Attribute Control Charts p and np Chart Assignment Guide

Four of the five problems ask a version of the same question, and the question contains a trap. "How is the hospital performing in relation to its stated goal?" and "is the hospital doing a good job?" are not questions a control chart answers. A control chart answers whether a process is *stable* — whether the variation in it is common cause or whether something special has intruded. Stability and performance are independent: a process can be beautifully stable and stably terrible, sitting at a 4% infection rate with every point inside its limits. The answer each of these problems wants is therefore two-part — first the chart's verdict on stability, then a separate comparison of the centre line against the stated benchmark. Students who give only the first half have answered the easier question and left the marks on the table.

Getting p and np the right way round is the cheapest mark on offer and the commonest thing dropped. A p chart plots the *proportion* nonconforming and tolerates a sample size that changes from subgroup to subgroup, because the control limits are recalculated for each point. An np chart plots the *count* nonconforming and requires a constant subgroup size, because a count is only comparable across subgroups when the denominator is fixed. That is why problem 1, with 40 cases sampled every week, permits both charts, and it is also why problem 5, where some months have fewer than ten patients, does not permit an np chart at all. The brief's instruction there — use 10, or use the average sample size — is a workaround for exactly this constraint and worth naming as one.

Problem 2 is the one that most rewards care, because it is a comparison and the two machines have very different inspection volumes. Machine A inspects around 483 units a month; Machine B around 149. Control limits on a p chart are a function of the subgroup size, so the smaller denominator produces wider limits, and a machine can look better behaved simply because its limits are more forgiving. The question asks whether either machine shows evidence of special cause variation, so the answer is about points relative to each machine's own limits and not about which machine has the lower defect rate. Machine A's May and October figures are the obvious candidates, and saying why they qualify — beyond a control limit, not merely higher than the others — is what demonstrates the concept.

Shewhart's distinction between common and special cause is the concept every one of these problems is testing, and it is worth stating in your own words rather than assuming it is obvious. Common cause variation is the noise inherent in the process as it is currently designed; it produces points that wander within the limits and it is reduced only by changing the process. Special cause variation is something that intruded — a new supplier, a broken instrument, an outbreak — and it produces signals. The practical consequence, and the reason this matters in a health services organisation, is that reacting to common cause variation as though it were special is a well-documented way of making a process worse, which is a stronger answer than "the chart shows the process is in control". Naming which of the five processes you would actually leave alone is a good way to show you mean it.

Problem 3 is the historically interesting one and it deserves a sentence of context in your interpretation. Semmelweis's maternal mortality data from 1846 to 1848 spans the period in which he introduced handwashing, so a p chart over the whole range is very unlikely to show a stable process — and that is the finding, not a failure of the analysis. The question asks whether the system was stable, and a chart with a visible step change answers it clearly. If you can, say roughly where the shift occurs and what it corresponds to, because a control chart that detects a known intervention is the clearest possible demonstration that the tool does what it claims. Treating the shift as an outlier to be excluded would be exactly the wrong instinct here. It is also worth saying that the chart is being used retrospectively here, to characterise a past series, rather than prospectively to monitor an ongoing one.

The mechanics are worth being deliberate about because the brief says show all work. For a p chart the centre line is the total nonconforming divided by the total inspected — not the average of the individual proportions, which differs whenever subgroup sizes vary. The limits sit three standard deviations either side, using the binomial standard error, and a lower limit that computes as negative is reported as zero because a proportion cannot be negative. For an np chart the centre line is the average count and the limits use the corresponding count-based standard error. Showing these calculations for at least one subgroup, rather than only pasting the SPSS output, is what "show all work" is asking for and it is quick to do. Doing it once and referring back to it for the other charts satisfies the requirement without repeating the arithmetic five times.

Two files are submitted, SPSS and Word, and they are graded together, so decide what each one carries. The SPSS file is the evidence that the analysis was run; the Word document is where the interpretation lives, and interpretation is where the marks concentrate. Paste each chart into the Word document beside its interpretation rather than leaving the reader to open the SPSS output alongside — five charts described in a document that contains none of them is a much harder document to mark. Label every chart with its problem number and its axis units, since a p chart and an np chart of the same data look similar at a glance and differ only in the scale on the vertical axis. Give each chart a title naming the process and the period, since problems 3, 4 and 5 all plot proportions over time and are otherwise hard to tell apart at a glance.

The page guidance is three pages for five problems, which is roughly half a page each and rules out long prose. Write each interpretation to a fixed shape: what the chart shows, whether the process is stable, how the centre line compares with the stated benchmark, and what you would investigate. That shape answers every part of every question asked here and keeps the whole thing inside the budget. Resist opening with a general account of statistical process control — the brief says to review this week's resources and mimic the development of the charts, which means the assessment is on execution and reading, not on whether you can define a control chart in an introduction nobody asked for. If something has to be cut, cut the description of what each chart looks like — the marker can see the chart — and keep the judgement about what it means.