Tensions in Practice: One workload summary in tension with comparable subgroup rates¶
Invented success counts for two task handlers
A succeeds on 9 of 10 easy jobs and 18 of 90 hard jobs. B succeeds on 72 of 90 easy jobs and 1 of 10 hard jobs. A’s success rate is higher in each group, yet B’s pooled rate is higher because B handled mostly easy jobs. The pooled summary answers what happened across each actual workload; the subgroup view exposes the composition behind that summary.
Report the actual total
Give one compact description of delivered outcomes.
Expose workload composition
Keep like-labeled task groups visible.
Why these aims pull against each other
A single rate combines performance with the mix of work. Exposing strata adds detail and requires meaningful group labels.
Choose an arrangement to see what changes and what remains difficult.
Rows identify the same task groups in both arrangements. Pooled rates remain unchanged while subgroup rates are revealed; no observations are added or removed. Shading marks the higher rate within each displayed comparison row; it is not a causal or overall quality judgment.
What this choice protects
What it costs
When it fits
Compare the arrangements
Report the pooled rates
Show total successes divided by total jobs for each handler.
| A | B | |
|---|---|---|
| Easy | Not shownRetained in total | Not shownRetained in total |
| Hard | Not shownRetained in total | Not shownRetained in total |
| Pooled | 27/10027% | 73/10073% |
- What it protects
- A compact workload-level outcome is available.
- What it costs
- It cannot support a within-group ranking and hides the very different mixes.
- When it fits
- Fits an explicitly descriptive question about these actual workloads.
Illustration note: Both totals are arithmetically correct; the omitted subgroup cells are not missing source data.
Show the subgroup rates
Keep easy and hard rows beside the unchanged pooled total.
| A | B | |
|---|---|---|
| Easy | 9/1090% | 72/9080% |
| Hard | 18/9020% | 1/1010% |
| Pooled | 27/10027% | 73/10073% |
- What it protects
- The reversal and its unequal weights can be inspected directly.
- What it costs
- More cells must be communicated, and subgroup definitions require justification.
- When it fits
- Fits when within-group comparisons matter and these labels are relevant.
Illustration note: No causal effect is identified by this table alone; subgroup comparability is not guaranteed by a shared label.
What this illustration does—and does not—establish
The source supplies the tension. The invented setting, alternatives and any numbers illustrate a limited comparison; each arrangement retains its stated costs and conditions.
- All counts are invented; no handler’s real effectiveness is estimated.
- The table proves an arithmetic reversal, not a causal explanation of success.
- The pooled answer is legitimate for its own descriptive question; stratifying mechanically on every variable is not a remedy.
Source entries
Simpson's Paradox
This source passage supplies the contextual tension. The concrete arrangements and schematic examples are editorial illustrations, not measured findings.
Aggregate Direction versus Subgroup Direction (sign/direction)
The same data can show one direction in every subgroup and the opposite in the pool, and neither number is arithmetically wrong.
The source operation
Simpson's paradox is the structural pattern in which a relationship between two variables *runs in one direction inside every subgroup of a population and in the opposite direction in the aggregate*, because the subgroups differ in size, in baseline rates, or in their joint distribution along a third variable that has been silently mixed away.