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Detecting an Improper Sampling Technique

Q: My X-bar control chart is signaling an out-of-control situation even though all of the points are very close to the center line. The rule violation is designated with the letter "I" and is recorded as "15 points in a row in Zone C (above and below the center)." Why is this happening when having all points close to the center line seems like it would be desirable?

A: This particular rule violation usually indicates a problem with the sampling and control chart procedures, rather than a problem with the process itself. Typically, an improper sampling technique called Stratified Sampling is indicated with this rule violation. It causes the calculated variability of the data to be inflated, which causes the control limits to be wider and the chart to be less sensitive to process changes.

Stratified sampling arises when the measurements of several processes (machines, workers, tools, etc.) are combined to form a subgroup for the control chart. In one example, there are 5 machines that all produce a particular part. Once an hour, a single part is taken from each machine to form a subgroup of size n = 5 to be put on the control chart. In another example, a painting machine has 4 heads that simultaneously spray paint on individual parts. At the end of each paint cycle, the thickness of the paint coating is measured on each of the parts and recorded as a subgroup of size n = 4 for quality control purposes. This type of sampling is known as Stratified Sampling, since each subsample includes an observation from each of the different processes (strata).

A common problem with stratified sampling situations is that the measured variance includes the differences between the strata, in addition to the common cause process variation. If there is a substantial difference between the mean value of each machine or head, this is included in each subgroup range or standard deviation. This increases the estimate of the process variation and the calculated control limits on the X-bar chart become wider. At the same time, the mean may stay fairly constant and thus near the center of the control chart.

The following chart shows the control chart pattern that typically indicates stratified sampling. This data was simulated with a standard deviation of 1.0, assuming five components with means of 1.0, 2.0, 3.0, 4.0, and 5.0, respectively. The mean of the X-bar chart is around 3.0, as expected. However, the standard deviation of the subgroups is 0.927, rather than

as would be expected with a subgroup size of n = 5. This more than doubling of the subgroup standard deviation is clearly reflected in the increased width of the control limits for the X-bar chart relative to the plotted points.

The control chart signals an out-of-control situation after there have been 15 points in a row, all within Zone C (within 1 standard deviation of the mean). The probability of having 15 consecutive points in a row within 1 standard deviation of the mean is very, very low for a process that is in statistical control. Thus, when this happens, it means that there is something unusual with either the process or the control scheme.

Due to the increased width of the control limits, the X-bar chart created under stratified sampling will be fairly insensitive to changes in the overall mean of the samples. It is even less sensitive to changes in an individual process mean since typically the overall mean will not change significantly. Rather than using a stratified sampling scheme, it is recommended that separate control charts be made for each process, machine or head. This will give the most information about the individual determinants of quality.