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One of the most poweful operational metrics I know of is Rolled Throughput Yield (RTY).  It’s used to assess the “true” yield of a given process.  This includes what we often call the “hidden factory” that plagues so many organizations… sucking profit right off their financial statements!

Traditional Yield

Let’s use an example to demonstrate how some, let’s call them, traditional manufacturing folks attempt to measure things. 

Say there is a manufacturing process with 3 steps - Processes 1, 2, and 3 (original, eh?).  Let’s also say that on a particular day they note the following performance:

• Process 1: 100 parts passed through this process and 84 “good” parts left this process (scrapped 16).
• Process 2: With some WIP laying around 110 parts passed through this process with 82 “good” parts passing (28 scrapped)
• Process 3: With even more WIP laying around this process they managed to produce 138 parts with 126 parts passing (12 scrapped).

Since the manufacturing manager only cares about “what goes out the door” the process they are most concerned with is the last one - process 3.  And since they had a great day (only scrapped 12 parts) they report a “yield” of 91% (126/138).  The manager even calls his buddy Sal, the sales manager, to brag about all the product they shipped!

Not So Fast Buddy

There are some fundamental flaws with this technique.  The most severe issue is the fact they are ignoring all the scrapped parts process 1 and 2 created.  This “hidden factory” is not known by Sal or really anyone else short of the folks on the line.

Here is how the manufacturing manager should be measuring the performance of their line.

RTY

Process 1 had a daily yield of 84% (84/100) while process 2 had a daily yield of 75% (82/110) and finally process 3 had a daily yield of 91% (126/138).

So, to calculate RTY we simply multiply these yields together giving us a composite yield for the day.  Doing this gives us:

• 84% x 75% x 91% = 57%

Our Opportunity

This value of 57% is a more accurate representative of how this production line is performing.  And more importantly this 57% is our opportunity as lean and six sigma practitioners!

Sure, the manager will probably not call old Sal telling him the latest RTY.  And that’s fine.  But we must not kid ourselves into believing we are performing better than we are.  By focusing on RTY we can be sure we stay focused on the true pulse of the organization.

Note: All the WIP I mentioned in the example is another problem!  But we will save that for another day. 

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If you hate statistics this post is for you. Why? Because it’s my intention to have you understand AND be in position to teach others one of the more complicated and misunderstood statistical concepts of our time - the central limit theorem (CLT) - by the end of this article.  If you are up for the challenge read on.

Central Limit Theorem – What is it?

OK, here is an official like CLT definition for the purists.

The central limit theorem (CLT) states that the means of random samples drawn from any distribution with mean m and variance s² will have an approximately normal distribution with a mean equal to m and a variance equal to s² / n.

Say what? I know… the people that write statistical text books need to join the rest of us on the planet earth. It’s like they get a kick out of making people wonder what the hell they are talking about.

All this confusing definition is really saying is that as n, or our sample size, increases just about any distribution (normal or non normal) will tend to behave normally.

How can it be?

The key to this theorem is the whole s²/ n part of the formula. As n, sample size, increases we see s², the variance, decrease. And less variance means a tighter, more normal, distribution.

Prove it to me

Remember, I told you that you will be able to teach others about this concept. So here are some teacher’s notes.  Around this time in your explanation the student or students will wonder who you are. They may even think you are on crack. That’s good. You have them right where you want as you are slowly setting the hook. Once they bite, and they will, you will then reel them with ease.  Let’s press on.

Time to Simulate

I came across this sweet little Java Applet tool that allows you to perfectly demonstrate the CLT. Just click on the “Start CLT Applet” button to launch the tool. This tool was developed by some folks at Seton Hall and from all I can tell is free for anyone to use.

Fun with the Weibull Distribution

So here is the situation. Let’s assume we have a process that exhibits a Weibull distribution which would fail to pass as “normal” data as it is skewed to the right. This means we can not use any so called “parametric” hypothesis tests. Often times we see a Weibull distribution with reliability/ failure analysis data.

Let’s now pretend we send someone out on 100 different occasions (trials) to collect data from this process we know to exhibit a Weibull distribution. Let’s also assume we tell this person to only collect “1” data point per trip/trial. After the 50th trial we take the 50 data points and study its shape (Top Figure).

The blue bars are our data and the yellow outline is what a typical Weibull distribution looks like. As you can see our distribution looks pretty Weibull-ish.

Now then, we then tell the person to go back 100 more times. Only this time we ask them to collect 5 samples instead of 1 sample each trip out. We will take the 5 data points from each trial and average them together. After the 50th trial we take the 50 data points (remember each data point is an average of 5 numbers) and study its shape (Middle Figure).

Notice how the distribution is beginning to behave a bit more normal like although still maintaining a little Weibullishness? Yes, that’s a word… at least in my dictionary.

Ok, we now attempt to really push our luck as we ask the person to go back out one last time. This time we ask them to collect 25 samples during each trial. We also buy them lunch at this point as we are beginning to whip them pretty good!

So, they go back out and collect 25 samples per trial. Again, we take these 25 samples and average them like in the second trial when we averaged 5 data points. After the 50th trial we take the 50 data points and study its shape (Bottom Figure).

Now we clearly see a normal, bell shaped, distribution beginning to appear. And all we did was increase the sample size, n, from 1 to 5 and finally to 25.

When you teach people this it’s at this point where you reel them in. Also, turn the simulation tool over to them so they can play around with it. Tell them to prove the theorem wrong if they can. No worries, they can’t.

Break out the dice

Another fun way to demonstrate the CLT is with fair dice. Simply have someone roll 1 die 50 times noting their results after each roll. When they graph this the distribution will be very flat. Then give them 2 die and have them roll them both at the same time 50 times (averaging the results each run). Finally, give them 5 die and repeat. You will see the distribution become more and more normal as the sample size, n, increases.

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Update: In case you don’t read the comments Rob, from LearnSigma, shared a link to the coolest dice game Applet.  So check it out.  Thanks Rob.

A Failure Modes Effect Analysis (FMEA) is an extremely powerful tool that all people can and will benefit from no matter your occupation or status in life.

Tonight, we shall discuss the history of the FMEA, the different types of FMEA, and finally how to actually construct one.  At the end of the post is a free FMEA template for your downloading pleasure. 

History

The FMEA is not a new tool.  The aerospace industry used the FMEA during the Apollo missions in the 1960s.  Later in 1974 the US Navy developed MIL-STD-1629 which discussed the proper use of the tool.  And around this time the automotive folks latched onto the tool and never let go.  Today, the FMEA is universally used by many different industries.

Type of FMEA

There are three main types of FMEA in use today.

1. System FMEA: Used to analyze complete systems and/or sub-systems during the concept of design stage.
2. Design FMEA: Used the analyze a product design before it is released to manufacturing.
3. Process FMEA:  Used to analyze manufacturing and/or assembly process.

The Process FMEA is probably the most commonly used and is also the least complex, in most cases. 

10 steps to creating a FMEA

1. List the key process steps in the first column.  These may come from the highest ranked items of your C&E matrix. 
2. List the potential failure mode for each process step.  In other words, figure out how this process step or input could go wrong.
3. List the effects of this failure mode. If the failure mode occurs what does this mean to us and our customer… in short what is the effect?
4. Rate how severe this effect is with 1 being not severe at all and 10 being extremely severe.  Ensure the team understands and agrees to the scale before you start.  Also, make this ranking system “your own” and don’t bother trying to copy it out of a book.
5. Identify the causes of the failure mode/effect and rank it as you did the effects in the occurence column.  This time, as the name implies, we are scoring how likely this cause will occur.  So, 1 means it is highly unlikely to ever occur and 10 means we expect it to happen all the time.
6. Identify the controls in place to detect the issue and rank its effectiveness in the detection column.  Here a score of 1 would mean we have excellent controls and 10 would mean we have no controls or extremely weak controls.  If a SOP is noted here (a weak control in my opinion) you should note the SOP number.
7. Multiply the severity, occurrence, and detection numbers and store this value in the RPN (risk priority number) column.  This is the key number that will be used to identify where the team should focus first.  If, for example, we had a severity of 10 (very severe), occurrence of 10 (happens all the time), and detection of 10 (cannot detect it) our RPN is 1000.  This means all hands on deck… we have a serious issue!
8. Sort by RPN number and identify most critical issues.  The team must decide where to focus first.
9. Assign specific actions with responsible persons.  Also, be sure to include the date for when this action is expected to be complete.
10. Once actions have been completed, re-score the occurrence and detection.  In most cases we will not change the severity score unless the customer decides this is not an important issue.

Dynamic Document

The single biggest failure people make with FMEAs is to spend time completing the document and then storing it in a file cabinet somewhere.  The FMEA is the ultimate dynamic document meaning it lives as long as the process or product it is associated with does.  Please use them!

Free Template

Here is a free FMEA Template for your use.  Simply “right click” the link and choose “Save Target As.” 

Feel free to share this template with as many people as you like.  Also, please email me if you have any questions about this or any other continuous improvement topic.

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Last night we discussed the history and background of the one sample t-test.  As promised, tonight we will discuss how it is you actually use the slick little hypothesis test.  At the end of this post is a free case study available for download.

When to use it

We may choose to use the one sample t-test when we want to compare a sample mean to a target value and we don’t know the true population standard deviation (s). Also, as we discussed last night, the one sample t-test allows us to work with smaller sample sizes. 

Assumptions

There are a few assumptions we need to consider prior to running with the one sample t-test. 

1. Our data should be stable and not trending.  If, for example, our data has been trending up for the last 3 months the one sample t-test should not be employed.  How to check this?  Throw the data into a control chart and see what it tells you.
2. The data should be normally distributed.  There are fancy statistical tests such as the Anderson-Darling test that can help us here.  I always recommend people first study the “shape” of the data in a simple histogram.  If the shape looks normal to the eye I say press on with the one sample t-test. 

State the null and alternate hypothesis

If we satisfy the assumptions it is now time to state the null (Ho) and alternate (Ha) hypothesis.  For the standard one sample t-test it will looking something like this, assuming our “target” value is 25 for the sake of this example.

• Ho: mu = 25
• Ha: mu not = 25

Determine the level of risk you are willing to take

With hypothesis testing we never accept anything.  Instead we either reject or fail to reject a hypothesis just like the American judicial system where we never prove someone innocent.  Instead, they are either guilty or not guilty beyond a reasonable doubt.  Just ask O.J. Simpson, he knows all about this aspect of hypothesis testing!

So with hypothesis testing we need to state the level of risk, or reasonable doubt, we are willing to take.  In most cases an “aplha risk,” as it is called, of 5% is commonly chosen.

Run the test and make a decision

Now then, we have met our assumptions and stated the level of risk we are willing to take.  Now all that’s left is to run the test and make a, gulp, decision.

When we run the test we will get a P value which is the is the probability of incorrectly rejecting the null hypothesis.  Just remember this saying, “if P is low, Ho must go.” 

So, we run the test and examine the P value.  If the P value is less than 5% we reject Ho and state that the alternate hypothesis is true at the confidence level of 100*(1-P value)%.  If it is greater than 5% we fail to reject the null hypothesis. 

We will also get information on confidence intervals which basically tells us a range of where we may expect to see our data. 

Case Study

Below is a fictitious case study demonstrating how this one sample t-test may be applied.  Since the document is free all I ask in return is for you to share it with as many people as possible.  That way, together we can get everyone hooked on hypothesis testing!

Click here to access the free case study.  Once the document opens in the window you can choose to save it.  You can also “right-click” the file and choose “Save Target As” if you wish.  Happy reading!

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Back in the early 1900s a certain W.S. Gosset, an Englishmen, was tasked with brewing better beer.  Really, I’m being serious. 

Gosset was a bright man, with two degrees from Oxford, and was hired by Guinness to help them brew the best beer using statistical methods instead of the “tribal knowledge” approach most brewing companies employed at the time.

Gosset’s main issue

Gosset’s main issue was sample sizes.  At the time the Z-test was the most prevalent test available for someone who wanted to compare a sample mean to some known target, or hypothesized mean.

The problem with the Z-test was it used the population standard deviation (s) which required large sample sizes.  This was impossible for Gosset as his resources were limited for economical reasons.  I mean there is only so much beer allowed for experimentation… wink, wink.

For example, in one experiment Gosset was attempting to determine the optimal type of barley to use.  Initially, these experiments started with 4 farms each growing one plot of each variety.  So assuming he knew the “population” standard deviation from this small sample was dangerous and wrong.  Lucky for all of us, Gosset challenged the status quo and went against the teachings of many well known statisticians of his time.

Introducing the one sample t-test

Gosset’s big, and somewhat controversial move, was to tweak the Z-test ever so slightly, creating what we now call the one sample t-test.  The tweak dealt with using the “sample” standard deviation (s) instead of the “population” standard deviation (s).  There are some mathematical reasons for this related to the fatness of the tails and other fun stuff but I will spare you from this discussion.  I’ll point to some additional reading at the end if you wish to read more.

The benefits

With the one sample t-test Gosset was now able compare a sample mean to some hypothesized mean (target).  And most importantly he did not have to worry about his small sample sizes.

Does it really matter?

A fair question, when comparing the Z-test with the one sample t-test, is does it really matter?  The best answer I can offer is - it depends.

If you are dealing with large sample sizes the results from a Z-test and one sample t-test are likely to be close to one another in which case it doesn’t really matter as much as some purists may say.

However, if you are dealing with smaller sample sizes the one sample t-test is likely the best choice due to the whole standard deviation conundrum.

Next up

Tomorrow night I will discuss the rubrics of the one sample t-test.  There are some assumptions we need to satisfy as well as some tricks we can play making this hypothesis test extremely powerful for both lean and six sigma practitioners alike.

Additional Reading

1. “Student” by R.A. Fisher
2. Guinness, Gosset, Fisher, and Small Samples by Joan Fisher Box
3. Everything you wanted to know about the one-sample t-test (but were afraid to ask) by Keith Bower

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Collecting data is tricky.  Many people think they can simply run off and grab some data, whip it into a spreadsheet, press some buttons and subsequently cure cancer.  I wish it worked this way… but it doesn’t.

Generally, in most training programs and text books out there you will hear about a 5 step data collection process.  I am not sure who first came up with these steps so if you do please leave a comment. 

Here they are with my own commentary.

1. Clarify your data collection goals.  Sounds straight forward enough but is often overlooked.  For example, what problem are you trying to solve by collecting this data?  Many people grow frustrated when they are asked to collect data and are not even told why.  Then, once this person has the data the person who asked for it in the first place can’t be bothered.  You will lose any ounce of credibility you may have had if you take this approach.
2. Develop operational definitions and procedures.  Here we need to be very clear as to what we are measuring, how it is to be measured, and who is to measure it. Often times we will employ sampling in which case we need to define a sampling plan.
3. Validate the measurement system.  Good golly Ms. Molly is this step ever butchered by most people!  True story… several years ago I was working with a supplier of my former company.  They made plastic parts.  They used this $300k “automated optical inspection” machine to measure critical “black diamond” dimensions.  They wanted my help with running a DOE.  I asked if a measurement system analysis had been done.  They assured me the machine had been recently calibrated to the “gold standard.”  I said, “that’s nice and have you done a MSA?”  Long story short we did an MSA and learned that due to a programming issue (a topic for another blog) their measurement system was useless.  They had been running like this for years supplying parts to a $50B market cap company and had no clue what dimensions these parts really were.  Ouch.  Moral of the story… confirm your measurement system!
4. Begin data collection.  Isn’t it funny how this 4th step of the 5 step process is where most people want to start?  Using all the knowledge from the previous steps we now go off and collect our data.
5. Continue improving measurement system and ensure people are following the data collection guidelines.  Measurement systems need to be verified often.  A good whack to a camera can really mess things up.  Also, as with anything related to continuous improvement sustaining a process is the hardest part.  Data collection is no different.

Many Six Sigma practitioners struggle to differentiate between a repetition and replication. Normally this confusion arises when dealing with Design of Experiments (DOE).

Let’s use an example to explain the difference.

Sallie wants to run a DOE in her paint booth. After some brainstorming and data analysis she decides to experiment with the “fluid flow” and “attack angle” of the paint gun. Since she has 2 factors and wants to test a “high” and “low” level for each factor she decides on a 2 factor, 2 level full factorial DOE. Here is what this basic design would look like.

Now then, Sallie decides to paint 6 parts during each run. Since there are 4 runs she needs at least 24 parts (6 x 4). These 6 parts per run are what we call repetitions. Here is what the design looks like with the 6 repetitions added to the design.

Finally, since this painting process is ultra critical to her company Sallie decides to do the entire experiment twice. This helps her add some statistical power and serves as a sort of confirmation. If she wanted to she could do the first 4 runs with the day shift staff and the second 4 runs with the night shift staff.

Completing the DOE a second time is what we call replication. You may also hear the term blocking used instead of replicating. Here is what the design looks like with the 6 repetitions and replication in place (in yellow).

So there you have it! That is the difference between repetition and replication.

Note: If you call these concepts by different terms please do share by leaving a comment.

Last night we discussed the Taguchi Loss Function and how Taguchi methods are more concerned with hitting the target compared to more traditional methods that often focus on keeping our data between the upper and lower specification limits.

Cpm

Staying with this theme I now want to introduce Taguchi’s version of Cp and Cpk which we call Cpm. Truth be told I don’t know where the “m” comes from. Perhaps it has ties to a Japanese word. I will default to someone like Jon Miller who knows a bit more Japanese than I. Also, if someone else knows where the “m” comes from please do share!

What I do know is why we actually prefer Cpm to Cpk in certain situations. It all has to do with the relationship between our “target” and the specification limits.

Example of when to use Cpm

For example, let’s say we are machining a part with a LSL (lower spec limit) of 10 mm and a USL (upper spec limit) of 20 mm. Let’s also assume our customer asks us to produce the part to 18 mm instead of the more typical 15 mm (dead center).

In other words, instead of being right between the LSL and USL our customer wants us to bias the dimension more towards the USL. While this biasing may not be the norm, it does occur. When I teach Cpm and ask a room of 25 for examples of where they have seen a bias towards one spec limit I normally get several examples. If you have an example please leave us a comment.

Cpk Makes no Sense

In this situation using Cpk makes little sense since we are purposely shifting the mean of our process a bit towards the upper specification limit per our customers request. In fact, assuming we are successful and are able to consistently produce a machined part of 18 mm Cpk would penalize us since the process is “shifted.”

A better method, in this example, would be to use Cpm which uses the “target value” in its calculation. In other words we will not be penalized for not being dead center between the upper and lower specification limit.

Math Geek Fix

For those interested in the math, the key difference between Cpm and Cpk has to do with the the way standard deviation is calculated. The traditional Cpk standard deviation is calculated by comparing each data point to the mean of the process. When calculating Cpm we use a different method to calculate the standard deviation. Instead of comparing the data points to the mean of the process we compare it to the target value.

Note that if the target for our process is dead centered between the LSL and USL Cpm and Cpk will be almost identical.

Well that is Cpm in a nutshell. I can’t promise a third straight day of Taguchi fun… then again you never can tell. So please stay tuned!

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Saying the words “Genichi Taguchi” to a hard core “western statistician” may get you some dirty looks. Actually, some of these crazy statisticians may want to strike you for saying this person’s name. Why the hate you may ask?  Good question.

Let me give you my take on it. Genichi Taguchi is a Japanese engineer and (close your eyes western stats geeks) statistician.

Hard core statistics nerds (most of whom never actually make it to the gemba) will tell you how Mr. Taguchi’s methods break all kinds of rules. They will spout out words like orthogonality, confounding, and all kinds of other gibberish in hopes you will turn your back on Taguchi methods.

Ignore them!

Fear not friends… those statistics friends of ours should be ignored and allowed to study their monitors all day long leaving everyone else alone. I am hear to tell you that Taguchi methods rock and I have used them many, many times successfully. My favorite Taguchi DOE is the L18. I will blog more about this in the future.

Tonight I want to introduce a key concept Taguchi teaches known as the “Loss Function.” It is at the core of all Taguchi methods and must be understood.

Traditional Bell Curve

Let’s start with our traditional “bell curve” approach to defects. Typically we see people draw in upper and lower specification limits (customer requirements). We then see a bell curve drawn in between these specification limits. If a data point falls outside a spec limit we have a defect. If all the data points are between the spec limits there are no defects. Simple as that, right?

Sort of.

Brain Surgeon Final Exams

Say you need to have brain surgery (I pray this never happens by the way). With something so serious it’s safe to assume you would want a top notch surgeon, right? Of course you would. But guess how surgeons get the right to slice into your head? They take exams, bunches of them, in medical school.

Imagine two nice fellows, Bob and Ted, are going through brain surgeon school together. Now imagine they are sitting for their FINAL exam. If they pass this exam they have the right to slice your head open.  After much study Bob and Ted take their exams.

Bob scores a 61% and Ted scores a 59%. Bob is celebrating and sharpening his scalpel as he “passed” the exam. Ted, poor guy, flunked and is looking into this new methodology called MVT as he hears it is replacing Six Sigma… this medical school stuff just isn’t working out for old Ted.

Is There Really a Difference?

But, really, is there really much difference between Bob’s knowledge and Ted’s knowledge? Not likely. Instead, what probably happened is Bob guessed right a few more times than Ted and earned the right to be a brain surgeon. Since his test score was “between” specification limits he passed. And since Ted’s score was outside the spec limit he is sent packing.

Enter the Loss Function

Genichi Taguchi realized this and hated it. So, he decided to turn that bell curve on its head – literally.

Taguchi said that having specification limits was all well and good. But what he wanted people focusing on was the “target” value. He stated that the further we drift away from the target value the more it costs the company. We want to aim for the target while doing all we can to reduce variation. The spec limits don’t get too much of our focus since we only want to nail that target and don’t stop until this is our reality.

This is not contradictory to what Six Sigma teaches. Six Sigma also aims to reduce variation while centering our process about the target. But if a Six Sigma practitioner ever becomes disillusioned with the fact that simply staying between the specifications limits is our goal explain the story of Bob the brain surgeon with the very expensive malpractice premium.

I will write more on Taguchi methods in the future. There are some really slick ideas I want to share.

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GE is arguably one of the best examples of Six Sigma excellence today. An often heard phrase is, “Motorola invented Six Sigma and GE perfected it.”A slick “variation weapon” GE has developed is called Span. I have never worked for GE but have worked with many former GE employees who have been kind enough to fill me in on this variation busting tool.

What is Span?

Span is another measure of dispersion much like the range. When calculating the range we simply take the difference between the largest data point and smallest data point. When dealing with non normal data using the range is likely safer than using the standard deviation. Read here for an explanation on why this is.

With Span we have another option for dealing with this non normal data, and let’s face it, much of the data out there is non normal. The secret to Span is that it ignores extreme outliers which can sometimes mislead us and instead looks at the difference between the 95th percentile and 5th percentile of the data set.

Specifically, Span is calculated as follows: 95th percentile – 5th percentile

Where can we use Span?

GE uses Span for things like focusing in on how well they are doing with on time delivery performance. There is an excellent article on iSixSigma about Span. In it you will hear from people like the former 80/20 man himself, Jack Welch, comment on how GE uses Span to focus in on their customers. You can access the article here.

If you have any real life experience using Span please do share.