I had a tough time to understand 6σ. I guess the best way is to understand it statistically.

The well-known 6σ is looking at DPMO below 3.401. But why the real 6σ corresponds to DPMO below 0.001?

The well-known 6σ is essentially 4.5σ. We are given a discount of 1.5σ. It is known as 1.5σ shift.

Some authors say the 1.5σ can shift in either direction. Again this is confusing.

It has to be single tail for the 1.5σ shift to be true. If it has two tails, the well-known 6σ is essentially 4.6452σ

and we are given a discount of 1.3548σ.

Note:

DPMO = Defects Per Million Opportunities