Question 4 of 6: Variables vs. Attributes Charts, and a $u$-Chart for Textile Imperfections

Question 4: Variables vs. Attributes Charts, and a $u$-Chart for Textile Imperfections (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) Why variables charts come in pairs, $p$-chart vs. $u$-chart, and monitoring $u$ under a varying sample size

A variable (continuous) measurement is fully characterized by two independent parameters — its central tendency and its dispersion — and a shift can occur in either one separately (a process can drift off-target while its spread is unchanged, or become more erratic while staying centered). One chart alone cannot see both, so variables control uses a pair: an $\bar X$ (or median) chart for the mean together with an $R$ or $s$ chart for the spread. An attribute (pass/fail, or a defect count) collapses to a single number per unit or per sample — a fraction nonconforming or a count — so a single chart (a $p$-, $np$-, $c$-, or $u$-chart) is sufficient; there is no separate "spread" to monitor because the underlying binomial or Poisson distribution ties its variance directly to its own mean.

A $p$-chart monitors the fraction nonconforming — each inspected unit is simply classified conforming/nonconforming (a Bernoulli outcome), and $\hat p=x/n$ is plotted; it answers "what fraction of units are defective?" and is used whenever a unit can only be counted once as good or bad, regardless of how many individual flaws it might have. A $u$-chart monitors the average number of nonconformities per inspection unit — a single item (e.g. one roll of textile, one printed circuit board) can carry several distinct imperfections, each counted, and $\hat u=x/n$ (defects divided by the number of inspection units in the sample) is plotted; it is used whenever defects can occur multiple times per item and every individual occurrence matters, not just whether the item is acceptable overall.

When the sample size (number of inspection units per sample) varies from period to period, the control limits are simply recomputed for each sample's own $n_i$: $UCL_i,LCL_i=\bar u\pm3\sqrt{\bar u/n_i}$, where $\bar u$ is the pooled average nonconformities-per-unit across all samples ($\bar u=\sum x_i/\sum n_i$, held fixed as the process estimate). A larger $n_i$ narrows that sample's limits (more inspection units average out sampling noise) and a smaller $n_i$ widens them, so the chart correctly demands stronger evidence of a shift from a small sample and less from a large one, while the center line $\bar u$ itself does not change with $n_i$.

(b) Why a positive LCL is preferred for a $c$-chart, and whether it is always achievable

A positive lower control limit lets the $c$-chart detect a decrease in the defect count, not just an increase. An unusually low nonconformity count is not automatically good news on its own account — it can signal that an inspector has stopped looking carefully, that a measurement or counting system has failed, or (more happily) that a genuine process improvement has occurred that should be identified, confirmed, and locked in as the new standard rather than treated as a random low day. If $LCL=0$ (or would be negative and is clipped to zero, as is common when $\bar c$ is small), the chart can never signal on the low side at all, and this diagnostic opportunity is lost entirely.

It is not always possible to have a positive LCL, and this is a real practical limitation, not just a design choice. $LCL=\bar c-3\sqrt{\bar c}>0$ requires $\bar c>9$: the process average count itself must be reasonably high (at least ten or so defects per inspection unit) before three standard deviations below the mean clears zero. For processes with a low average defect count (the common, and generally desirable, case for a well-controlled process), $\bar c<9$ and the standard $c$-chart simply has no active lower limit. When it is achievable, it is done exactly the way part (b) of the previous question found the analogous condition for a $p$-chart: by using a larger inspection unit (aggregating more product per sample) so that the pooled average count $\bar c$ (or $\bar u\times n$ for a $u$-chart) is inflated above the $\bar c>9$ threshold.

(c) $u$-chart for 20 days of textile-roll imperfections

Given. Twenty days of inspection, rolls produced and total imperfections recorded per day (sample size $n_i=$ rolls produced/day $\in\{20,25\}$):

Find. Trial 3-sigma $u$-chart limits, any revision needed, and the estimated $\lambda$ (expected nonconformities per roll).

Approach. Pool to get a trial $\bar u$, compute step limits at $n_i\in\{20,25\}$, flag and remove any point outside its own day's limits as an assignable cause, then re-pool and re-check the remaining points.

1. Trial center line. $N=\sum n_i=445$ rolls, $\sum x_i=214$ imperfections, so $\bar u=214/445=\boxed{0.4809}$ imperfections/roll.
2. Trial control limits. $UCL_i,LCL_i=\bar u\pm3\sqrt{\bar u/n_i}$: for $n=20$, $UCL=0.4809+3\sqrt{0.4809/20}=\boxed{0.9461}$, $LCL=0.4809-3\sqrt{0.4809/20}=\boxed{0.0157}$; for $n=25$, $UCL=\boxed{0.8970}$, $LCL=\boxed{0.0648}$. Checking each $\hat u_i=x_i/n_i$ against its own day's limits, day 19 ($x=30$, $n=20$, $\hat u=1.5$) is far above its $UCL=0.9461$ — an assignable cause — while all 19 remaining days fall inside their limits.
3. Revise. Remove day 19 and re-pool the remaining 19 days: $N_{rev}=445-20=425$, $\sum x_{rev}=214-30=184$, so $$\bar u_{rev}=\frac{184}{425}=\boxed{0.4329}\ \text{imperfections/roll}.$$ Revised limits: $n=20$: $UCL=0.4329+3\sqrt{0.4329/20}=\boxed{0.8743}$, $LCL$ raw $=-0.0084\to\boxed{0}$; $n=25$: $UCL=\boxed{0.8277}$, $LCL=\boxed{0.0382}$. Re-checking all 19 remaining points against these revised limits, every one now falls inside — no further revision is needed.
4. Estimate $\lambda$. $\lambda$, the expected number of nonconformities per roll under control, is estimated by the retained (revised) center line: $\hat\lambda=\bar u_{rev}=\boxed{0.4329}$ imperfections per roll.