Z-Scores and Process Sigma

Process Sigma and Z Values
Z score to standard distribution to Sigma to percentage

Z score to standard distribution to Sigma to percentage

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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