Process charts: test extra variation before blaming a weekly signal

Last updated: 7 Oct 20265 min read

tutorial

When observed defect fractions vary more than an independent binomial model predicts, standard p-chart limits can create misleading alarms.

Look beyond one plotted point

A weekly parcel audit may sample many parcels from the same truck, supplier lot, or route. Those parcels share handling conditions, so their defect outcomes can move together. A binomial chart that treats them as independent has limits that are too narrow. Different inspectors may also apply the damage definition differently; a sudden apparent improvement could be a labeling change. Compare subgroup composition, route mix, inspector assignments, and audit coverage before attributing a signal to operations. The p-chart lesson states the simple reference model.

Calculate a rough dispersion diagnostic

For each week, compare its defect count with n times the baseline fraction, scaled by the binomial variance n p(1-p). Summing these standardized squared residuals and dividing by a stated degrees-of-freedom value gives a rough dispersion statistic. Values well above one suggest more variation than that simple model expects. The code uses k minus one because the pooled baseline is estimated from the same k weeks; it is a screening calculation, not an automatic formal test or a license to widen limits until anomalies disappear.

Separate stable heterogeneity from a real shift

Extra variation may reflect stable route differences, an unstable process, or data collection artifacts. Stratify by prespecified operational groups or use an approach that respects clustering, then rebuild a stable baseline. If the process itself drifted throughout the baseline, a larger dispersion factor hides the instability rather than correcting it. The clustering lesson connects shared units to understated uncertainty; serial dependence matters across weeks.

Monitor the measurement process too

Audit a sample of images under one damage rubric and track disagreement by inspector and week. A process chart on defect labels cannot distinguish product quality from a new interpretation of damage without such checks. Report both the operational trend and the measurement audit. The project holds a quality claim if route clustering or a rubric switch has not been reconciled, even if a simple control chart appears favorable.

Implementation

def rough_binomial_dispersion(weekly_counts):
    if len(weekly_counts) < 2 or any(n <= 0 or not 0 <= defects <= n
                                      for defects, n in weekly_counts):
        raise ValueError("at least two valid weeks required")
    total_defects = sum(defects for defects, _ in weekly_counts)
    total_inspected = sum(n for _, n in weekly_counts)
    baseline = total_defects / total_inspected
    if not 0 < baseline < 1:
        raise ValueError("baseline needs both outcomes")
    pearson_sum = sum((defects - n * baseline) ** 2 /
                      (n * baseline * (1 - baseline))
                      for defects, n in weekly_counts)
    return pearson_sum / (len(weekly_counts) - 1)

assert rough_binomial_dispersion([(4, 100), (4, 100)]) == 0
assert rough_binomial_dispersion([(1, 100), (9, 100)]) > 1

Performance and operating cost

The diagnostic is O(w) time and O(1) extra space for w weeks. Better chart design may need route-level records and a different variance model. Inflating bands after seeing an inconvenient alarm is cheap but erases the intended monitoring rule.

Common Mistakes

Treating many parcels from one truck as independent observations.
Calling every out-of-limit week a process failure without checking sampling.
Using overdispersion to justify hiding a baseline that was already drifting.
Ignoring inspector or rubric changes when labels define the measured outcome.