Six Sigma

The equation for standard deviation is shown below.

$\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}}$ )                  (13.3.2)

where $\sigma$ = standard deviation, and the other variables are as defined for the average.

For each measurement, the difference between the measured value and the average is calculated. This difference is called the residual. The sum of the squared residuals is calculated and divided by the number of samples minus 1. Finally, the square root is taken.

The standard deviation is the basis of Six Sigma. The number of standard deviations that can fit within the boundaries set by your process represent Six Sigma. If you can fit 4.5 standard deviations within your process specifications then you have obtained a Six Sigma process for a long term scale. However, the number of errors that you can have for your process as you move out each standard deviation continues to decrease. The table below shows the percentage of data that falls within the standard deviations and the amount of defects per sigma, in terms of "Defects Per Million Opportunities" or DPMO. The percentage of errors that you are allowed is one minus the percentage encompassed by the percent of the total.

The image below shows an example data set with lines marking 1 to 6 standard deviations from the mean. In this example, the mean is approximately 10 and the standard deviation is 1.16.