1.5 Sigma Process Shift Explanation

iSixSigma released a process sigma calculator which allows the operator to input process opportunities and defects and easily calculate the process sigma to determine how close (or far) a process is from 6 sigma. One of the caveats written in fine print refers to the calculator using a default process shift of 1.5 sigma. From an earlier poll, greater than 50 percent of polled quality professionals indicated that they are not aware of why a process may shift 1.5 sigma. My goal is to explain it here.

I am not going to bore you with the hard core statistics. Every Green, Black and Master Black Belt learns the calculation process in class. If you did not go to class (or you forgot!), the table of the standard normal distribution is used in calculating the process sigma. Most of these tables, however, end at a z value of about 3. In 1992, Motorola published a book (see chapter 6) entitled Six Sigma Producibility Analysis and Process Characterization, written by Mikel J. Harry and J. Ronald Lawson. In it is one of the only tables showing the standard normal distribution table out to a z value of 6.

Using this table you’ll find that 6 sigma actually translates to about 2 defects per billion opportunities, and 3.4 defects per million opportunities, which we normally define as 6 sigma, really corresponds to a sigma value of 4.5. Where does this 1.5 sigma difference come from? Motorola has determined, through years of process and data collection, that processes vary and drift over time – what they call the Long-Term Dynamic Mean Variation. This variation typically falls between 1.4 and 1.6.

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After a process has been improved using the Six Sigma DMAIC methodology, we calculate the process standard deviation and sigma value. These are considered to be short-term values because the data only contains common cause variationDMAIC projects and the associated collection of process data occur over a period of months, rather than years. Long-term data, on the other hand, contains common cause variation and special (or assignable) cause variation. Because short-term data does not contain this special cause variation, it will typically be of a higher process capability than the long-term data. This difference is the 1.5 sigma shift. Given adequate process data, you can determine the factor most appropriate for your process.

In Six Sigma, The Breakthrough Management Strategy Revolutionizing The World’s Top Corporations, Harry and Schroeder write:

“By offsetting normal distribution by a 1.5 standard deviation on either side, the adjustment takes into account what happens to every process over many cycles of manufacturing… Simply put, accommodating shift and drift is our ‘fudge factor,’ or a way to allow for unexpected errors or movement over time. Using 1.5 sigma as a standard deviation gives us a strong advantage in improving quality not only in industrial process and designs, but in commercial processes as well. It allows us to design products and services that are relatively impervious, or ‘robust,’ to natural, unavoidable sources of variation in processes, components, and materials.”

Statistical Take Away: The reporting convention of Six Sigma requires the process capability to be reported in short-term sigma – without the presence of special cause variation. Long-term sigma is determined by subtracting 1.5 sigma from our short-term sigma calculation to account for the process shift that is known to occur over time.

Comments 11

  1. mark casey

    Well explained and the last paragraph “stastical take away” summarizes perfectly.

  2. william harper

    so, is there anyone who takes it all into account and does not use any fudge factors?

    cuz that is who i want to work for. . .

  3. Rajeev

    Perfect explanation – Buy-in communication about the subject

  4. Vinod Bhardwaj

    Nicely explained.

  5. vivek kestur

    Excellent explanation.. very simple and informative

  6. Antonio Belardo

    Very clear and effective explanation!

  7. George Ferver

    The author seems to have mixed what occurs in processes over time, notably in the ‘Statistical Take Away’ – probably a typographical error, but it needs correction as there are readers who don’t understand either processes or statistics.
    To correct the error, please replace the Long Term sigma sentence with: “Long-term sigma is determined by /adding/ 1.5 sigma /to/ our short-term sigma calculation to account for the process shift that is known to occur over time.”
    This correction reflects that long-term error has all the short-term common cause variability, plus the long-term shifting that occurs. And that’s why AIAG manuals state short term capability may be 1.33 Cpk, but Long Term capability may be 1.66 Cpk (yes, the AIAG math adds 1 sigma, rather than 1.5 sigma).

  8. Manny Fernandez Jr.

    I have read the above, I just don’t buy into the math. Here is the problem: By definition a process that is in statistical control and models a normal distribution will find that 99.7% of output falls within the curve and + – 3 standard deviations. If the process goes beyond +- 3 standard definitions then technically it is not longer in statistical control. So when you add 1 1/2 std definition to the curve on both sides giving you a +-4.5 curve you no longer have a statistically controlled process. I understand the logic, it just does not pan out. Yes we need to make our processes robust to handle variation and still deliver. I am trying to explain this to my Green Belt Class. I don’t really want to tell them to ignore it all. But I also do not want to provide incorrect information.

  9. Brandon Thomas

    ok so I have a question, I am currently in my green belt course and heard a 6sigma level translates to 3.4ppm rejects or a 99.73% success rate. why at the 8 sigma level does it translate to 64ppm rejects? if your sigma level improves I thought your rejects went down.

  10. Raj Mitra

    Z-shift is one of the most misunderstood concepts in Six Sigma.
    In reality, the 1.5 sigma shift was part of Shewhart’s methodology for computing Control Limits before the time of computers. Shewhart recommended the use of a sample size of 4 to make calculations easier for control limits (Control Limit = Mean +/- 3*Standard Deviation / sqrt (N, or Sample Size))
    With individual data, the Control Limit would be 3 Sigma, however with a sample size of 4, the control limit must change, because using the mean would reduce the observed variation. Applying the precise calculation would mean a control limit of 1.5 Sigma. A well written paper to further explain can be found here:

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