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A small inspection group naturally produces a more variable defect-density estimate than a large one under the same count model. One fixed band can make small groups look unstable and hide meaningful changes in large groups. After validating the inspected-area denominator, use a variable-area u-chart to make the monitoring threshold follow the actual observation opportunity rather than yesterday's sample size.
Measurement purpose
Detect prospective departures in a comparable event rate while respecting changing inspected area.
Specimens and conditions
- Comparable population
- Fixed event definition, material state and detection method within each modeled stratum.
- Chronological observation
- Count, positive eligible area, timestamp and process identity retained.
Equipment and records required
- Qualified inspection: Stable detection and registered observable-area mapping.
- Versioned calculation: Reproducible baseline, point-specific limits and alarm rule.
Method sequence
- Establish
Review baseline comparability and count-model assumptions.
Record: Baseline version, original data and justified exclusions.
- Monitor
Apply each new point's actual area to its limits.
Record: Count, area, density and limit comparison.
- Respond
Preserve signals and investigate the process interval.
Record: Cause evidence, containment and authorized action.
Decision and uncertainty
A chart signal prompts investigation; drawing acceptance and lot disposition remain separate.
Clustering, sparse counts, serial dependence, changed detectability and estimated baseline affect signal behavior.
Inspection-method owner defines the model; authorized quality owner controls disposition and restart.
Traceable outputs
| Record | Required contents |
|---|---|
| Monitoring history | Events, areas, times, baseline and point-specific limits. |
| Signal response | Preserved alarm, investigation, affected interval and approved action. |
Method review decisions
- Freeze a comparable chronological baseline before assessing new observations.
- Calculate each subgroup's signal limits from its actual inspected area.
- Investigate statistical signals without automatically changing product disposition or widening the baseline.
Separate prospective monitoring from density comparison
A descriptive comparison asks how many defined events were observed per valid area. Ongoing monitoring asks whether a new chronological observation is unusual relative to an established process model. The latter needs a baseline, a signal rule and a response procedure. Calculating a density alone does not answer that time-dependent question.
This method starts after event counting and observable-area mapping have been resolved. It does not redefine a defect or infer the condition of hidden material. Retain count, eligible area, timestamp and process identity for every subgroup. Keep product disposition separate: a statistically ordinary count can still contain an individual defect forbidden by the drawing. A green chart therefore cannot authorize shipment by itself.
Freeze a comparable baseline before judging new subgroups
Select baseline observations representing the same event definition, detection method and relevant process condition. Different film families, functional regions or inspection sensitivities may need separate strata. Investigate documented special causes without deleting points simply because they widen the limits. Preserve raw data and the reason for each modeling exclusion so a later reviewer can reproduce the decision.
Under a homogeneous Poisson rate model, estimate baseline density ubar as total eligible events divided by total inspected area. Do not average subgroup densities equally when their areas differ. Freeze the accepted baseline version for prospective use. Recalculating the centre line after every high observation can absorb deterioration into the definition of normal, leaving an apparently quiet chart that no longer answers the original monitoring question.
Give each subgroup its own statistical scale
For subgroup area Ai, the model expects count ubar Ai and count variance ubar Ai. Dividing the count by Ai gives a density with modeled variance ubar/Ai. The conventional three-sigma u-chart half-width is three times the square root of ubar/Ai. Larger areas therefore produce narrower density limits under these assumptions, despite sharing the same centre line.
Do not plot a negative lower limit. These boundaries are a signal convention, not a material tolerance or universal confidence coverage. A fixed-area count chart describes a different sampling arrangement. If the outcome is nonconforming parts rather than countable events that can occur several times on one part, select a model appropriate to that outcome instead of silently reusing the Poisson rule.
ui=Ci/Ai; UCLi=ubar+3 sqrt(ubar/Ai); LCLi=max(0,ubar-3 sqrt(ubar/Ai))
- Ci: eligible event count for chronological subgroup i.
- Ai: positive eligible inspected area, in the declared units.
- ui and ubar: observed and baseline event rates per area.
- UCLi and LCLi: point-specific signal boundaries, not drawing limits.
Comparable detection and event definitions, representative baseline and an appropriate homogeneous independent Poisson event model. Sparse counts, clustering and baseline estimation limit the approximation.
Understand why equal densities can produce different signals
Assume an illustrative accepted baseline of 0.20 events per square millimetre. At fifty square millimetres the expected count is ten, the density standard deviation is approximately 0.06325 per square millimetre, and the three-sigma upper boundary is approximately 0.38974. At two hundred square millimetres the expected count is forty and the upper density boundary is approximately 0.29487.
Now observe sixteen events in fifty square millimetres and sixty-four in two hundred. Both densities are 0.32, but only the larger group's result exceeds its own upper boundary. The larger observation provides stronger evidence of a rate departure under the model. This is not inconsistent product grading: the observations contain different amounts of information and the chart answers a process-monitoring question.
| Area | Count and observed density | Calculated density limits | Upper-limit rule |
|---|---|---|---|
| 50 mm² | 16; 0.32/mm² | 0.01026 to 0.38974/mm² | No signal |
| 200 mm² | 64; 0.32/mm² | 0.10513 to 0.29487/mm² | Signal |
Do not borrow the precision of a different inspection area
Applying the larger group's narrow band to both example results would signal both. The smaller observation would be judged by precision it did not provide. Applying the smaller group's wide band to everything would miss the larger group's departure. A convenient straight line therefore changes the operational decision in either direction; its visual simplicity is not evidence that it is the right rule.
Display point-specific limits or a validated standardized presentation while retaining original counts and areas. The interface must identify the limits belonging to a selected point. Do not use the final subgroup's limits as the legend for the entire sequence. Convert area, density and boundaries consistently when changing units so an unchanged dataset does not appear different merely because the report switched from square millimetres to square centimetres.
Check clustering and sparse counts before trusting alarm behavior
One contaminant can produce several nearby marks, and a printing disturbance can affect adjacent panels. These mechanisms can violate independence and create more variation than the Poisson model predicts. Examine spatial and chronological patterns with process events. Frequent signals can reflect an actual disturbance, an inappropriate model or changed inspection; the frequency alone does not establish which explanation is correct.
Small expected counts make symmetric three-sigma limits a rough convention. Integer count thresholds have discrete tail probabilities. Where the consequences require controlled false-alarm behavior, evaluate the actual count-tail rule or another justified model, including uncertainty in the baseline. Do not promise a fixed normal-tail alarm percentage for every area. Adding several trend rules also changes overall signal behavior and must be declared before monitoring, not chosen after viewing a troublesome sequence.
Preserve the signal and investigate its process interval
Retain the original count, area map, images and timestamp when a new subgroup crosses its boundary. Verify classification and registration before changing the process. Check inspection settings, material routing, handling and maintenance events. An unexpectedly low count may also deserve investigation if fields were missed or detection sensitivity fell; a lower plotted number is not automatically a process improvement.
Bound potentially affected material using the last supported process state and the event history. The quality owner decides containment from functional risk and actual evidence, not the chart alone. Repeating inspection until a point falls inside the band and then discarding the first result invalidates the monitoring procedure. Keep confirmation observations linked to the original signal and record unresolved causes rather than adjusting the centre line to eliminate them.
Version the monitoring rule without changing the drawing requirement
A verified permanent process or inspection change may justify a new baseline. Compare the states deliberately, retain the previous limits with their validity dates and document the reason for change. Revising a chart does not make earlier nonconforming parts acceptable. Likewise, a stable sequence does not prove that a process meets every insulation, geometry or electrical requirement of the ceramic circuit.
Deliver a chronological count-and-area file, baseline selection, model checks, limit calculations and response records. For a thick-film inspection review, ChipSimple needs the event definition and drawing consequence alongside the statistics. The practical output is an interpretable prospective monitoring system with explicit action ownership, not a green dashboard made by forcing different inspection areas into one uncertainty band.
Review variable-area defect monitoring
Send chronological observations rather than only a chart screenshot.
- Event definition and drawing consequence.
- Counts, eligible areas, images and timestamps.
- Baseline scope and inspection changes.
- Signal rule and alarm-response ownership.
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