**Between variation** - variation due to the variation between subgroups. If the process is in statistical control this variation should be zero.

Those definitions are correct for the generic case of a control chart and the resulting standard deviations. (Bob was discussing sources or inputs of variation as well as additional components of variation such as what one would see in a multi-vari)

Two points of clarification - and these are important to understanding real world control charts:

Even with a homogenous process stream that is in statistical control, there will be some between subgroup variation. (You can prove this to yourself using a random number generator for a perfect Normal distribution). In this case the between subgroup variation will be sampling error or the standard error of the mean. It is S_average = S_Population/sqrt

. until n = infinity or the full population size S_average (=S_between) cannot be zero.

Most processes don't have homogenous streams and many subgrouping schemes are not rational.

**A rational subgroup will contain the variation within the subgroup that is to be "controlled" between subgroups**. With homogenous process streams the largest component of variation is piece to piece. So the rational subgroup is to sample several sequential pieces (WITHIN) in each subgroup. Then wait some time (BETWEEN subgroup) and sample the same number of sequential pieces. This is the condition that you described so succinctly.

If you have a non homogenous process stream: out of statistical control, with one or more systemic 'natural' sources of variation such as Bob's now infamous tool wear, natural seasonal effects, large measurement error, largest component of variation is the within piece or lot to lot variation, etc. we will not have 'statistical control' in the standard subgrouping scheme even when our process is perfectly stable and predictable and even capable. We need to change our subgouping scheme to be rational if we want to apply a control chart type monitor. This will result in a more complex set of components of variation. And it is this real world complexity that many new or casual users of SPC adn Capability indexes don't comprehend and thus they can't understand their results because they don't look like the text book examples that they learned...