Six Sigma Explained

Oh, another "n things" today, but this one is really on statistical side. Notice the "sigma" word, yes it is the $\sigma$ as the standard deviation in statistics as we all familiar with.

Six sigma is an improvement technique to deliver high quality products and services whist removing inefficiency.  First we need to understand the DMIAC approach.

Six sigma methodology improved "existing" process by contantly reviewing and re-tuning the process using DMAIC.

Now, what about the "sigma"?
A good brief introduction here give an example. Consider that you run a pizza delivery business and you set the target of delivering pizza's within 25 minutes of receiving the order. If you achieve that 68% of the time, that is 1 sigma. If you can do that 99.9997% of the time (late 3.4 times in one million orders), it is 6 sigma.

In terms of defects, Six Sigma quality means there are no more than 3.4 defects per million opportunities (DPMO). The sigma is the standard deviations away from the mean point in a bell curve (normal distribution).

Got it?, No I did not.

Actually,many things are wrong here. First the 68% is actually the area of the curve between -1 sigma to 1 sigma around the mean as shown by this MATLAB code:

>> normcdf(1,0,1) - normcdf(-1,0,1)

ans =

   0.682689492137086

or

>> p = normspec([-1,1],0,1)

p =

   0.682689492137086

For the pizza problem, if we deliver pizza ways under 25 minutes (the means), says 10 minutes, it is acceptable. So it is one-tailed not two-tailed problem. The right answer for pizza problem is:

>> norminv(.68,0,1)

ans =

   0.467698799114509

Or, if we deliver pizza 68% within the order time (no matter what), it is 0.47 sigma.

For one sigma, we need to do it 84% of the time.

>> normcdf(1,0,1)

ans =

   0.841344746068543

The second mistake is, even we do two-tailed version of six-sigma, we are not going to get the 3.4 defects ppm.

>> 1-(normcdf(6,0,1) - normcdf(-6,0,1))

ans =

    1.973175400848959e-009

Actually, for six-sigma, we are going to get 2 defects per billion opportunities.

Aha, something is fishy here. Everyone accept that six sigma means 3.4 ppm. How's so?

A perfect statical explanation is found here. The truth is, 3.4 ppm comes from the assumption that the process mean might shift 1.5 sigma either to the left or right of the original mean, which is normal for a process in a long run.

In this case, we can again use MATLAB to check our understanding.
>> normcdf(-6,-1.5,1)

ans =

    3.397673124730062e-006

That's it!!!. That the 3.4 ppm everyone is talking about.

In the meantime, the 2 defects per billion opportunities we got before is for the six sigma process with 0.0 shift in the mean.

In manufacturing the variation in measurement in processes tend to fall into a normal distribution, for example, the dimensions of parts.

However, an article published in Fortune stated that "of the 58 large companies that have announce Six Sigma programs, 91 percent have trailed the S&P 500 since". The analysis is that, while six sigma is good what it is at, which is "fixing an existing process", it does not help in coming up with new products or disruptive technologies.