Grades 9–12
What can a score hide about the people behind it?
A score summarizes a record, so its meaning depends on what was counted. A completion rate divides completed tasks by attempted tasks. When easy and hard tasks are combined, each group contributes according to the number of attempts it contains. Different mixes can produce a surprising overall comparison.
In the diagramGrouped and combined rates can reverseFictional puzzle completions / attemptsDifficultySystem ASystem BEasy81 / 9019 / 2090%95%Hard1 / 1016 / 8010%20%Combined82 / 10035 / 10082%35%Easy-puzzle share: A 90/100; B 20/100.B is higher in both groups; group sizes differ.
In the pictured fictional records, B completes 95% of easy tasks and 20% of hard tasks, above A’s 90% and 10%. Yet A completes 82% overall while B completes 35%. A has many more easy tasks in its mix: 90 of its 100 attempts, compared with 20 of B’s 100. The combined rate gives those easier tasks greater weight in A’s score.
The chosen measure can affect behavior. If a toy reward pays only for submission count, increasing submissions may earn more points without improving the intended goal of verified work. That result follows from the stated reward rules; real people may respond in many different ways.
Wording can emphasize one side of a record: “80 successes” and “20 failures” may describe the same 100 attempts. Timing can also tempt a stronger conclusion than the evidence supports. An increase after a reward begins does not isolate the reward’s effect without a suitable comparison and attention to other changes. A useful score needs its denominator, context and limits.
Sources
- Pennsylvania State University, STAT 504 — Three-Way Tables: Types of Independence — Simpson’s paradox
- Stanford Encyclopedia of Philosophy — Simpson’s Paradox
- OpenStax, Rice University — 8.5 Individual and Group Incentive Plans
- OpenStax, Rice University — 2.4 Agency Issues: Shareholders and Corporate Boards
- National Council for the Social Studies — C3 Framework for Social Studies State Standards
- Amos Tversky and Daniel Kahneman, Science — The Framing of Decisions and the Psychology of Choice (1981)
- National Park Service — Historic Places and the Inquiry Method
- OpenStax — Scatter Plots, Correlation, and Regression Lines