Shouldn't a two sigma process be better than six sigma process? Going by basic statistics and common sense, I think a two sigma process should be better than six sigma process because it will be closer to mean and have less deviation. The number of defects in a six sigma process are less because we have taken a huge range around the mean for consideration as compared to two sigma deviation. For example if diameter of a pipe should be 3.50 cm then the two sigma process would have range of 3.49 to 3.51 as compared to 3.46 to 3.54 in case of six sigma. Aren't we performing better when we are more close to the standard of 3.50 which is two sigma in this case?
I know I am wrong here and but what is it that I am missing ?

 A: I believe what you're missing is that the tolerance outside which items are considered defective is fixed by prescription.
If the prescribed acceptable tolerance is $\ m-\delta\ $ to $\ m+\delta\ $, then a $\ d$-sigma process is one whose standard deviation is $\ \sigma=\frac{\delta}{d}\ $, and whose mean $\ \mu\ $ satisfies $\ m-\frac{3\sigma}{2}\le\mu\le m+\frac{3\sigma}{2}\ $.  In the worst case, when the process mean lies at one of its allowable extremes, the fraction of defective items produced is $\ \mathcal{N}\left(d+1.5\right)-\mathcal{N}\left(-d+1.5\right)=$$\mathcal{N}\left(d-1.5\right)-\mathcal{N}\left(-d-1.5\right)\ $, where $\ \mathcal{N}\ $ is the standard normal distribution. For $\ d=6\ $ this evaluates to approximately $\ 3.4\times10^{-6}\ $, whereas for $\ d=2\ $ it's approximately $\ 0.31\ $.
A: You're worrying about the wrong error.
You'll never get a 3.5 cm pipe. A real pipe has an irrational diameter that varies depending on where you measure it. In fact, its impossible to say "the diameter is ___". We can, however, say "the diameter isn't ___". As in hypothesis testing, we hope for a process that minimizes the size of the set of "the diameter isn't 3.5 cms" pipes. This size of set error, not the difference from 3.5 cm, is the important error.
By assumption, regardless of process, all the pipes' diameters are normally distributed. We want to get a set of pipes that are 3.5 cm, give or take. The two sigma process throws away many more mistakes than the six sigma process.

No, I used to candle eggs at his farm. Do you know what that is? You hold an egg up to the light of a candle and you look for imperfections. The first time I did it he told me to put all the eggs that were cracked or flawed into a bucket for the bakery. And he came back an hour later, and there were 300 eggs in the bakery bucket. He asked me what the hell I was doing. I found a flaw in every single one of them - you know, thin places in the shell; fine, hairline cracks. You look closely enough, you'll find that everything has a weak spot where it can break, sooner or later. - Anthony Hopkins as Ted Crawford, Fracture (2007)


Edit: its worth underlining lonza leggiera and almagest’s point: it makes the most sense to compare processes with the same fixed tolerance. The 2-sigma and 6-sigma processes have different distributions/sigmas.
