Dec
20
Tools You Should Know - Defect Densities and the Poisson distribution
Filed Under Tools you should know, Uncategorized
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 fx 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 -
| Defects | Expected Percentage | Expected / 1000 Units | Cumulative Percentage |
| 0 | 0.000000% | 0 | 0.00% |
| 1 | 0.000004% | 0 | 0.00% |
| 2 | 0.000041% | 0 | 0.00% |
| 3 | 0.000275% | 0 | 0.00% |
| 4 | 0.001374% | 0 | 0.00% |
| 5 | 0.005496% | 0 | 0.01% |
| 6 | 0.018321% | 0 | 0.03% |
| 7 | 0.052347% | 1 | 0.08% |
| 8 | 0.130867% | 1 | 0.21% |
| 9 | 0.290815% | 3 | 0.50% |
| 10 | 0.581631% | 6 | 1.08% |
| 11 | 1.057510% | 11 | 2.14% |
| 12 | 1.762517% | 18 | 3.90% |
| 13 | 2.711565% | 27 | 6.61% |
| 14 | 3.873664% | 39 | 10.49% |
| 15 | 5.164885% | 52 | 15.65% |
| 16 | 6.456107% | 65 | 22.11% |
| 17 | 7.595420% | 76 | 29.70% |
| 18 | 8.439355% | 84 | 38.14% |
| 19 | 8.883532% | 89 | 47.03% |
| 20 | 8.883532% | 89 | 55.91% |
| 21 | 8.460506% | 85 | 64.37% |
| 22 | 7.691369% | 77 | 72.06% |
| 23 | 6.688147% | 67 | 78.75% |
| 24 | 5.573456% | 56 | 84.32% |
| 25 | 4.458765% | 45 | 88.78% |
| 26 | 3.429819% | 34 | 92.21% |
| 27 | 2.540607% | 25 | 94.75% |
| 28 | 1.814719% | 18 | 96.57% |
| 29 | 1.251530% | 13 | 97.82% |
| 30 | 0.834354% | 8 | 98.65% |
| 31 | 0.538293% | 5 | 99.19% |
| 32 | 0.336433% | 3 | 99.53% |
| 33 | 0.203899% | 2 | 99.73% |
| 34 | 0.119940% | 1 | 99.85% |
| 35 | 0.068537% | 1 | 99.92% |
| 36 | 0.038076% | 0 | 99.96% |
| 37 | 0.020582% | 0 | 99.98% |
| 38 | 0.010833% | 0 | 99.99% |
| 39 | 0.005555% | 0 | 99.99% |
| 40 | 0.002778% | 0 | 99.997457% |
| 41 | 0.001355% | 0 | 99.998812% |
| 42 | 0.000645% | 0 | 99.999457% |
| 43 | 0.000300% | 0 | 99.999758% |
| 44 | 0.000136% | 0 | 99.999894% |
| 45 | 0.000061% | 0 | 99.999955% |
| 46 | 0.000026% | 0 | 99.999981% |
| 47 | 0.000011% | 0 | 99.999992% |
| 48 | 0.000005% | 0 | 99.999997% |
| 49 | 0.000002% | 0 | 99.999999% |
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 –
| Defects | Expected / 1000 Units | Actual / 1000 Units |
| 0 | 0 | 0 |
| 1 | 0 | 0 |
| 2 | 0 | 0 |
| 3 | 0 | 1 |
| 4 | 0 | 4 |
| 5 | 0 | 8 |
| 6 | 0 | 17 |
| 7 | 1 | 29 |
| 8 | 1 | 44 |
| 9 | 3 | 58 |
| 10 | 6 | 70 |
| 11 | 11 | 76 |
| 12 | 18 | 76 |
| 13 | 27 | 70 |
| 14 | 39 | 60 |
| 15 | 52 | 48 |
| 16 | 65 | 36 |
| 17 | 76 | 26 |
| 18 | 84 | 17 |
| 19 | 89 | 11 |
| 20 | 89 | 6 |
| 21 | 85 | 5 |
| 22 | 77 | 3 |
| 23 | 67 | 3 |
| 24 | 56 | 4 |
| 25 | 45 | 4 |
| 26 | 34 | 6 |
| 27 | 25 | 7 |
| 28 | 18 | 10 |
| 29 | 13 | 12 |
| 30 | 8 | 14 |
| 31 | 5 | 17 |
| 32 | 3 | 19 |
| 33 | 2 | 20 |
| 34 | 1 | 21 |
| 35 | 1 | 22 |
| 36 | 0 | 22 |
| 37 | 0 | 22 |
| 38 | 0 | 20 |
| 39 | 0 | 19 |
| 40 | 0 | 17 |
| 41 | 0 | 15 |
| 42 | 0 | 13 |
| 43 | 0 | 11 |
| 44 | 0 | 9 |
| 45 | 0 | 7 |
| 46 | 0 | 5 |
| 47 | 0 | 4 |
| 48 | 0 | 3 |
| 49 | 0 | 2 |
| 50 | 0 | 2 |
| 51 | 0 | 1 |
| 52 | 0 | 1 |
| 53 | 0 | 1 |
| 54 | 0 | 0 |
| 55 | 0 | 0 |
| 56 | 0 | 0 |
| 57 | 0 | 0 |
| 58 | 0 | 0 |
| 59 | 0 | 0 |
| 60 | 0 | 0 |
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.
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