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  • June 11, 2008
  • By Marcia Gulesian
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Quantitative Methods Employed in Six Sigma

Assuming normal distributions, look at some quantitative methods employed in your use of Six Sigma. A brief discussion of mean (µ) and standard deviation (σ) used in this section is given in Appendix 1.

First, consider Process Capability (Cp):

Cp = where

USL = An upper specification limit that is a value below which performance of a product or process is acceptable.


LSL = A lower specification limit that is a value above which performance of a product or process is acceptable.

Click here for a larger image.

Figure 5a: Cp for different USL-LSL to 6σ ratios

The capability index Cp measures the ratio of the width of a specification to the width of a process. This is useful in telling you your process's ability to meet specification, if you get the process output to average right in the center of the specification. If your natural tolerance is exactly equal to your specification width, Cp = 1 and the process is said to be "potentially minimally capable" of meeting specifications. "Potentially" because it might be terribly off target and making 100-percent scrap but with variation equal to the specification width. More usually, companies have a goal of getting all their capability indexes equal to 1.33.

Many processes will only have one limit: either an upper or lower control limit. These are sometimes called 'One Sided Specs'. A Cp cannot be calculated for one-sided specs.

Figure 5b: Normal (Gaussian) curve showing mean and standard deviation

The normal distribution curve (the bell curve in Figures 5a and 5b) is also called Gaussian. A consistent inaccuracy will displace the curve to the left or right of the nominal value, while a perfectly accurate machine will result in a curve centered on the nominal. Repeatability, on the other hand, is related to the gradient of the curve either side of the peak value; a steep, narrow curve implies high repeatability. If the machine were found to be repeatable but inaccurate, this would result in a narrow curve displaced to the left or right of the nominal. As a priority, machine users need to be sure of adequate repeatability. If this can be established, the cause of a consistent inaccuracy can be identified and remedied.

Figure 5c: Small standard deviation does not guarantee conforming to specifications

Consider the possibilities of accuracy versus repeatability. Suppose you measure the offset error 10 times and plot the 10 points on a target chart (refer to Figure 5c). Case 1 in this diagram shows a highly repeatable machine because all measurements are tightly clustered and on target.

The average variation between each point, known as the standard deviation, is small. However, a small standard deviation does not guarantee an accurate machine. Case 2 shows a very repeatable machine that is not very accurate.

As the bulk of the measurements are clustered more closely around the target, the standard deviation becomes smaller and the bell curve will become narrower.

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