When monitoring solution characteristics from chemical batches there are three sources of variation - average over time, within-batch and batch-to-batch. In this example we are mixing paint. The key characteristic is percent solids. The specification limits are 7.7 ± 0.3. To verify homogeneity, ten samples (Table 1) are taken from each batch at various locations inside the mixing vessel (Figure 1). We will first look at the most obvious, but incorrect, way to analyze the data. Next, we will analyze the same data with an ingenious method that has been around for years but is largely not supported in off-the-shelf SPC software.
Since this alternative approach is not well known, it still enjoys the status of "trade secret." However, usingInfinityQS SPC, the methods described in this example are fully supported.
The Incorrect Analysis - Using Traditional Control Chart
If we only had traditional control charts available to us, we would be inclined to treat the ten readings within each batch as a subgroup of size 10. The control chart would most likely be the X-bar and S chart (Figure 2).
Figure 2: Incorrect Analysis. The control limits on the top chart are too tight because they are incorrectly based on within-batch variability instead of the batch-to-batch variability.
This approach incorrectly uses the within-batch sigma (based off the centerline on the sigma chart) in the control limit calculations in the batch-to-batch chart (the X-bar chart). This common mistake typically results in the pattern illustrated in Figure 2 - the control limits on the X-bar chart are not representative of the plot points. In this case, the control limits are very tight. This indicates that the within-batch variability is much smaller than the batch-to-batch variability.
The Correct Analysis - Using a 3D Control Chart
What we really need is two charts to track the variability - a moving-range chart for batch-to-batch variability and a sigma chart (since the within-batch subgroup size is 10) for the within-batch variability. We now use the centerline off the batch-to-batch moving-range chart in the calculations for the batch-to-batch X-bar chart. Figure 3 shows the same data, but with the additional moving-range chart and the correct control limits on the X-bar chart. (Because of how InfinityQS SPC processes the data, the top chart is now treated as an individuals chart - IX).
Figure 3: Correct Analysis. 3D control chart. The Moving R chart displays the batch-to-batch variability and the S(within) chart displays the within-batch variability. Control limits on the top chart are based off the Moving R centerline. The statistics on the right side of the chart are broken down into batch-to-batch (Piece), within-batch (Within) and total variability components.
The vertical (tier) lines on the IX chart’s plot points represent the within-batch variability. These lines are displayed only for illustration purposes. Notice that the within-batch variability of batch #7 is much greater than the others. However, this variability is limited to within the batch and does not show up as a special cause on the IX chart or the Moving R chart.
There are many cases where both within subgroup and between subgroup sources of variability need to be monitored. In those cases, the 3D chart is the solution.
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