1.5 σ Shift in the Six Sigma Process

The Sigma (σ) of a process is calculated based on the ability of the process to stay within the specification limits.

Assuming that the mean of the process is in the centre of the Upper Specification Limit (USL) and Lower Specification Limit (LSL), the process has the same probability of staying inside the limit on EITHER side.
Now in most cases, the process mean doesn’t stay exactly in the centre of the specification limit. The actual location of the mean is closer to one side than the other. This is called the Shift of the Mean.
This means that the distance between the mean and specification limit is not the same on both sides. I.e. the process has more leeway on one side than the other and the probability of the process to stray outside the specification limit is higher on one side than the other.

Since the performance of the process is measured based on its capability to stay inside the specification limit on EITHER side, the σ value is calculated by looking at the SHORTER distance between the mean and the specification limits.

I.e. Sigma level = Lower of [(USL – Mean) OR (Mean – LSL)] / σ

The SHORTER gap divided by σ will give the sigma level of the process.

E.g. let’s consider a 6 σ process. Here 6 σ is the theoretical σ level of the process, i.e. the distance from mean to LSL and USL to mean is the same and is 6 σ. The mean is assumed to be in the centre of the specification limits.

Let’s say that the actual mean of the process, when measured, appears to have shifted 1.5 σ to one side.

Now, the new Sigma level of the process is,

Theoretical Sigma Level – Shift of the Mean
i.e. 6 σ – 1.5 σ = 4.5 σ

So the process is now a 4.5 σ process instead of a 6 σ process and the defect level is 3.4 ppm.

6 σ and 3.4 ppm

There is a notion that the 6 σ process yields a defect rate of 3.4 ppm. This is not entirely true as a real 6 σ process yields a defect rate of 0.002 ppm.

The 3.4 ppm concept came from a Motorola process where it was decided that a process will be called “6 σ Process” if

1. The (USL – LSL) / 2 σ = 6 i.e. Theoretical sigma level = 6, and
2. The Mean has shifted by 1.5 σ, yielding an actual sigma value of 4.5 and corresponding defect rate of 3.4 ppm.