1.5 standard deviation shift, moving mean, on target, variation of the mean

No process stays perfectly on target all of the time. Six Sigma allows the mean to shift plus or minus 1.5 standard deviations.
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A plus or minus 1.5 standard deviation shift of the mean is allowed with six sigma.

Why Does Six Sigma allow a 1.5 SD shift?

The most confusing elements in Six Sigma is the 1.5 standard deviation shift. Many people do not have any understanding of why it is used or even any basis for 1.5 standard deviations. Why not use 1 standard deviation or maybe even 2?

It seems intuitive to almost everyone that a process will move some. We all know that there is some variation in the output of every process. Even if the process is exactly centered over the target value, there will be times when the process produces output that is above the target value and other times when the output is below the target value. If we take a samples of the output and calculate the average for these samples it seems intuitive that they also will not all be exactly on the targeted value. The real question is how to account for this shifting of the mean.

Anyone that has ever tracked a process using statistical process control chart knows that the sub-group averages are selected by the control chart designer and while it is hoped that there is logic in the selection of the subgroup size (rational subgroups), two people doing an analysis on the same process may select different subgroup sizes. Subgroup sizes can vary but one of the more common subgroup sizes is 4. That is four independent samples are taken and the average of the four independent samples is used for the X Bar chart.

With an X Bar R chart the plots are of the average of the subgroups and the range of each subgroup. Control limits are set at 3 standard deviations of the subgroup averages on the X Bar chart (using the average range for the calculation).

Any basic statistics book will include the calculation of the Standard Error of the Mean or sometimes called the Standard Error.

SEM=SDavg = Standard Deviation of the Individuals

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SQRT (n)

The standard deviation of the averages is equal to the standard deviation of the individuals divided by the square root of the sample size. Where n is the sample size.

You instantly recognize that 3 standard deviations of the averages are the control limits on an

X Bar chart. Using that knowledge and a sample size of four we get

3SDavg = 3 SD of individuals = 3 SD of individuals = 1.5 SD of individuals

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SQRT (4) 2

If you are tracking individuals, and defects are measured as individuals

(which is the normal case for measuring defects), then the action level

is at the same value as the control limits for a control chart with sub group size of 4.

There is statistical logic behind selecting the 1.5 SD shift!

Cary W. Adams

10A Bayou RD

Lake Jackson, TX 77566