STAT 5003 Module 2 Probability and Sampling Analysis Example

The Best Store and the Worst Store Were Both the Smallest: Probability, Sample Size and Eight Monthly Return Rates

Student Name

American College of Education

STAT5003: Business Statistics: Data-Informed Decision-Making

Module 2 Assignment

Instructor Name

May 19, 2025

The Number Being Quoted

Riverbend Home Supply, the composite home goods retailer used in this course, tracks the share of large-item orders that customers return within 30 days. Returns are expensive, since each one requires a truck, a crew and often a markdown, so the regional manager ranks the eight stores each month by return rate and discusses the results on a call with store managers. Last month, 285 of 4,600 large-item orders were returned, a company rate of 6.2 percent. Wexford had the lowest rate, 4.1 percent, and was praised; Stillwater had the highest, 9.3 percent, and its manager was asked for an improvement plan.

Wexford and Stillwater are also the two smallest stores, with 220 and 150 large-item orders that month, against 1,100 at the largest. That pattern is the starting point of this analysis. When the best and worst performers are both the smallest units, the first suspect is not management but arithmetic.

A Probability Model for Returns

A simple model treats each order as an independent trial with the same probability of being returned. If every store shared the company rate of 6.2 percent, the number of returns at a store with n orders would follow a binomial distribution, and its return rate would have a standard error equal to the square root of p times the quantity one minus p, divided by n, where p is 0.062 and n is the store's number of orders. The formula matters less than what it implies, which is that the uncertainty in a rate depends on how many orders produced it. The model's assumptions are worth stating. It assumes orders are independent, which is reasonable for different customers, and that every store has the same underlying rate, which is exactly the claim being tested. It also assumes the store's product mix is similar, which is roughly true across Riverbend's stores.

The standard error shrinks as volume grows, but only with the square root. At Lakeside, with 1,100 orders, the standard error is about 0.73 percentage points, so chance alone would usually keep the monthly rate between about 4.7 and 7.7 percent, two standard errors either side of 6.2. At Stillwater, with 150 orders, the standard error is about 1.97 points, and the usual range runs from about 2.3 to 10.1 percent. The same process, with no difference in management at all, produces a range more than two and a half times as wide at the smallest store as at the largest.

Testing the Two Extremes

Stillwater's 14 returns on 150 orders is a rate of 9.3 percent. Under the binomial model with p equal to 0.062, the probability of 14 or more returns in 150 orders is about 0.08. Wexford's 9 returns on 220 orders is a rate of 4.1 percent, and the probability of 9 or fewer is about 0.12. Neither result is unusual for a single store in a single month; each would occur by chance roughly once a year at a store of that size, even if its managers did nothing differently.

The ranking makes the problem worse, because it looks across eight stores at once. If every store truly shared the company rate, the chance that at least one of the eight would show a result as far into the upper tail as Stillwater's, a probability of 0.08 or less, is about one in two, treating the stores as independent. Some store will look bad almost every other month by chance. Wainer (2007) called the formula for the standard error of a mean the most dangerous equation, because ignoring it leads decision makers to see talent or failure in what is only the greater variability of small units, and he documented the error in settings from school rankings to cancer rates. Tversky and Kahneman (1971) found that even trained researchers expect small samples to resemble the population much more closely than they do, a belief they called the law of small numbers.

What the Ranking Is Doing

Deming (1986) distinguished variation from common causes, built into a stable process, from variation with special causes that can be traced and fixed, and warned that treating common-cause variation as if it had a special cause leads managers to blame and reward people for outcomes they did not control. The monthly ranking does exactly that. Of the eight stores last month, all eight fell inside the two-standard-error range for their size. The manager at Stillwater has been asked to explain a number that the model says could easily be chance, and the manager at Wexford may draw lessons from a good month that will not repeat.

None of this proves that the stores are identical. A store could have a real problem, such as a delivery crew that damages items, and still fall within the range in a given month. But a single month at a small store cannot separate a real problem from chance, and the ranking gives both the same weight.

Sampling to Find Out Why

If the rates cannot say which store has a problem, the reasons for returns might. Riverbend records a reason code, but store staff choose it quickly, and the most common code is simply customer changed mind. A better approach is to draw a random sample of returned orders from the last six months and have one trained reviewer read each order's notes, photographs and delivery record to classify the true reason. To estimate the share of returns caused by delivery damage within plus or minus 3 percentage points with 95 percent confidence, using the conservative assumption that the share is near one half, the sample needs about 1,068 returns. If the share is thought to be nearer 30 percent, about 897 are enough.

Riverbend had about 1,700 large-item returns in the last six months, so a sample of about 900 is feasible but large. A smaller sample of 400, giving a margin of about plus or minus 4.5 points at a 30 percent share, would still separate a problem affecting one return in three from one affecting one in ten. The sample should be drawn by random selection of return numbers across all stores, so that the result describes the company and, with enough returns per store, allows the largest stores to be compared.

What the Manager Should Do Differently

Three changes follow. First, stop ranking stores on one month's return rate. Report each store's rate against its own expected range, so that a store is flagged only when its rate falls outside what chance would produce at its volume. Second, judge small stores over longer periods: six months of orders at Stillwater, about 900, would narrow its standard error to about 0.8 points, similar to a large store in one month. Third, use the reason audit, not the rate, to decide where to act, since knowing that a store's returns come mostly from delivery damage says what to fix in a way that a rate never can.

References

Deming, W. E. (1986). Out of the crisis. MIT Press.

Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76(2), 105-110. https://doi.org/10.1037/h0031322

Wainer, H. (2007). The most dangerous equation. American Scientist, 95(3), 249-256. https://doi.org/10.1511/2007.65.1026