Question 2 of 6: SPC Diagrams, Traditional vs. Special Control Charts, and Attributes Charts

Question 2: SPC Diagrams, Traditional vs. Special Control Charts, and Attributes Charts (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Location diagram vs. Pareto diagram

A location (defect-concentration) diagram is an outline or photograph of the physical part or product, on which every observed defect is marked at the actual location it occurred. Its value is spatial: patterns of WHERE defects cluster (e.g., all scratches near one corner, all voids on the weld seam) point directly at a physical/process cause — a die edge, a fixture contact point, a specific tool station — that a tabular defect count could never reveal. Example: on a stamped sheet-metal panel, marking every scratch's location on a diagram of the panel might reveal that 90% of scratches cluster along one edge, immediately implicating the conveyor guide rail that contacts that edge rather than the stamping process itself.

A Pareto diagram is a bar chart of defect (or cost, or downtime) categories sorted in descending frequency, with a cumulative-percentage line overlaid, built on the empirical observation that a small number of categories ("the vital few") typically account for most of the problem, while many categories ("the trivial many") each contribute little. Its value is prioritization: it tells the team WHICH category of problem to attack first for the largest return on improvement effort. Example: tabulating all defect types on a batch of castings might show that porosity and short-fill together account for 75% of all rejects, while five other defect types combined account for the remaining 25% — directing the improvement team's limited time squarely at porosity and short-fill.

Used together, the Pareto diagram answers "which defect category matters most," and the location diagram then answers "where on the part is that category occurring" — a natural two-step drill-down in any process-improvement investigation (this same logic extends to cause-and-effect/fishbone diagrams for "why," completing the classic seven basic quality tools).

(b) Traditional vs. special (EWMA/CUSUM) control charts

Traditional Shewhart charts (the $\bar X$, $R$, $S$, $p$, $np$, $c$, $u$ family) plot each individual sample statistic and react only to information in the current sample; they are preferable when the process is expected to run with occasional large, sudden assignable-cause shifts (a tool breaking, a wrong material substituted, an operator error), because a Shewhart chart's 3-sigma limits are highly sensitive to LARGE shifts (typically detected within one to a few samples) and its logic and construction are simple enough for shop-floor personnel to apply and interpret without statistical training. They are the correct default chart for the great majority of routine process monitoring.

Special charts — the CUSUM (cumulative sum of deviations from target) and EWMA (exponentially weighted moving average) — incorporate the history of previous samples into each plotted point, which makes them far more sensitive than a Shewhart chart to SMALL, sustained shifts (roughly $0.5$–$1.5\sigma$), because a small shift accumulates detectable evidence over many samples even though no single sample looks unusual on its own. Example uses: a CUSUM chart on the fill weight of a packaged product, where a slow calibration drift of the filling head (a small, sustained shift) would go undetected for many samples on an $\bar X$ chart but accumulates a clear upward or downward trend on a CUSUM/tabular V-mask chart; an EWMA chart monitoring a chemical process's impurity concentration, where the smoothing parameter $\lambda$ can be tuned to be as sensitive to small drifts as a CUSUM chart while remaining easy to compute and, unlike a raw Shewhart chart, is also robust to mild non-normality in individual measurements because the EWMA statistic itself is closer to normal by a central-limit-type averaging effect.

Nominal (target) values should be used to CENTER a control chart only when the process is known to be capable of running on target and the objective is to detect any departure from that target — the classic case for CUSUM/EWMA charts, which are specifically designed to be sensitive around a target value. For a standard Shewhart chart used for ongoing process monitoring, however, the chart should ordinarily be centered on the process's own estimated mean $\bar{\bar x}$ (or $\hat\mu$) from actual in-control data, not the nominal/specification target: centering on a nominal value the process cannot actually achieve manufactures false alarms (points routinely outside limits centered on an unattainable target) and defeats the purpose of a control chart, which is to distinguish the process's own natural variation from assignable-cause variation, not to test conformance to a target that must instead be pursued through capability improvement (recentering only after the process has actually been shown able to run there).

(c) Trend charts, variables vs. attributes chart pairs, p- vs. u-charts, and demerit charts

A trend chart is a control chart whose center line is not flat but follows a predictable, systematic trend over time (e.g., linear tool wear, seasonal drift) — the control limits are constructed around this moving center line (rather than a constant mean) so that the chart correctly signals departures from the EXPECTED trend, rather than flagging every point as "out of control" simply because the process is legitimately, predictably drifting (as would happen if a standard flat-centerline chart were misapplied to a genuinely trending process, e.g., monitoring tool diameter as the tool progressively wears between scheduled replacements).

Variables data (a continuous measurement) carries two independent pieces of information per sample — where the process is centered (location) and how spread out it is (dispersion) — and a single chart cannot detect both an out-of-control mean and an out-of-control variance simultaneously, so variables monitoring always uses a PAIR of charts, one for location ($\bar X$ or individuals) and one for dispersion ($R$ or $S$). Attributes data (a count of defectives or defects) is a single number per sample, and its natural sampling distribution (binomial for defectives, Poisson for defect counts) has a variance that is a fixed, known FUNCTION of its own mean ($np(1-p)$ for the binomial count, $c$ itself for the Poisson count) — there is no independent dispersion parameter to track separately, so a single chart on the count (or proportion) fully captures the process's behaviour.

A p-chart monitors the FRACTION defective, $\hat p=D/n$, where $D$ is the number of defective units (each unit is simply classified good/bad) in a sample of $n$ units, and is used when the sample size varies from sample to sample (control limits $\bar p\pm3\sqrt{\bar p(1-\bar p)/n}$ recompute with each sample's own $n$). A u-chart instead monitors the average number of DEFECTS per inspection unit, $\hat u=c/n$, where a single unit can contain any number of individual defects (e.g., several scratches on one panel) and $n$ is the number of inspection units in the sample (control limits $\bar u\pm3\sqrt{\bar u/n}$). The essential distinction is defectives vs. defects: a p-chart classifies each unit as pass/fail (binomial), while a u-chart counts how many flaws each unit has, allowing more than one defect per unit and more information per inspection unit (Poisson-based).

A demerit chart (or demerit-per-unit chart) extends the u-chart concept by weighting different defect classes according to their severity (e.g., "critical," "major," "minor" defects assigned demerit weights such as 100, 25, 5) before summing and charting the total weighted demerit score per unit, $U=(w_A c_A+w_B c_B+w_C c_C)/n$, so that a unit with one critical defect is treated as far worse than a unit with several trivial ones — a refinement over a plain u-chart when defect severity varies widely and a simple defect count would understate the impact of the most serious flaws.