I still do not understand

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    Rachel Kozlowski

    I have worked with means and standard deviations for years in medical technology.  However, what is a sigma? I have never heard this term before.  How does this shake down with means and standard deviations?  I hope that this does not mean that qc results should fall between six standard deviations, you might as well draw a bullseye and blindfold yourself.  As long as it is near the target it would be in and I find that unacceptable.  I am a student still so please excuse my ignorance and educate me.  I need more information on what a sigma is.



    I can empathise with where you are coming from – but 6S is to standard deviation as Emmental is to Cheddar – both are palatable, but one is more preferable.
    Perhaps the following may help put things in perspective
                                                                    Short Term Failure Rate Failure  PPM              CpK             6 Sigma Value >1 in 2  > 50%            > 500000    < .33          <1.6  1 in 3  < 33%   .51                      2 1 in 8  < 12.5%  .67            2.75 1 in 20  < 5%    .67            3.15 1 in 80  < 1.25%  .83            3.75 1 in 400 < .25%   1.00          4.3 1 in 2,000 < .05%   1.17          4.8 1 in 15,000 < .0067%  1.33           5.3 1 in 150,000 < .00067%  1.50         5.85 1 in 1,500,000 < .000067%  1.67           6
    Sorry the formatting hasn’t quite worked out during copy/paste (6S in software – that’ll be the day)
    The choice of Sigma as a name is I believe, an error, as it is very similar to the std. dev. usage – however its application is different.  Some practioners have made comparisons with SD such as “If we looked at six SD then 99.9999999997% of our product would satisfy customer expectations” – which only helps to cloud the water even more.
    6 Sigma sigma is only a value – treat it as such, and life gets a lot easier.
    Does this help?


    Ken K.

    I’ll take a crack at this too . . .
    If you are familiar with means and standard deviations then you’ve got a good start.
    First, lets talk about means. Suppose you wanted to measure the amout of radiation each patient receives from a CAT scan from you rmodel XY321 CAT scan equipment. Certainly there is some variation there, and I could see where the distribution might be normally distributed. For the rest of this discussion I am assuming normality.
    So the POPULATION we are focusing on is “all people who receive CAT scans from a model XY321 scanning of equipement”. If we could gather the measurements from every patient who will ever use this equipment (impossible task) and average them, that value would be the POPULATION MEAN, represented by the Greek letter MU.
    Since we can’t really gather every patient, instead we take a SAMPLE of patients – say we measure the exposure for 200 patients. We take those 200 values and calculate the sample’s mean. This mean is our best estimate of the true, but unknowable, population mean, mu.
    Now, we can do the same thing with variation. If we had every patient, we could calculate the POPULATION VARIENCE using the sum of the squared deviations from the mean, divided by n (I’m using n for the population variance). The square root of the population variance is the POPULATION STANDARD DEVIATION, represented by the Greek letter SIGMA.
    Since we can’t really gather every patient, instead we take that SAMPLE of patients – the 200 patients – and calculate the sample’s standard deviation (you know this formula – dividing by n-1 for a sample). This standard deviation is our best estimate of the true, but unknowable, population standard deviatoin, sigma.
    Now, for a normal distribution, it turns out that the standard deviation has some pretty cool properties related to the width of the distribution:
    1. The inflection points on either side of the peak a normal distribution – the places where the slope changes from increasing to decreasing on the left side and visa versa on the right side – will always be exactly one standard deviation away from the mean. At least I think that is kind of nifty.
    2. Regardless of the exact values of the mean and standard deviation, we can make some general statements about the probability of observing certain values from the distribution:
    38.3% of the data will fall within 0.5 standard deviations of the mean     (the range defined by (mu – 1/2sigma) & (mu + 1/2sigma)
    68.3% of the data will fall within 1 standard deviations of the mean
    95% of the data will fall within 1.96 standard deviations of the mean
    95.4% . . . . . 2 standard deviations
    99.7% . . . . . 3 standard deivations  . . . etc…
    In Six Sigma world of quality, we also talk about a quality metric called “Sigma” (I tend to capitalize the term when referring to the metric and use lower case when referring to the population standard deviation).  This is easiest to explain by giving a specific example:
    Six Sigma: In this scenario the upper & lower spec limits are sitting precisely at (target + 6*sigma) and (target – 6*sigma). If our process is exactly centered on the target, we will only see 0.002PPM falling outside our spec limits (to get PPM we multiply the proportion of defects by one million, just to make these tiny numbers easier to work with).
    But, if our process shifts by 1.5 sigma, our mean will be sitting 1.5 sigmas above or below the target. The spec limits don’t shift, so one spec limit will now be sitting at 7.5 standard deviations from the mean – we will see essentially zero defects that far out. The other spec limit will now be sitting at 4.5 standard deivaitons – we expect to see 3.4PPM beyond that spec limit.
    So, in a 1.5 sigma shifted – six sigma scenario, we would expect a total of 3.4PPM defect rate. This is how we get 3.4PPM related to Six Sigma.
    For a three sigma scenario, the spec limits are sitting at +&- 3 standard deviations (sigmas) from the target. If the distribution is shifted 1.5 standad deivaitons (remember the spec limits don’t shift), then one spec limit is now at 4.5 standard deviations from the mean and the other spec limit is at 1.5 standard deviations. We can expect 3.4PPM beyond the 4.5 sigma spec limit and we can expect 66,807.2PPM beyond the 1.5 sigma spec limt. Combined we expect 66,810.6PPM from the 1.5 sigma shifted – three sigma scenario. This is how 66,811PPM is associated with 3 Sigma – although many people, for some reason, ignore the 3.4PPM on the one side and use simply 66,807PPM.


    Erik L

    If you’re comfortable with means and standard deviation, then it won’t be too much of a stretch to incorporate sigma levels into your vernacular. The easiest way to visualize the linkage is through the Cpk calculation. The only requirement will be that you know the specification limits (upper and lower). Once you have that and your calculations for mean and std deviation, you can get the sigma level by taking 3*Cpk(min). For simplicity you can get the sigma level by taking Zmin.



    From one neophyte to another: 
    Assume I am an Olympic archer.  I shoot 100 arrows and miss the bullseye on averge by 3 inches .  Say the std dev is .5 of an inch.  Now call the std dev a “sigma” and not a “std dev”.  I then move 3 sigmas   (3 x std dev) off the 3 inch mean.  If my technique (aka process) does not change and was stable and well defined to begin with, I can expect 99.7% of all future shots to fall within 3 sigmas of the mean, or within “6 sigma” (3 per side of the mean).  The use of sigma allows us characterize our process as to its stability, or its ability to perform in an expected way.  Note that it has nothing to do with meeting product specifications.  I could shoot all day long and the closest I will come using the current process is 1.5 inches OFF target.  Only by introducing something new (i.e. new equipment, different technique, etc) can I hope to move my process average closer to the target and reduce the variation.  You would have to ensure the process is capable of hitting the target and then properly aim at it.  I must be stable and capable and centered in order to hit my target, and there are statistical methods to determine these things.  Hope it helped……I know I took a few liberties, but if I have stated anything that is GROSSLY incorrect, somebody please correct me. 



    Rachel,I decided to look at the definition of “sigma” provided by the iSixSigma dictionary. Generally, their definition does a reasonable job, as follows:”Sigma is a statistical term that measures how much a process varies from perfection, based on the number of defects per million units.(I would drop the last part in this definition)Essentially, “sigma” value is a quality measure. While “sigma” is the name of the variable that stands for the measure of variation, aka. standard deviation. “Sigma” value can be thought of as a relationship between the requirements and the process variability. Sometimes this relationship has also been referred as the Voice of the Customer(Requirements) and the Voice of the Process(Variability). As process variablity decreases, with fixed requirements, the sigma value increases. Another way to look at “sigma” value is as the level of control available to the process, or as the process elbow room. Generally, a 4-sigma process will have specifications located 4 standard deviations from the process target. Following this example, a six sigma process will have specifications located 6 standard deviations from the target. What standard deviations do we speak of here? Ans: the standard deviations of the process.If we assume the process produces values whose histogram looks like a bell-shaped curve, and the process maintains the average output within +/-1.5 standard deviations from the target, then we can compute the expected defect level produced from the process for any given sigma value, as shown in the table below:One Sigma = 690,000 per million units
    Two Sigma = 308,000 per million units
    Three Sigma = 66,800 per million units
    Four Sigma = 6,210 per million units
    Five Sigma = 230 per million units
    Six Sigma = 3.4 per million unitsHope this clears up the confusion…Ken Myers

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