November 17th, 2011

In both cases of Process Sigma and Process Capability we are talking about performance relative to the Customer’s requirements. Is there a difference? Is one measure better than the other? These are good questions which you will be able to answer by reading on.

The difference between the Six Sigma metric of Process Sigma and Process Capability relates to the definitions of these performance metrics. In both cases the Customer Specifications are compared to process performance. For metrics that can be measured on a continuous scale the process mean and standard deviation must be calculated. If the metrics from the process are discrete, or attributes, then the percent defective is calculated. In either case the process performance as measured on a control chart should exhibit only natural random patterns of variation, or what is often called common cause variation over time.

Process Sigma is defined by numeric levels that are related to a process’s output of defects per million opportunities. Defects are defined as any failure to meet the customer’s specifications. Process yield is used to look up the Process Sigma level from a table. Yield is based on Defects (D), Units Processed (N), and the number of Opportunities (O) for a defect to occur. Once the yield is calculated the Process Sigma can be found in the table below.

Find the Process Sigma level in the table above.

If the process output can be measured on a continuous scale the process average and standard deviation are used in a formula that compares the average to the closest specification, whether it is the Upper Specification or the Lower Specification we choose the one closest to the process average. Basically we are making this a one sided specification. The standard normal distribution is used to estimate the defect rate which can then be converted to yield and use the above table to look up the Process Sigma level.

The other method is to calculate the Z statistic which estimates the number of standard deviation units the average is away from the closest specification which is based on the standard normal distribution. We then add a 1.5 sigma shift to the calculated Z statistic which gives us the Process Sigma level directly. The 1.5 sigma shift has been a matter of contention, but it is part of the definition of Process Sigma. The following is the formula.

The definition of Process Sigma is that at a level of 6 on the scale there will be only 3.4 defects per million opportunities. The process is assumed to remain stable over time and if it drifts in one direction at a time of 1.5 sigma there will be no more than 3.4 defects per million under the tail of the distribution outside of the Customer’s Specification. When we calculate the Z statistic we add the 1.5 sigma shift to give us the Process Sigma level directly.

Process Capability assessment begins with control charts to evaluate the stability over time for the process. For the Process Capability study to be meaningful the process must exhibit only common cause variation, which are natural patterns of random variation.

In the case of Attribute, or discrete variables, data the Process Capability becomes the centerline of the Control Chart, or the average p, c, or u depending upon the chart being used. In the example below the average p, or fraction non-conforming, is 0.076, or 7.6% defective. The chart is in statistical control so the capability is the overall process average of percent defective.

In the case of continuous variables data the mean and standard deviation from the stable, or in statistical control, process comes into play. We will describe 2 of the many Process Capability Indices. The first is the Cp which compares total process variation to the total width of the specification. The second is Cpk which takes into consideration centering of the process within the specification so we look at ½ of the process variation compared to the closest specification limit.

The Cp formula below divides the specification width by the measured process spread of 6 standard deviations. A fully capable process has a Cp = 1.33 or greater.

The Cpk formula below divides the absolute value of the difference between the process average and the closest specification by ½ of the measured process spread of 3 standard deviations. A fully capable process has a Cpk = 1.33 or greater.

The Cp value indicates how capable a process can become if it is perfectly centered. The Cpk value indicates how much work is needed to get centered. In the following table notice that for a Cpk = 1.33 the Process Sigma = 5.5. A process that operates at a Process Sigma = 6 has a Cpk = 1.50. The Z Value is the number of standard deviation units away from the specification when the data is converted to a standard normal distribution.

Table of conversions from Z to Cpk and Process Sigma.

Both measures of performance use that same statistics to be computed. Whether you prefer Process Capability or Process Sigma the key is to apply the metrics consistently. To be valid your process must be in a state of statistical control and exhibit only common cause variation.

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