Statistics in the Real World: Three Full Worked Case Studies

Individual formulas and calculators are easier to understand once you've seen how they combine to answer a genuine question someone actually needed answered. This guide walks through three full case studies — one from business, one from healthcare, one from sports — each using real statistical methods covered elsewhere on this site, worked through start to finish with actual numbers, not just described in the abstract.

Case Study 1: Business — Setting Inventory Safety Stock

The problem: A retailer sells a product with average weekly demand of 120 units and a standard deviation of 18 units (demand varies week to week due to normal fluctuation). It takes 2 weeks for a new order to arrive from the supplier. How much extra "safety stock" should the retailer keep on hand to avoid running out, while still being confident of meeting demand 95% of the time during that vulnerable reorder window?

This is a direct application of the normal distribution and z-scores: demand over the 2-week lead time is treated as approximately normal (a reasonable assumption when demand is the sum of many independent daily purchases, per the Central Limit Theorem), and "95% confident of meeting demand" translates directly into a z-score from the standard normal table — z = 1.645 for a one-sided 95% threshold.

Demand's variability over 2 weeks doesn't simply double the weekly standard deviation — variances add for independent periods, so the standard deviation scales by the square root of the number of weeks:

Safety stock = 1.645 × 18 × √2 = 1.645 × 18 × 1.414 ≈ 41.9 units (round up to 42, since a fraction of a unit can't actually be stocked).

Adding this to the expected demand over the lead time (120 units/week × 2 weeks = 240 units) gives the full reorder point: 240 + 42 = 282 units. When inventory drops to 282 units, it's time to reorder — the extra 42 units of buffer exist specifically to absorb the demand variability that a z-score of 1.645 says will be exceeded only 5% of the time. A retailer choosing 99% confidence instead (z ≈ 2.33) would need a noticeably larger buffer — the tradeoff between holding cost (excess inventory) and stockout risk is exactly what this calculation makes explicit and adjustable.

Case Study 2: Healthcare — Does the Medication Work?

The problem: A clinic tests a new blood pressure medication on 8 patients, measuring each patient's systolic blood pressure before starting the medication and again after four weeks of treatment. Because the same patients are measured twice, each patient acts as their own baseline — this calls for a paired-samples t-test, the version of the t-test built specifically for before/after or matched-pairs data, which strips out person-to-person variation that has nothing to do with the treatment itself.

A paired t-test works entirely on the differences, not the raw before/after values — reducing an 8-patient, two-measurement dataset to a single 8-number list. The mean difference is 7.5 mmHg, with a standard deviation of the differences of approximately 3.42 mmHg.

Standard error = 3.42 / √8 ≈ 1.21. t = 7.5 / 1.21 ≈ 6.20, with df = n−1 = 7.

A t-statistic this large, on only 7 degrees of freedom, is far beyond the critical value of roughly 2.365 needed for significance at α = 0.05 (two-tailed) — the exact p-value comes out to approximately 0.0004, well under the conventional 0.05 threshold. The conclusion: the average 7.5 mmHg drop in systolic blood pressure is extremely unlikely to be due to chance alone, given how small the variability between patients was relative to the size of the drop. This doesn't yet prove the medication caused the improvement in some broader population — a genuine clinical trial would need a control group receiving a placebo, to rule out effects like patients naturally improving over time or simply being more careful about diet during the study — but within this specific before/after design, the statistical evidence for a real change is strong.

Case Study 3: Sports — Comparing Players Across Eras

The problem: Two baseball players from different decades are being compared. Player A batted .320 in a season where the league average was .260 with a standard deviation of .030. Player B batted .300 in a different era where the league average was .230 with a standard deviation of .020. Whose season was more statistically impressive, relative to their own era?

Comparing the raw batting averages directly (.320 vs. .300) would seem to favor Player A — but raw numbers from different eras aren't directly comparable, because the league-wide context (average and spread of performance) differed. This is exactly the situation z-scores exist to fix: converting each player's average into "how many standard deviations above their own era's mean" puts both performances on the same standardized scale.

Player A: z = (0.320 − 0.260) / 0.030 = 2.0

Player B: z = (0.300 − 0.230) / 0.020 = 3.5

Once standardized, the comparison reverses: Player B's .300 average was 3.5 standard deviations above their era's mean, versus Player A's 2.0 standard deviations — meaning Player B's season was substantially more exceptional relative to the competition they actually faced, despite the lower raw number. This is the same core idea behind the normal distribution and empirical rule: a z-score of 3.5 is a genuinely rare outcome (well under 1% of a normal distribution falls that far out), while a z-score of 2.0, though still strong, is meaningfully more common. Sports analysts use exactly this technique — sometimes formalized further into metrics like OPS+ or ERA+ — specifically to make cross-era comparisons fair.

What These Three Cases Have in Common

Despite covering completely different domains, all three case studies follow the same underlying pattern: start with a genuine decision or question that needs answering, identify which statistical tool matches the structure of the data (a normal-distribution threshold for the inventory problem, a paired test for the matched before/after clinical data, standardization for cross-context comparison), and interpret the numeric result back in terms of the original real-world question rather than stopping at the formula's output. This translation step — from a t-statistic or a z-score back into a decision or claim that means something to a non-statistician — is very often the hardest and most valuable part of applying statistics, more so than the arithmetic itself.

Common Mistakes

Skipping the "so what" step. Computing a correct t-statistic or z-score isn't the end goal — translating it into "the medication likely works" or "Player B had the more impressive season" is what actually answers the original question, and it's easy to stop short of that translation.

Treating statistical significance as proof of real-world importance. The blood pressure case study's p ≈ 0.0004 is strong evidence the change is real, but whether a 7.5 mmHg drop is clinically meaningful is a separate, domain-specific judgment that statistics alone doesn't settle.

Forgetting that variances add, not standard deviations, when combining independent periods. The inventory case study's √(lead time) scaling factor is a common point of confusion — doubling the time period does not simply double the standard deviation.