causes: Record these as subcategories.

• For each subcategory ask “Why” several times to drill down to sub-subcategories and sub-sub-subcategories.

• 4. Identify Most Probable Root Causes: Reach a consensus on the most likely causes.

• 5. Verify Most Probable Root Causes: Use DOE or other statistical techniques to verify the causes of the problem.



• The Cause-and-Effect Diagram consists of a main arrow pointing to a problem. This arrow branches off into main categories of potential causes (or solutions).

• Typical categories are equipment (or machinery), personnel (or manpower), methods (or procedures), materials, and environment.

• Teams can customize these categories to fit their needs.

• Smaller arrows identifying subcategories branch off each main branch; causes related to further subcategories are identified by arrows pointing to the higher level category, etc.

• The following slide provides an example of the basic structure of a fishbone diagram:



Generic Cause-and-Effect Diagram:

Problemor Effect

MethodMaterial

Machine Manpower

Cause

Cause

Cause

Cause

Cause Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

Cause

CauseEnvironment

Cause

Cause



In the following example, ‘variation in coating thickness’ was identified as the number one customer complaint.

The improvement team started by reviewing the definition of ‘variation in coating thickness’ to ensure that all members understood the purpose of the cause-and-effect analysis.

The team then agreed to use the major categories of causes: Materials, methods, machinery, environment and manpower.

Possible causes were brainstormed, and the ideas were recorded in the categories that seemed most suitable.

The team reviewed the diagram, and agreed that the most probable causes of incorrect thickness were ‘inconsistent methods’ and ‘insufficient training’.



Cause-and-Effect Diagram for ‘Variation in Coating Thickness’:

Varia tionin C oatingTh ickness

M ethodsM ateria l

M ach inery M anpower

Varia tion

N ot C a lib ra ted

Insuffic ientT ra in ing Absentee ism

ProceduresU nclear

P rocessingSpeed

Too l W ear

Age

Environm ent

Form ula tion

H um id ity

Tem pExperience

Inconsisten tM ethods


Cause-and-Effect DiagramsCause-and-effect diagrams have a number of benefits:

• They help identify root causes;

• They are a helpful way to brainstorm ideas;

• The fishbone diagram helps categorize ideas;

• They are a useful way to document initial ideas and point teams in the direction of possible causes.

The limitations of cause-and-effect diagrams include:

• They are based on theories, which must be tested;

• They cannot substitute for the use of real data;

• Teams may get caught up in the details, such as which category to use, instead of focusing on generating ideas.



An effective approach to generating multi-level cause-and-effect diagrams is known as “The 5 Whys”.

The 5 Whys helps a team get to the root cause of a problem.

For each cause listed, the team should ask “why”? Example:

Problem: Low machine through-put

Why? Machine downtime

Why? A critical component breaks down frequently

Why? The component is not properly lubricated

Why? Lack of maintenance

Why? The component is not part of the preventativemaintenance schedule


Cause-and-Effect Matrix

Cause-and-effect matrices (XY Matrices), introduced earlier, are another tool for identifying potential root causes of problems.

Unlike cause-and-effect diagrams, XY matrices allow us to identify potential causes for multiple problems, and provide a formal method for evaluate potential causes.

XY matrices can also be used to identify the most important inputs to a process.

Causes (or process inputs) are rated on the basis of ‘impact’ on the individual effects (or process outputs).

Causes are rated from 0-3, where 0 is ‘no impact’ and 3 is ‘high impact’. Effects are weighted based on importance to the customer.


Cause-and-Effect Matrix

Here, Effect 1 is considered the most important to the customer.

The ratings for each cause are multiplied by the weight for each effect. These are then summed for each cause.

The “Total by Input” helps us target factors that are believed to have the biggest impact on the outputs. In this example, Cause 1 and Cause 2 appear to have the greatest impact overall.

Effect 1 Effect 2 Effect 3 Effect 4 Effect 5 Effect 6

Weight 4 2 0.5 1.5 1 1 10

Cause 1 2 3 2 3 2 0 21.5

Cause 2 3 2 1 0 0 3 19.5

Cause 3 1 2 1 3 2 0 15

Cause 4 1 3 1 2 2 0 15.5

Cause 5 2 2 0 3 0 0 16.5

Effects - OutputsTotal by

Input (Weighted)

Causes - Inputs


Skills Practice

• In this exercise, we will once again break up into teams.

• Each team will develop a cause-and-effect diagram for the following problem: Delays in getting to work on time.

• Refer to the section, “Cause-and-Effect Diagram” for details.

• You will have approximately 40 minutes.

• Be prepared to give a brief presentation to the group.


Continuous Data

Estimator and Control Charts: Continuous Data


When Is An Estimator Good?

Accurate but not as

Precise

NotAccurate

and Not

Precise

Accurate and

Precise

Ouch!!!

NotAccurate

but Precise

Whew!

Biased!



An estimate is biased if, when averaged over all possible samples, it is centered at a value other than the one it is estimating.

The bias is the difference between the value of the parameter being estimated and the average value of estimates that might be obtained from different samples.

Suppose that a research group sends customers questionnaires to assess their satisfaction with the company’s service, and that the customers who are most dissatisfied do not respond to the survey.

Then the estimate of satisfaction will overstate satisfaction, even though the sample is chosen at random. It will be biased.


Precision refers to obtaining consistent estimates in repeated sampling.

The precision of an estimate almost always depends upon the sample size.

The larger the sample, the more precise the estimate (equivalently, the smaller the error or estimation).

We are interested in whether the estimate comes close to the parameter that it is estimating; in other words, we are always interested in the precision of our estimates.

A sample gives only partial information about the population, and so the resulting estimate hardly ever equals the population parameter of interest.



To understand this concept at a deeper level, one needs to understand the Rule of Averages.

The Rule of Averages is a result that helps us determine how close an estimate is to the parameter value it is estimating.

It gives a “yardstick” for calculating how far off the estimate is likely to be (how large an error) at a given probability level.

The Rule of Averages states that an estimate is more likely to be close to the population mean if it is based on a larger sample than if it is based on a smaller sample.

The Rule of Averages



Suppose that the population of interest consists of 100,000 sales of a certain type, and that we are interested in cycle time.

Cycle time is defined as the number of days between the customer’s first inquiry and delivery to the customer.

Suppose that, were all of these sales to be examined, the distribution of cycle times would be given by the histogram to the left.

Note that the mean cycle time is 190.33, and that the distribution is skewed.



However, it might be too time consuming to determine cycle times for the entire population.

Suppose that we examine a random sample of these sales.

We consider selecting a random sample of size 10, and using it to estimate the mean.

To see the general behavior of such samples, 50 samples of size 10 were selected, and their means computed.

On the following slide, we have plotted the raw values and means for each of the 50 samples.



Compare the raw values to the sample means.

Note how the means are “in the middle,” less variable, and more normal than the raw values.

Day

s

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Sample

Oneway Analysis of Days By Sample


The Rule of AveragesCompare the histogram for the sample means to that of the raw

values. Where are the sample means centered? What is their spread? What is the shape of their distribution?

Distribution of Raw Values

Sample Means for 50 Samples of Size 10


The Rule of AveragesNow we examine the distribution of sample means for samples of

size 10, 25, 50, and 100.Each histogram is based on the sample means for 100 samples of

the specified size.


The Rule of AveragesWhatever the sample size, the sample means are roughly centered at the

population mean.

As the sample size increases, the sample means tend to be less spread out and closer to the population mean.

As the sample size increases, the sample means have a distribution that is more and more normal, despite the fact that the underlying population is quite skewed.

Even for relatively small sample sizes, the distribution of the sample means is approximately normal.

It is this result that allows us to compute how far from the population mean a sample mean is likely to fall.



The following slide shows two histograms.

The one at the far left consists of the 100,000 observations on our hypothetical population.

Note that the population mean, for all intents and purposes, is 190, and that the population standard deviation is 115.

Samples of size 25 are generated, their means computed, and the distribution shown in the second histogram.

Note that the sample means are centered, essentially, at 190.

Also note that the standard deviation for the sample means.



115 / 25 23.0

The means of the samples of size 25 are centered at the mean of the population.

The standard deviation of the sample means is approximately the standard deviation of the population divided by the square root of 25:



Specifically, the Rule of Averages (or the Central Limit Theorem) states:

Suppose that X has a distribution with population mean and population standard deviation .

Then sample means of samples of size n drawn from the distribution of X:

(a) Have population mean ;(b) Have population standard deviation ; (c) Have a distribution that becomes more normal as the

sample size n gets larger.

/ n



Why is this rule important?

It explains why we so often average sample data:

The standard deviation of the sample mean is always less than the standard deviation of the raw data (by a factor of the square root of the sample size).

We get more precise estimates of population behavior by taking samples. Larger samples give more precise estimates than smaller samples.

Averaging is a very effective noise filter.

Sample means are normally distributed.


The Standard Error

As we have seen, sample estimates vary about the population values they are supposed to be estimating.

Once we have a sample estimate, we need to know how far off that estimate might be from the parameter it is estimating.

The standard deviation and the Rule of Averages give us a way to measure this distance.

Recall that the Rule of Averages says that sample means are approximately normally distributed, and that they tend to get closer to the population mean as the sample size increases.

The rule states that the standard deviation of a sample mean based on a sample of size n drawn from a continuous distribution is approximately , where n is the sample size./ n


In practice, we don’t know what is, so we estimate it with S, the sample standard deviation.

The quantity , called the standard error of the mean, estimates the standard deviation of the sample mean.

/S n

The Standard Error

The standard error takes the sample size into account. The standard error is smaller for larger sample sizes, reflecting

increased precision of estimation, or equivalently, smaller error.

Data Type Parameter Standard Error

Continuous Mean /S n

58 2002 North Haven Group 58

Introduction

When we are working with variables (continuous) data, and our sample or subgroup size (n) is greater than one, Shewhart or X-bar charts are often used to monitor the process average.

Range or Sigma charts are used along with X-bar charts to monitor the process variability.

In this section, we will focus our attention on using X-bar and R, X-bar and S


Understanding Variation - RevisitedIn previous sections we have attempted to characterize the

random behavior of a variable of interest, such as the diameters of piston rings, in terms of two characteristics:

The accuracy of the process.

An accurate process is on target. An inaccurate process is off target. Accurate processes are also said to have ‘zero bias’.

The precision of the process.

A process is said to be precise if the standard deviation, or average distance from the mean, is small. Precision = 1/S.

A process is said to be imprecise if the standard deviation is large.


Understanding Variation - RevisitedBased on 1000 measurements, we determined that the piston ring

process was on target, but found that approximately 5% of the rings were outside the specifications of 75mm 0.5mm.

0.05

0.10

0.15

0.20

0.25

74.3 74.5 74.8 75.0 75.3 75.5 75.8 76.0


Understanding Variation - Revisited

We now examine a third dimension of data: Time.

Data that has a natural time ordering is often referred to as a time series data.

If we collect a large amount of data on a process during a brief time period, then how well will our estimates of the process average and standard deviation relate to the process over time, say some months after we collected the original data?

Are the process average and standard deviation likely to have the same value as we estimated some months previous?

We cannot easily answer this question unless we have some mechanism by which we can monitor the process average and standard deviation over time.



No process left to its own desires will stay at a fixed mean level for very long.

There are simply too many sources of variation that will cause the mean to shift about over time. So, we are not only interested in the overall mean and standard deviation of a process, we are also interested in the behavior of the process over time.

Through the use of control charts, we will attempt to monitor the mean and try to detect when a special or systematic source of variation has significantly shifted that mean.

The shift may be an abrupt change, or it may be a slow trend or drift.



We return to the piston ring example. Suppose that for 100 days a production supervisor takes one piston ring as a sample of the output for that day.

Based on this histogram, what can we conclude about the performance of the process with regard to the target and the specifications (75mm 0.5mm)?


Understanding Variation - RevisitedSuppose that the supervisor uniquely marked each ring to keep

track of the day on which it was collected.

Does a time series plot provide any extra information?



It can be difficult to say whether or not we are observing a stable process over time.

It is not always easy to determine if a process is stable or unstable, because all observations over time exhibit random behavior.

We expect variation in the observations or means of observations over time.

The challenge is to determine when the underlying mean and/or standard deviation have significantly changed.

‘Significantly’ implies that the shift must be greater than can be explained by common cause variation alone.


Understanding Variation - RevisitedDr. Edwards Deming stated that one should never make

adjustments to a stable process, because such adjustments will only make the process more variable or volatile.

Tampering is the act of making process adjustments or changes in response to observed random variation (tweaking in response to noise).

Tampering tends to be based on “feel”, with little or no data to validate the need for change or the type of change that is being made.

Deming considered tampering as the preeminent cause of poor quality and poor profitability. In some industries, so much tampering occurs that completely unstable (inconsistent) manufacturing processes result.



Recall the three categories of process variation:

Common Cause Variation: Natural background variation or process noise that is always present and is inherent to the system. This is small scale variation.

Special Cause Variation: Large scale variation that may permanently change the process due to some assignable or special sources of variation. The causes are external.

Systematic Variation: Large scale variation that systematically changes the process mean and is inherent to the process.

A stable or in control process is subject to only common cause variation. An unstable or out of control process is subject to common cause, special cause and/or systematic variation.



Deming and Shewhart pointed out that trying to adjust a process when only common cause variation is present is tampering and can only make things worse.

The difficulty is for the observer to understand when the process is stable and when special causes have crept into the process.

Statistical Process Control (SPC) and Control Charts are a set of statistical tools which attempt to determine when special or systematic cause variation occurs over time.

A control chart is nothing more than a signal detection device, which attempts to detect special and systematic causes of variation against the background noise of common cause variation.


Rational Subgrouping

The basic idea of control charting is to monitor a process over time by collecting samples of data, called subgroups, from the process at some regular time interval.

Based upon the sample average and standard deviation (or range) we attempt to draw conclusions as to whether the underlying process mean and the underlying process standard deviation have changed over time.

For a stable process the subgroup averages will vary randomly about the process mean .

If special cause variation enters into the process, then the underlying mean changes and the shift should be detectable in our plot of subgroup averages.



Since the subgroup averages already vary randomly, it is not quite so straightforward to determine when a true shift in the mean has occurred.

The subgroup range or standard deviation is similarly used to monitor for significant changes in the underlying process standard deviation, .

A fundamental question in the control chart is how to collect subgroups of data for each sampling period.

One strategy is to go out to the production line and grab a random sample of n parts that have been produced since the last sampling period.

Another approach is to take a nonrandom sample of n parts.


Rational SubgroupingIn this case we attempt to select our sample such that the parts or

units are most alike.

For example, the sample of parts may be taken in sequence as they come off the production line.

Shewhart referred to the nonrandom sampling method as rational sub-grouping.

He made a case that a rational subgroup strategy would give the control chart more power to detect shifts in the mean:

If a subgroup is taken by randomly sampling from all parts or units made since the last sampling period, then there is a good possibility that the subgroup of parts contain units before and after a special cause of variation has appeared.



By taking the sample such that the parts are most nearly alike, we minimize the possibility that our subgroup of parts contain the effects of a special cause of variation.

The variation observed with a rational subgroup is considered to be purely common cause. We take the parts made close together in time such that there is little or no possibility of a special cause of variation occurring while taking the sample.

Shewhart correctly reasoned that one can pool together observed variation within the subgroups to get a very good estimate of the common cause variation (noise) in the system.

To illustrate the need for rational sub-grouping, we return to the piston ring example shown earlier.



OD readings for the last 30 days. A downward shift in the mean diameter occurred between days 6 and 7.



We collect a subgroup of 5 piston rings right after ring 21 is completed (rings 17 through 21 will make up our subgroup).

We calculate the standard deviations, S, for a random sample of the piston rings and for the rational subgroup (next slide).

Notice that S for the random sample is much larger than S for the rational subgroup.

The S for the random sample probably incorporates a special cause (the shift between rings 6 and 7) and is NOT measuring common cause variation.

As we will see, this inflated standard deviation would cause our control chart to be insensitive to shifts or changes in the process.



1674.94

1274.62

1074.59

675.22

174.91

Ring Number

Diameter

Random Sample of 5 piston rings

2174.69

2074.78

1974.54

1874.82

1774.94

Ring Number

Diameter

Rational Subgroup of 5 piston rings

74.85

0.259

X mm

S mm

74.75

0.150

X mm

S mm


Rational SubgroupingNote the difference in the control limits using the two estimates

of the standard deviation.

Smaller S, tighter control limits (more sensitive).

Larger S, wider control limits (less sensitive).


Rational SubgroupingControl charts consists of 3 basic elements:

1. A Time Axis to show the time ordering of the samples.

2. A Center Line representing the overall process average.

3. Control Limits

Recall that 99.73% of all values will fall within 3 standard deviations of the mean.

Control limits are set up so that 99.73% of subgroup means will fall within 3 standard deviations of the overall mean.

In order to create an X-bar and R (or S) chart we typically require an initial ‘calibration’ data set of 25 to 30 subgroups from which we estimate the control limits and centerline.


Control Charts for Continuous Data

When calculating control limits and the center line, we assume that the subgroups in this initial data set come from a stable process, and that the mean and standard deviation are representative of the overall process mean and standard deviation.

Any subgroups where a special cause may have occurred should not be included.

If these special causes are included in the calculations, our control limits will be wider than they should be. As a result, the control charts would be less sensitive to process changes.

This usually requires us to recalculate control limits after special causes have been eliminated. (We may do this several times).


Control Charts for Continuous DataTo set up the charts, we use an estimate of the process

standard deviation, which is calculated by pooling together the subgroup ranges or standard deviations.

As we discussed in the previous section, subgroup ranges or standard deviations provide our best estimate of common cause variation.

For typical subgroup sizes, around n = 5, it makes no difference if we use the range or the standard deviation for the estimate of the process standard deviation.

As a rule, if the subgroup size n <10, then the subgroup ranges are typically used to estimate . When n 10, the subgroup standard deviation is typically used.


Control Charts for Continuous Data

Calculating Control Limits for an X-bar and R chart:

1. Collect an initial set of 25-30 subgroups of size n.

2. Calculate the grand average of the data, written . This is the center line for the X-bar chart.

3. Calculate the range R for each subgroup.

4. Calculate the overall average of the ranges, written . This will serve as the centerline of the R chart.

5. Estimate the process standard deviation using the average range.

6. Calculate the upper and lower control limits (UCL and LCL) for the X-bar chart.

7. Calculate the control limits for the R chart.

X

R


Control Charts for Continuous DataThe control limits for sample averages can also be based upon the

subgroup standard deviations.

In these cases, we use an S chart or Sigma chart, rather than an R chart, to monitor the process standard deviation.


Detecting Process Shifts or Changes

Control charts useful process monitoring devices.

However, they do have some weaknesses that are not readily apparent.

All control charts will generate false positives.

A false positive is an occasion where the chart will signal an out of control condition, indicative of a special cause, when in fact only common cause variation is present.

Control charts usually do not signal immediately when a special cause of variation enters the process.

This is particularly true if there is a slow drift, where something is causing the mean to slowly move over time.



Depending upon the magnitude of the special cause, the chart may take some time to signal the presence of the special cause.

In order to discuss the capabilities of control charts for all types of processes, we usually talk about special causes in terms of Sigma Shifts in the mean.

The number of standard deviations the process mean has been moved by the special cause.

X-bar charts are relatively insensitive to shifts of 1 or less.

For shifts of 1.5 or greater, the X-bar chart is very sensitive and will signal quickly.


Detecting Process Shifts or ChangesSince the subgroups of data contain random variation, the time

required for a control chart to signal after the onset of a special cause is also random.

The random variable Run Length is a measure of the time required for the control chart to signal once a special cause appears. (This topic is commonly covered in SPC courses - we will only mention it briefly here).

For a small shift, for example 0.5, the run length for the X-bar chart is long - it will take a long time to detect a small shift.

One technique for increasing the sensitivity of X-bar charts is to increase the subgroup size.

Larger subgroup sizes result in narrower control limits.



Larger subgroup sizes make control charts more sensitive to shifts or changes.

A number of rules have been developed to make X-bar charts more sensitive to small shifts and subtle “out of control” behavior.

These rules were developed by Western Electric, and are often referred to as the Western Electric Rules.

Although the rules do make the X-bar chart more sensitive, they also increase the likelihood of false positives.

In other words, the additional rules signal more often, causing the chart to generate more false positives as well.


Detecting Process Shifts or ChangesTest 1: One point more than three sigmas from the center

line detects a shift in the mean, an increase in the standard deviation, or a single aberration in the process. For interpreting Test 1, the R chart or S chart can be used to rule out increases in variation.

Test 2: Nine points in a row on the same side of the center line detects a small shift in the process mean.



Test 3: Six points in a row all increasing or all decreasing detects a trend or drift in the process mean. Small trends will be signaled by this test before Test 1.

Test 4: Fourteen points in a row, alternating up and down detects systematic effects such as two alternately used machines, vendors, shifts or operators.



Test 5: Two out of three points more than 2 sigmas from the center line (same side) detects a shift in the process average or increase in the standard deviation. Any two out of three points provide a positive test.

Test 6: Four out of five points more than 1 sigma from the center line (same side) detects a shift in the process mean. Any four out of five points provide a positive test.


Detecting Process Shifts or ChangesTest 7: Fifteen points in a row within 1 sigma of the center

line - “Hugging the center line”. Detects stratification of subgroups when the observations in a single subgroup come from various sources with different means.

Test 8: Eight points in a row more than 1 sigma from the center line (either side) detects stratification of subgroups when the observations in one subgroup come from a single source, but subgroups come from different sources with different means.


Detecting Process Shifts or ChangesWhich rules are violated for the forged piston ring data?

1 2002 North Haven Group Lean Six Sigma DMAIC: Measure Phase.

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Transcript of 1 2002 North Haven Group Lean Six Sigma DMAIC: Measure Phase.