3.4 PER MILLION
3.4 PER MILLION
What Is 3.4 per
by Joseph G. Voelkel
n 1969, the Chicago Transit Authority released the song “Does Anybody Really Know What Time It Is?” The lyrics read:
As I was walking down the street one day,
A man came up to me and asked me what the time was that was on my watch,
And I said,
Does anybody really know what time it is?
Along the same lines, does anybody really know what 3.4 parts per million (ppm) is? To see if you do, take the quiz in the box below. No looking at the answers first.
When Motorola was connecting the notion of 6∋ to ppm levels, it assumed a process might shift 1.5∋ from its target. So Motorola based 3.4 ppm on such an assumption. I say “assumption” because I am unaware of any formal analysis done to substantiate this claim. In fact, such a shift should be considered very unlikely, as I will explain later.
So, the 3.4 ppm is connected, in some way, with a putative 1.5∋ shift. Now, let’s review the possible quiz answers.
If you answered (a), you may be a Six Sigma novice. The fraction that falls outside this range is actually 0.27%. This sounds small when expressed as a percentage, and in fact there are many process owners who would be delighted at such a defect rate. However, when expressed as ppm, it amounts to a whopping 2,700 ppm and is definitely not Six Sigma quality.
If you answered (b), you may know enough about Six Sigma to be dangerous. The actual ppm here is a small 0.002 ppm. Though this would be a much better ppm number to associate with Six Sigma, it is definitely not 3.4 ppm.
So, does this mean if you answered (c) you would be correct? No. The fraction that falls outside that range is actually 6.8 ppm. The correct answer is (d) “none of the above.” The actual value of 3.4 ppm refers to the fraction that would fall either above µ + 4.5∋ or below µ – 4.5∋.
Confusing, isn’t it? If you didn’t answer this question correctly, you are not alone. Ironically, part of the very nature of Six Sigma, precision of operation, is abandoned by this awkward and tenuous connection.1
Now let’s consider three simple process scenarios to see which ppm levels would be generated by each while keeping the 1.5∋ shift in mind. For each scenario, assume the process, if running ideally, is centered within two-sided specification limits that are symmetrical and precisely at µ ± 6∋. That is, the process is on target and is 6∋ away from each limit. This ideal process would produce the 0.002 ppm defect rate.
Scenario one: Let’s say the process is always running either 1.5∋ above the target or 1.5∋ below the target. In such a case, the 3.4 ppm defect rate would occur, but it is totally unrealistic for what is supposed to be a Six Sigma process (see second scenario).
Scenario two: What if the process suddenly undergoes a 1.5∋ shift? This is not too realistic, but could happen if, for example, a machine were set up improperly. Let’s say the shift happened at the beginning of the week on a process that is run for eight hours a day, seven days per week, and is monitored every two hours using a standard X–/R control chart with a subgroup size of n = 4. A mathematical analysis shows a 50% chance of detecting the 1.5∋ shift in each subgroup using the standard ±3∋ control chart detecting rule.
Suppose once this out of control condition is detected the process is corrected back to target. This means the process would run at a 3.4 ppm defect rate for two hours with a 50% chance of detecting the 1.5∋ shift, for four hours with a 25% chance, for eight hours with a 12.5% chance and so on. Then, even if such product were released, the average ppm defect rate for the entire week turns out to be only 0.24 ppm or 240 parts per billion (ppb). This is because the monitoring process would most likely catch the out of control condition early in the week.
If you consider different amounts of sudden process shifts in this scenario, you would need to have a sudden process shift of 2.25∋ to have a 3.4 ppm for the entire week. This is shown in Figure 1 (p. 63), where the results are plotted on a ppb scale for better readability.2 Remember, no shift corresponds to 2 ppb, and a 1.5∋ shift corresponds to 240 ppb. Even if a sudden 2.25∋ shift were to occur, this would shift the process up 4.5∋ on the chart, would be detected with a 93% chance at the first subgroup and would quite possibly result in some action to prevent the release of the material produced during that short timeframe.
Scenario three: Let’s say the process drifts around. A controller is attached to the process, and it keeps the process on target on an average long-term basis; however, the process mean drifts in a sinusoidal (sine wave) pattern of ±k∋ around the target. (See Figure 2 for an example of a pattern of ±1∋.) Such controllers are common in industry and may be thought of as engineering process control instead of statistical process control.3
If the maximum process shift that occurs under this scenario is 1.5∋, the resulting defect rate is 1.02 ppm. This defect rate is higher than the rate due to the sudden process shift scenario of 1.5∋ for two reasons:
1. There is no detection step that would lead to adjusting the process back to the target value.
2. The nature of sine waves means the process spends more time near the extreme values of the drift than in the center of the process. For example, in Figure 2, the process is more than 0.5∋ from the target—half of the maximum drift—for two-thirds of the total process time.
Even if this is the case, it would take a maximum process drift of 1.78∋ to produce a 3.4 ppm defect rate. This value, as well as those corresponding to 0∋ and 1.5∋, is highlighted in Figure 3.
Evidently, there is no rational place for 3.4 ppm in a Six Sigma world even though it is no doubt here to stay. This is OK—the need for process improvement is more compelling than the need to correct values that have only a secondary role in such improvement—but it is worth keeping in mind.
The next line in the Chicago Transit Authority’s song asks:
Does anybody really care?
Well, I care, and I hope you care, too. And if enough of us do care, we will have no need for the next line:
If so I can’t imagine why.
REFERENCES AND NOTES
1. A similar point is made by Kenneth S. Stephens in “Your Opinion: What Is the Future of Six Sigma?” Six Sigma Forum Magazine, November 2001, pp. 47-48.
2. Details on finding ppm values under scenarios two and three may be found in “What Is 3.4 Parts per Million?” a technical report by Joseph G. Voelkel published by the Center for Quality and Applied Statistics, www.rit.edu/eng/cqas.
3. For an understanding of both control methods from a statistical perspective, see George E.P. Box and Alberto Luceño, Statistical Control: By Monitoring and Feedback Adjustment, Wiley-Interscience, 1997.
JOSEPH G. VOELKEL is graduate program chair and associate professor at the John D. Hromi Center for Quality and Applied Statistics at the Rochester Institute of Technology in Rochester, NY. He earned a doctorate in statistics at the University of Wisconsin-Madison. Voelkel is a Fellow of ASQ.
The 3.4 per Million Quiz
If the quality characteristic being measured has a normal distribution with mean µ and standard deviation ∋, then 3.4 parts per million is:
a. The fraction that
outside µ ± 3∋.
b. The fraction that
outside µ ± 6∋.
c. The fraction that
outside µ ± 4.5∋.
d. None of the above.
This term is based
on the assumption
a process might shift 1.5∋ from its target.
Weekly Defect Rate vs. Amount
Of Sudden Process Shift
QUALITYPROGRESS I MAY 2004 I 63
3.4 PER MILLION
64 I MAY 2004 I www.asq.org
Example of Process Mean
Drifts From Target Over Time
Defect Rate vs. Maximum Amount of Off-Target Drift of Process Mean From Target
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QUALITYPROGRESS I MAY 2004 I 65