# Calculating the Six Sigma Level of Processing

## Six Sigma

## Introduction

Process sigma is a measure of the variation in a process relative to requirements. Requirements are set as an upper tolerance limit (UTL) and a lower tolerance limit (LTL). The term process sigma is really the same as the statistical z-score. With an assumption that a process being measured falls under a normal distribution, the distance between the mean of the process and UTL or the LTL is the process sigma. So, the higher your process sigma is, the better your process is. The reason for this is because the distribution is narrowing at the mean as the process sigma increases.

Six Sigma is one of the best tools to keep processes from varying. You must have an acceptable quality range, the mean, and a calculated standard deviation (__σ__), which is used to quantify the amount of variation or dispersion of your data. With this data the process sigma (z-score) can be calculated. To find this measure either the UTL or the LTL can be used. The equations to find process sigma can be found below.

## Process Sigma Equation

## Example

Measurements for a part that needs to be between the range of 2 and 10 inches. The LTL will be 2 and the UTL will be 10. From past data collected on this process, the mean of the process is 6.5 and the standard deviation is 1.75. Let's first use the UTL to calculate the sigma level.

(10 - 6.5) /1.75 = 2.0

Process sigma for the UTL is 2.0

(6.5-2) /1.75 = 2.57

Process sigma for the LTL is 2.57

**Process Sigma = 2**

We choose 2 because 2 is closer to the mean than 2.57 giving us more defects. If we want to go a few steps further we can calculate a few more items. In order to find the area of defective parts outside of the upper tolerance limit we must take the process sigma (z score) "2.0" and find 2.00 on a normal distribution chart. So as you can see on the chart below to find the z score you must find 2.0 on the horizontal axis then find .00 on the vertical axis. The vertical axis is for the hundredths place only. Find the z-scores in the chart below.

## Normal Distribution Table

**Upper Tolerance Limit**

Process sigma for the UTL is 2.0

Probability of good outcome = .9772

Probability of a bad outcome 1- .9771 = .0229

**Lower Tolerance Limit**

Process sigma for the LTL is 2.57

Probability of good outcome = .9949

Probability of a bad outcome 1-.9949 = .0051

After finding the z scores on the chart we can say when operating at a level above the level of the mean we have a 97.72 percent chance of having a good outcome with the UTL and a 99.49 chance of having a good outcome with LTL. The probability of having defective products can be found by simply by subtracting the probability of good outcome by from 1. Adding these probabilities together will give you the upper and lower defect percentage (.51 + .0229 = .028 or 2.8%). Now if you multiply the probability of having bad parts to 1 million we can now say that this process has 2,800 defective parts per million Opportunities (DPMO). See the visual representation below for the outcome percentages.

## Use of Microsoft Excel to Find Process Sigma

Microsoft Excel can be utilized to find process sigma using a series of steps. You can download the Microsoft Excel example here. As you can see in the picture below all calculations can be made in Microsoft Excel.

Microsoft Excel Statistical functions used to calculated process sigma:

=STANDARDIZE (Calculates process sigma)

=NORM.S.DIST (Left-tail defective percentage from negative z-score)

=NORM.S.DIST (UTL good parts percentage)

=NORM.S.DIST(ABS (UTL good parts percentage from the negative z-score)

=1-NORM.S.DIST (Right-tail Defective percentage)

## References

Boyer, K. & Verma, R. (2010). *Operations & supply chain management for the 21st century*. Mason, OH: South-Western.

## Questions & Answers

**© 2018 Joshua Crowder**

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