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About this lesson
The Benchmark Z provides a second method for determining process capability. The 1.5 sigma shift accounts for long-term stability within the process variation.
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Quick reference
Benchmark Z and 1.5 Sigma Shift
The Benchmark Z provides an alternative method for determining process capability based on Z scores. The 1.5 sigma shift also relies on a Z score analysis to account for long-term stability within the process variation.
When to use
The Benchmark Z and 1.5 sigma shift are normally used during the Analyze phase to provide greater insight into process performance.
Instructions
The 1.5 sigma shift and the Benchmark Z score rely on the Z transformation methodology. They related a process capability measure known as process sigma and the Z transformation to provide insight into process performance.
1.5 Sigma Shift
The 1.5 sigma shift is based on an observation from Motorola. Even though they improved a process so that in the short term it achieved a process capability level of 2.0 or greater, over time the process would shift slightly and the capability would erode to approximately 1.5 without the introduction of any new special causes. A process capability of 2 implies that the mean is at least 6.0 sigma away from the closest spec limit, hence the name Six Sigma. However, instead of claiming that the process would have a yield associated with a Z score of 6, which is 99.9999998% (2 in a billion), Motorola claimed that the long-term yield would be in line with a Z score of 4.5, which is 3.4 defects per million. There is no formal theory behind the 1.5 sigma shift, rather it is based on practical real-world experience.
Benchmark Z
Benchmark Z is a technique for expressing process capability when the process creates defects on both ends of the allowable specification limits. Benchmark Z, also known as Bench Z or Z Bench, combines the percentage yield associated with both out-of-spec conditions and converts it to one Z score. This Z score can be treated as a type of process capability score. The Benchmark Z can be done for either a short-term sample or a long-term full population.
To calculate the Benchmark Z, first the mean and standard deviation of the distribution representing process performance must be determined. With these, the Z score for the upper spec limit and the Z score for the lower spec limit can be calculated. These are then used with the Z Transformation lookup tables to determine a percentage defective at each spec limit. The two percentages are added together and an overall yield percentage can be determined. This yield percentage is used with the right tail Z Transformation table to find a Z score that is associated with this percentage. That is the Benchmark Z.
Hints & tips
- Some statistical theoreticians do not acknowledge the 1.5 sigma shift, but the practical Lean Six Sigma black belts and green belts recognize the validity.
- Benchmark Z is closely coupled with Attribute process capability which also relies on percentage yield to determine process capability.
- 00:04 Hi, this is Ray Sheen.
- 00:06 I want to take a few minutes and dig into some advanced concepts for
- 00:10 how we determine process capability.
- 00:13 First will be the 1.5 Sigma shift,
- 00:15 the concept that I briefly alluded to when discussing process capability.
- 00:20 For starters, if you actually do the math with a frequency distribution and
- 00:25 a process operating at the 6 sigma level,
- 00:27 it turns out that the yield is close to 2 in a billion.
- 00:31 But we only claim that a 6 sigma process can achieve 3.4 defects per million.
- 00:37 If you go to a version of the Z lookup table that has at least 7 significant
- 00:42 digits, you will find that this corresponds to a sigma of 4.5.
- 00:46 That's quite a difference.
- 00:48 So why do we slide the statistic value from 6 sigma back to 4.5?
- 00:54 This is based upon real world data, not a arcane statistical theory.
- 00:59 When Motorola started tracking process sigma level,
- 01:02 one of the things they noticed is that when they would run a new sample and
- 01:06 a process, they would get slightly different descriptive statistics.
- 01:10 This is normal, samples are only a subset of the full population.
- 01:15 So they will be different, because they don't have the exact same data points.
- 01:19 In Motorola's experience, the subset would drift about 1.5 standard
- 01:24 deviations up or down the scale, hence the term 1.5 sigma shift.
- 01:29 The implication is once you get a process to a 6 sigma level,
- 01:33 the long-term sustainability of that process is only 4.5 sigma.
- 01:37 The difference in sigma level accounts for
- 01:40 why when we say the process is a Six Sigma process,
- 01:44 we only give the process quality credit for a 4.5 sigma level yield.
- 01:50 Let's look at the implication of this shift.
- 01:52 In this table, the first three columns are the sigma level, the normal curve,
- 01:57 calculated yield, and the attribute data, DPMO for that sigma level.
- 02:02 The last two columns are the interesting ones.
- 02:06 The fourth column is the DPMO if we shift 1.5 sigma levels, and
- 02:10 the fifth column is the yield associated with that shifted level of DPMO.
- 02:15 Let's look at an example with the sigma level of 2.
- 02:18 The short- term yield is 22,700 DPMO.
- 02:22 But when the 1.5 sigma shift is put in place, the long-term process capability,
- 02:28 DPMO is 308,000, which comes from the 0.5 sigma level.
- 02:33 This means that the long-term yield is only 69.2%.
- 02:38 The same pattern follows with all of the other sigma levels.
- 02:42 So when we look at the last line in the table,
- 02:46 the six sigma line, we see that the DPMO and
- 02:50 yield are shifted down from the 4.5 sigma level and
- 02:55 we have a DPMO of 3.4 and a yield of 99.99966%.
- 03:00 One additional topic in this general area is the Benchmark Z.
- 03:05 The Benchmark Z, which is sometimes called Z Bench or Z-Benchmark, all these
- 03:11 terms are used for the same thing, is another way to express process capability.
- 03:16 The Benchmark Z is determined by combining the number of out of spec parts
- 03:22 all on one side of the graph and then using the Z-score for that point.
- 03:27 Let me explain.
- 03:28 Let's say your process is close to being centered, but
- 03:32 the sigma level is not very good.
- 03:34 Your process is stable, but sometimes it creates parts that are above the upper
- 03:39 spec limit and some parts that are below the lower spec limit.
- 03:42 Rather than trying to work with two different Z-scores, one upper and
- 03:47 one lower, we just put all the points on one side and use one Z-score.
- 03:52 Incidentally, this is the principle behind the attribute data process capability.
- 03:58 The short-term Benchmark Z is based on the sample data,
- 04:02 and therefore, the sample standard deviation.
- 04:05 The long-term Benchmark Z will be based upon the standard deviation for
- 04:10 the full population.
- 04:11 You may recall that the formula for sample and
- 04:14 population standard deviations were slightly different.
- 04:18 The difference between the long-term and
- 04:21 short-term Benchmark Z is referred to as the Z Shift.
- 04:25 That value can be significantly affected by the number of data points in
- 04:29 the sample.
- 04:30 The 1 .5 sigma shift is a little squirrely,
- 04:33 but it does represent the reality of most manufacturing processes.
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