# sigma levels and central limit theorem

Six Sigma – iSixSigma Forums Old Forums General sigma levels and central limit theorem

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• #43004

howe
Participant

Can someone provide a brief explanation as to the difference between std deviation relating to the central limit thoerem (68.26; 95.46; 99.73) and sigma levels (69.16; 99.37; 99.9997)?
It gets confusing to someone new to six sigma when talk is about standard deviation yet there are two different values depending on the conversation.
Thanks

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#136071

Craig
Participant

Central limit theorem indicates that regardless of the underlying distribution, you can take samples of size n, and the distribution of the averages will be normal.
I am not sure where your numbers came from, but it looks like they represent  % under the normal curve  at 1,2, and 3 sigma.  Can you elaborate on where you got the 2 sets of numbers?

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#136072

howe
Participant

The numbers I provided (68.26/95.46 and 99.73) are the percentages found under a normal distribution.  I heard someone calling those standard deviations.  At the same time, someone else was talking about sigma levels and using the term standard deviations, yet when compared to the normal distribution percentages, the percentages used with sigma claculations are slightly different (69.15, 99.37 and 99.9997).  Is this just a misuse of the term standard deviation?  If not, how can I be at 68.26% under a normal distribution (+/- 1 std dev) and be at 69.15% at a two sigma level (+/- 1 std deviation)?  Seems confusing.
Thanks

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#136313

XXXX
Member

Hi Mike
If your process produces 68.27 defect free product out of 100, you are at 1 sigma level
Similarly if your process produces 95.45 defect free product out of 100, you are at 2 sigma level, similarly it proceeds.
This is the percentages you have written there.
The formula to calculate sigma level are as follows.
If your data is continous, first you find the probability that are out of the target by normdist function in excel which is normdist(X, mean, stdev, 1)
Then using normsinv function you can find the sigma level which is 1.5+normsinv(first time right proportion)
Suppose if your process produces 99 defective pieces out of 1500 pieces then the sigma level for that process is
X=99/1500=0.066
Sigma level = 1.5+normsinv(1-0.066) = 3.00
You are at 3 sigma level.
Regards

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#136330

Ropp
Participant

It’s important to realize that the Central Limit Theorem is a large sample concept and only applies as n becomes large and only if sigma is known.
Dave

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#136331

Craig
Participant

The numbers in the table are the areas under the normal curve between +/- a prescribed number of standard deviations. (Z score). Regardless of how you estimate the standard deviation, you can look up probabilities from the table. Make sure you understand if the table is giving one tail of the distribution or 2.  Basically, the table is for answering questions like:
If the average height of the adult male population is 70 inches and the standard deviation is 2 inches, what is the probability that a randomly selected adult male will be taller than 80 inches?
The first step is to obtain the Z-score: (80-70)/2 = 5.  The question can be re-phrased as: What is the probability of selecting an adult male who is 5 standard deviations above the average height? Find Z=2 on the table and obtain the associated probability. Make sure you know how to read the table. If the values max out at 0.5000, the table represents the left or right area under the curve.  You will most likely have to subtract the value found at Z=2  from 0.5000 to get the right-tail probability.
Sigma does not have to be known for the central limit theorem (CLT) to apply. As sample size increases, the distribution of the averages of becomes more normal. This discussion and the one above are really not related, so don’t confuse them. The discussion above assumes that the distribution of heights of individual males is approximately normal. The clt allows you to do stats on sample averages even though the underlying distribution of individual values may not be normal.
The confusion comes in with the 1.5 sigma shift in the Six Sigma world. If you were to take the right tail probability from the example above and try to pin a sigma value on it, the straighforward answer would be 5 sigma.  If you plugged the probability value into a sigma conversion chart, you might get 6.5 sigma. Give it a try.

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#136445

Jonathon Andell
Participant

Use raw data to estimate process sigma (“Z”) or process capability (Cpk). That gives you the probability that a given unit of output will (or will not) meet customer requirements.Use Central Limit Theorem to conduct a hypothesis test of whether a process change has shifted the mean of the process, even if that shift lies inside the typical span of process observations.

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