Referring back to the short- and long-term behavior of the process mean, there are 2 values for Z, short-term Z, or Z st, and long-term Z, or Z lt. Similarly, for a 3-sigma quality level, the process sigma must be: In general, the Z formula can be rearranged to calculate the maximum allowable process sigma, or standard deviation, for any sigma level.įor example, given a mean of 25 seconds and SL of 31 seconds, for a Six Sigma quality level, the required process sigma is calculated as: In order to bring the process to the golden Six Sigma quality level, the process sigma would have to be reduced to 1. Therefore, the process is at a 3-sigma quality level. In this example, the process sigma level for a specification limit of 31 seconds is: Given the specification limit, SL, the process sigma level, or process Z, is:

The one-sided defect rate is applicable to any capable process with 1-sided or 2-sided SLs, even at a 3-sigma level. The 1.5-sigma shift makes defects approach 0 on the opposite side of the shift even at lower sigma levels. Figure 2: Process Mean Shift of 1.5 Sigma and Defect Rate Corresponding to 4.5 Sigma A Six Sigma process is actually 4.5 sigma in the long term, and the 3.4 PPM defect rate is the 1-sided probability of having a data value beyond 4.5 sigma measured from the short-term mean. A similar argument applies to the extreme case of 1.5-sigma shift to the left. The overall defect rate, therefore, is 3.4 PPM. The left specification limit is at 7.5 sigma from the mean with a defect rate of 0 PPM. The right specification limit is at 4.5 sigma from the mean with a defect rate of 3.4 parts per million (PPM). The red lines in Figure 2 (below) show the extreme case of 1.5-sigma mean shift to the right.

In a stable process, the mean naturally shifts as much as 1.5 sigma in the long term on either side of its short-term value. (See Figure 1.) Figure 1: Normal Distribution With Mean, Z-score and Six Sigma Specification Limits Therefore, a process data point can be 6 standard deviations from the mean and still be acceptable. The farther the specification limits are from the mean, the lower the chance of defects.Ī Six Sigma process has a specification limit which is 6 times its sigma (standard deviation) away from its mean. Any value beyond the specification limit indicates a defect or unacceptable result. In a process, deviations from the target or mean are accepted to a certain value defined by the specification limits (SL) around the mean. Z = (31- 25) / 2 = 3 Specification Limits and Defect Rates This count is denoted by sigma level, Z, also known as Z-score, as shown below. If the standard deviation is 2 seconds, the same point is 6/2 or 3 standard deviations away from the mean. This distance can also be measured by counting the number of standard deviations in the distance. For example, a data point with a value of x = 31 seconds is 6 seconds away from a mean value of 25 seconds. The distance from the mean μ to a data value in terms of data units can be measured.

Process data usually has a normal distribution. A larger standard deviation indicates that a data set has a wider spread around its mean. Standard deviation shows the extent of variation or spread of data.

3 sigma vs 2 sigma