Z scores and process sigma help us know what chance a process has of meeting its target. This article discusses the relation between the two concepts.

Z Scores and Process Sigma Overview

We’ve covered the basics of Z scores here. Basically, we use them to transform a given standard distribution into something that is easy for us to calculate probabilities on. Why? So we can determine the likelihood of some event happening.

In practical applications, we often have specification limits for our process. For example, we need to create physical widgets with a length between 5 cm and 5.5 cm.  It’s important to know what the percent chance your process has of meeting that kind of target.

Assuming your process follows a normal distribution, the distance between the sample mean of your process and the upper (or lower) bound is the process sigma. The process sigma metric is essentially a Z equivalent.

Remember that a sigma score tells you how many standard deviations can fit between the process mean and specification limit of your process. The better your process, the more sigmas. This is a case of more is better!

When we covered the basics of Z scores, I recommended drawing a picture. That holds here, too.

Let’s look at some scenarios:

What is the Z score of a process that can NEVER meet specification?

That means you have a 100% error rate. We would be looking for an area of zero under the curve. On the chart, you’d have a negative infinity Z score.

A low sigma value means that a significant part of the process output is not inside specification limits.

What is the Z score of a process that meets specification half of the time?

Looking at the Z table, you’d have a 1.5 Z score. In other words, 1.5 standard deviations of your process would fit in the specification limits.

What does a low Sigma or Z score mean for your process?

A low sigma (Z) score means that your process has a lot of defects. If we made a graph of this, a significant part of the tail of the distribution extends past the specification limit.

What Z score would your process need to be at a 6 Sigma level of accuracy?

Remember, a z value is a standard deviation. The answer here is simple; a Z score of 6.

The higher the Z score, the fewer the defects there are in your process. Graphically, you would see the variation in your process  a safe distance away from the specification limit. 6 Standard deviations has been considered an industry standard as a safe distance – some industries require a greater that 6 sigma performance, though.

What Z score would reflect zero errors in the process?

This would be all of the area under the curve, or a positive infinity Z score.

How do you Calculate a Z Score from DPMO?

Ok. Let’s say you’ve calculated your baseline sigma. Let’s use the 333,333 DPMO we calculated in the example on the baseline sigma page. That’s a ratio we can express as 0.3333.

If you look at the Z Score table, you’ll see that 0.3333 falls between a Z score of  -0.43 and -0.44 (because of the 0.3330 and 0.3336.) Since it is right in the middle, we can say that a DMPO of 333,333 translates to a Z Score of -0.435.

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Comments (4)

Two related questions: If a question doesn’t specify, do we assume a 1.5sigma shift or not? example question “what is the DPMO of a 6sigma process?” traditionally is reported with the shift to give 3.4, but if you were just asked “what is DPMO of a 3sigma process?” would you still assume the shift or use the traditional 99.7% of a 3 sigma process to 300,000 DPMO?

Similarly, when converting DPMO to Z-score and vice versa, how do we know when to use the are under a single tail or both ends? Example sigma =3 gives an area of 0.00135 to the either side of a normal distribution, so times 2 rounds to 0.003 and we get 99.7% under the curve within +/-3 sigma. But in other cases, like when there is a 1.5 sigma shift, only a one sided area is used to get DPMO (like 6sigma shifted to 4.5sigma uses one sided area to get 3.4 DPMO.) How do you know which case applies?

Matthew,
These are both good questions, and I would encourage you to always get clarification when using these in a professional setting as some companies do interpret things a little differently. It is my experience that you only apply the shift when referring to 6 Sigma industry standard level. At 3 sigma, the impact of that shift is substantially larger and really needs to be called out if it one of the assumptions. On the one vs two tail, this is really a matter of context. If defects are of the nature that can be both over and under specification limits then two tails makes sense.

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