Which Store Is Letting Us Down?

Capstone 5 ran a one-way ANOVA on exam scores where every assumption held. This is the same shape of question with none of them holding, and it closes the loop the last two chapters opened: Mann-Whitney for two groups, Wilcoxon for two paired measurements, and now the rank-based answer for three or more.

Four stores collected customer experience ratings on a 1-to-10 scale over one quarter. The regional manager wants to know which locations need attention. That is not one question but two, and they need separate machinery.

The null hypothesis is that all four locations' rating distributions are the same. The alternative is that at least one differs, which is the most an omnibus test can deliver. Identifying which is a second step, with six pairwise comparisons and its own error-rate problem, exactly as in Capstone 5.

Cleaning removed three duplicate records, three blanks, and one impossible rating of 11, and normalized the location labels, leaving 194 ratings. Group sizes range from 43 to 57, because each store collected what it could. That is untidy and perfectly acceptable for these methods.

Harbour Point's spread is a finding in its own right. A store whose ratings run from the bottom of the scale to the top is delivering an inconsistent experience rather than a uniformly poor one, and those two problems have different causes and different fixes.

Normality falls in three of the four stores. No single location looks wildly non-normal, but with ratings bunched in the upper half of a bounded scale the departures are systematic rather than accidental. Equal spread falls too, and decisively.

The four locations are not interchangeable. Epsilon-squared says location accounts for around 16 percent of the variation in ranks, which is a moderate effect for a factor as blunt as which building a customer walked into.

What the test does not say is which stores differ. It is tempting to look at the mean-rank ordering and read the gaps between adjacent stores as established. They are not, and the next section is where that becomes concrete.

Four groups make six pairwise comparisons. Running six tests at 0.05 apiece would push the chance of at least one false alarm to roughly 26 percent, the same arithmetic that drove Capstone 4. Dunn's test is the standard follow-up to Kruskal-Wallis: it compares mean ranks pairwise using the ranking from the full sample, and the p-values are then adjusted, here by Holm's method.

The ordering is real. The gaps between neighbors mostly are not resolvable at this sample size.

It is not a test of medians

Northgate and Old Town both have a median of 7. Dunn's test still puts 21 points of mean rank between them, because Old Town has far more customers up at 8 and above: 48 percent against Northgate's 33 percent. Kruskal-Wallis, like Mann-Whitney in Capstone 14, asks whether one group's values tend to be larger. It is not comparing any single summary statistic, and two groups can share a median while being clearly distinguishable.

Significance does not chain together

Riverside differs from Northgate (p = 0.002). Old Town sits between them and differs from neither (p = 0.14 and p = 0.10). Read as logic that looks like a contradiction. It is not.

Customer experience genuinely varies across the four locations, and about a sixth of the variation in ratings tracks with where the customer shopped. Harbour Point is the clear problem: lowest ratings, widest spread, and reliably below the two best stores. Riverside is the strongest. Old Town and Northgate sit in the middle and cannot be told apart from each other.

The operational reading is to put the attention on Harbour Point, keep an eye on Northgate, and build no league table out of the middle two. The wide spread at Harbour Point is its own signal, pointing at an inconsistent experience, which usually means staffing or scheduling rather than the site itself.

• Customers who leave feedback are not customers. Every rating here is volunteered, and people with strong feelings volunteer more readily. If Harbour Point's unhappy customers are also its most vocal, part of its gap is a response-rate artifact rather than a service difference.
• The stores are not otherwise identical. They serve different neighborhoods, sizes, and footfall patterns. This compares locations, not management. Attributing the gap to the staff at Harbour Point needs evidence this design does not contain.
• Ranking people on this would be a mistake. A result where the middle two stores are indistinguishable cannot support individual performance judgments. It supports a question worth asking at one location.
• Do not average the ratings for a dashboard. The composition chart in section 2 is the honest display for an ordinal scale.
• Group sizes reflect collection effort, not the customer base. Northgate contributed the most responses because it collected the most, which is fine for these tests but means the pooled figures are not a picture of the population.

Kruskal-Wallis completes a set the last two chapters started, and the symmetry is worth committing to memory. Every parametric comparison has a rank-based counterpart that asks the same structural question with weaker assumptions.

Friedman is the one this part did not run. It is Kruskal-Wallis for repeated measures, and it would have been the fallback in Capstone 6 had the sphericity correction not been available.

Three of the six comparisons came back "not distinguishable". That phrase does a lot of work and hides an important distinction, because a pair can be indistinguishable for two very different reasons: the stores really are alike, or the study was too small to tell them apart. Intervals separate those cases; p-values do not.

The shift estimates are coarse, and the reason is instructive. Every one comes out at a whole number of rating points, because the ratings themselves are whole numbers. On a 10-point scale a shift statistic cannot resolve anything finer, so it is a blunt instrument here. The interval widths still separate the unresolved pairs: Northgate against Harbour Point spans 0 to 2 points, so that pair is probably different and this quarter could not confirm it, while Riverside against Old Town spans 0 to 1, a genuinely narrow gap. Both were reported as "p above 0.05"; they are not the same situation.

The top-box shares are where the resolution actually is. 63 percent of Riverside's customers rate 8 or above against 23 percent of Harbour Point's. That is the comparison to put in front of an operations team, because it is expressed in customers rather than in ranks and it does not require averaging an ordinal scale. Epsilon-squared gets a range too, so "location explains about a sixth of the variation" is fair and anything more precise is not.

Eight questions on this capstone, from the omnibus-then-post-hoc discipline to why significance refuses to chain. Answer them, hit Check Answers, and keep refining until you score 100%. Your progress is saved.