Recap: Allegory of the Paste WeightsAs you recall from Part I, we discussed two things.
- How to detect assignable causes of variation, using short term variation within rational subgroups to gage long term variation between subgroups.
- How to put data from distinct causal regimes onto a common scale for comparison by subtracting strata means from the data for each stratum.
|There is more variation from group to group than there is on the average within groups.|
- Collect small subgroups in such a way that major sources of variation are constant: same operator, same material, same tooling, same time. This usually means a few consecutive cycles of the process.
- A good baseline should run for 20-30 subgroups, where possible, at appropriate intervals. This should be enough to get a decent average range.
- Calculate the internal variation within each subgroup (range or standard deviation). This represents natural process variation.
- Calculate the average subgroup range, which will be used as a gage. Check individual ranges to see if there is any evidence of short-term assignable causes.
- Now use that average range to calculate limits on the subgroup means. Plot the long-term variation against the limits calculated from the short-term variation.
- The chart asks the musical question: Is there more variation from group to group than there is on the average within a group?
- All points within the limits. Most of the process variation takes routinely in the short term, within groups. This probably means that major factors are designed into the process.
- Some points outside the limits. There are important factors that affect some groups differently than other groups, between groups.
The Allegory of the Cookies!
|Each subgroup consists of a cookie from the left, center, and right sides of the oven.|
But notice the rationale of the subgrouping: the ranges do not represent short-term variation, but variation across the oven belt. The chart asks the question: Is there more variation from time-to-time than there is on the average from side-to-side. The answer is no, most of the variation takes place side-to-side across the oven.
Time series for each position from oven. Breakdown reveals stratification.
|Adjusted data now reveals time-related assignable causes.|
However, the dip below the probability limits says that the data series is incompatible with the hypothesis of a single mean. Hence, our adjustment, which was to subtract a single mean from each of the zones, is now Highly Suspect. In particular, if the latter portion of the data were really more variable, the two points at 13-14 might be beyond the limits not because a change in temperature but because the temperatures had become more variable and the calculated limits are too narrow! Perhaps limits should be calculated separately for Samples 1-6 and 7-15. But at this point we really do not have enough data (6 points and 9 points, resp.) for a good analysis.
|Reconstructed paleotemperatures for Holocene Epoch|
Central Greenland Ice Cores