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These are fairly simple relationships. With the exception of %Tolerance, all the other metrics are just different manipulations of the same numbers.

s²_Total = s²_parts + s²_{Measurement System} 

s²_{Measurement System} = s²_{Repeatibility} + s²_{Reproducibility}

s²_{Reproducibility} = s²_Operator + s²_{Operator*Part}

P/T = %Tolerance = 6* s_{Measurement System} / (USL-LSL)

P/TV = %Study Variation = s_{Measurement System} / s_Total

%Contribution = s²_{Measurement System} / s²_Total

Distinct Categories = Round-down ((s_{Measurement System} / s_parts) * 1.41)

And for Don Wheeler fans -

Discrimination Ratio = Square Root ((s²_{Measurement System} / s²_parts) * 2 – 1)

Look at the table showing how the numbers change depending on how much variation the measurement system is contributing and the tolerance. Distinct Categories and Discrimination Ratio are basically the same number. %Study Variation is simply the square root of %Contribution.

Most are introduced to the ideas of DPU and RTY as part of Six Sigma training. Where

                           RTY = e^-DPU

Most were not exposed to the assumptions behind this or how to use this as an analytical tool.

This relationship is based on defect densities follow the Poisson distribution and this is simply the equation for the Poisson when defects = 0. RTY is the proportion of units that will come through a process with no defects. 1 – RTY is the proportion that comes through a process with 1 or MORE defects. This assumes that the defect density is predictable.

When this idea was first put out as part of a DFA course sponsored by Bill Smith in 1987, several of us took a lot of data to convince ourselves this was true. We found that on our well-controlled processes it was true. What we found was when it was not true, there were always easy fixes possible that followed an idea put out by Juran several decades earlier. That is the idea of splitting the analysis into streams. 

How to Find the Expected Defect Distribution –

Find out you DPU for the process you are interested in. Either use a Poisson table from a stat book or Excel.

If you are using a stat book, you’ll find the first column labeled np which is your DPU. Go down until you find the closest match. Then read across. These are cumulative tables so if you want to find the probability of a specific number of defects it will be the probability of that number – the probability of (1- that number).

If you are using Excel (preferred), start by creating a column labeled defects and go from 0 to about 4xDPU (Poisson is skewed right with a fairly long tail).  Label the next column expected proportion. Use the f_x function (called Paste Function in a mouse over) and go to Statistical – Poission and click on it. The x it asks for is the defect number (cell A2 if you started in the upper left hand corner of the worksheet). The mean is your DPU and for cumulative put false (you will get back the exact probability of the individual defect counts). Fill in actual proportions from your data and compare. A chi-squared test could be applied, but usually the numbers line up nicely or they are way out of wack.

A Hypothetical Example (based on an actual improvement project) –

A process is run making structural components for the automotive industry. Every component has defects, but components with x or less defects are further processed, components with > x defects are scrapped. The scrape rate is high, but more importantly the components that are further processed have defects detected all the way out to the end user.

The process is to run several components of a given length and when enough components are collected, multiple components are put through a heating process where there are multiple locations. The heating process joins together two metal pieces to give the component its structural integrity. No care is taken to know what pieces pass through which location in the oven. After the heating the components are put through a non-destructive test that is looking for voids in the joining process. The number of voids is collected.

It is found that the DPU detected by this test is 20. What the Poisson would predict for this is -

So a real quick interpretation of this – 

1)   If I were placing +/- 3s limits on this like an SPC chart, they would be at 8 and 34. On my average daily production of 1,000 pieces, we would expect 2 pieces per day to be beyond the limits. This makes sense.

2)   The most likely number of defects in a single piece is 19 or 20 (equal to 8 significant digits), but numbers between 11 and 29 will show up quite often.

3)   Most important, I will never expect numbers like 0 – 3 or anything greater than 41. These numbers should occur less than once in our expected annual volumes.

The next thing to do is compare actual defect counts per unit with the theoretical counts –

Wow, something is wrong!

The team is reconvened to go back through the process maps to specifically brainstorm where there is clearly a consistent input at work and where there may be variation across the input. Things like material batches, the front end of the process where the metal pieces are straightened and fixtured to go to the heating process, and the cooling process prior to the non-destructive test. After discussion it is understood that the data is only one days production and the metal supplier was consistent making the front end of the process consistent (relatively speaking). After much discussion, it is agreed to keep track of the position inside the heating process. Long story short, the team finds that the middle 2/3’s of the oven is running at 12 DPU and the outer 1/6 on each side is running at 36 DPU.

The team decides on a containment action to only run in the center 2/3’s of the oven while further studying consistency of the oven.

There is a lot more to the study as the original containment did not work (the physics of it not working made sense), but the team learned to make the containment work achieving 12 DPU but production capacity was reduced. With about $5K worth of modifications, learned to run full loads again at 5 DPU.

There are frequent questions on the discussion boards asking for things like a z table from -6 to +6 standard deviations – or a derivative of it a sigma table from 0 to 6 sigma (the old 1.5 sigma shift). I have also helped people through things like the CQE or CSSBB exams where they are carrying multiple books just to get access to statistical tables. With a little knowledge of Excel, you can create all of these things for your self.

Excel basics – all of what you will need is contained in the f_x (mouseover and it is called paste function) button. When you click on it, you will be given several options, choose statistical. You will find multiple distributions covered there, certainly all of the ones used in Six Sigma training and loads you have probably never heard of. Think about how table are laid out, the convention in Excel is to call having a stat looking for a probability (distribution)dist. For example the standard normal where you are looking for a probability given a z value is called normsdist. The convention for having a probability looking for a stat (distribution)inv. For example, the standard normal distribution where you have a probability looking for the associated z value is normsinv.

Setting up a z (standard normal) table from -6 to +6 standard deviations –

1)    Start in cell A1 and type in z value

2)    In cell A2, type -6. In cell A3, type -5.9. Then highlight cell A2 and A3, go to the bottom right of cell A3 until the cursor becomes a small cross. Drag down to cell A122.

3)    In cell B1, type .09. In cell C1, type .08. Then highlight B1 and C1, go to bottom right of cell C1 until the cursor becomes a small cross. Drag across to cell K1.

4)    In cell B2, open f_x by double clicking. Go to statistical, normsdist. And for the z value, type in $A2+B$1. Go to the bottom right hand corner of B2 until the cursor becomes a small cross and drag across to K2. Release the mouse and while cells B2 through K2 are still highlighted go to the bottom right of cell K2 until the cursor becomes a small cross and double click. That will fill out the whole table from -6.09 to +6.09.

5)    All of the values listed are correct, but for clarity we need to reverse the column headings starting in row 64. Go to row 64, highlight both the 64 and 65 right click and insert. This will give you two blank rows.

6)    In cell A65, type 0.

7)    In cell A64, type z value. In cell B64, type .00. In cell C64, type .01. Then highlight B64 and C64, go to bottom right of cell C64 until the cursor becomes a small cross. Drag across to cell K64

    Modify the formula in B66 to =NORMSDIST($A66+B$64). Go to the bottom right hand corner of B66 until the cursor becomes a small cross and drag across to K66. Release the mouse and while cells B65 through K65 are still highlighted go to the bottom right of cell K65 until the cursor becomes a small cross and first drag up to row 65. Release the mouse and while cells B65 through K66 are still highlighted go to the bottom right of cell K66 until the cursor becomes a small cross and double click. That will fill out the whole table from 0.01 to +6.09.

9)    Label the tab for the worksheet z table. 

Your table is done. 

Setting up a Sigma Table table from 0 to 6 Sigma –

1)    Start with a blank worksheet.

2)    Start in cell A1 and type in Sigma value

3)    In cell A2, type 0. In cell A3, type .01. Then highlight cell A2 and A3, go to the bottom right of cell A3 until the cursor becomes a small cross. Drag down to cell A62.

4)    In cell B1, type 0. In cell C1, type .01. Then highlight B1 and C1, go to bottom right of cell C1 until the cursor becomes a small cross. Drag across to cell K1.

5)    In cell B2, type in =(1-NORMSDIST($A2+B$1-1.5))*1000000. Go to the bottom right hand corner of B2 until the cursor becomes a small cross and drag across to K2. Release the mouse and while cells B2 through K2 are still highlighted go to the bottom right of cell K2 until the cursor becomes a small cross and double click. That will fill out the whole table from 0 to +6.09.

6)    Label the tab of the worksheet Sigma value (1.5 sigma shift). 

Your table is done.

Gary

There are many tools from the basic quality toolset omitted from Lean and Six Sigma training. One of them is the OC Curve which basically defines the protection you get when doing any kind of acceptance sampling. It can be correctly argued that acceptance sampling has no place in a truly Lean organization, but the reality is that that is the end result of years of work and most have some form of this going on. It may be called Incoming Inspection and it might be called an audit, but it’s going on.

We’ll take the most common example to show how the tool works – sampling from a finite population. The appropriate distribution is the hypergeometric, and while it can be approximated by the binomial or poisson, why bother? We have software  available to us.

Let’s take the example of where we have a population of 3201 and we are going to sample 125 from that population and accept on 3, reject on 4. This is a common 1% AQL plan. 

Using Excel, the insert function (f(x)), statistical, and hypgeomdist we can create a table to give us the Probability of Acceptance (P(a)) of a given quality level of the population.

The resulting curve is easy to plot in Excel or Minitab.

Interpretation of the curve is simple and usually involves the P(.95) and the P(.10) points. This particular plan has a 95% probability of accepting a 1.1% defective lot and a 10% probability of accepting a 5.2% defective lot. Not very good protection which is usually the real discovery - acceptance sampling is not a good control! Read Deming.

Gary