M11 · optional contextual support
Separate a comparison from a causal claim
Change cohort composition and diagnose an aggregate reversal.
Start with arithmetic, percentages and basic algebra. Open any section directly. Practice and completion here do not change any AI course requirement or the original five-lesson mathematics pilot.
1. Try a concrete case
Treatment succeeds in 80% of easy cases and 20% of hard cases; control succeeds in 70% and 10%. Treatment helps within both groups, but giving it mostly hard cases can make its overall rate look worse.
2. Predict before changing a value
Give treatment fewer easy cases than control, then equalize the mixtures. Explain the reversal without changing any within-group rate. Name the quantity you expect to change and one you expect to stay fixed before using the controls.
3. Experiment
Change one quantity, follow every calculation
- Observed treated success
- 0.32000
- Observed control success
- 0.58000
- Equal-mixture treated
- 0.50000
- Equal-mixture control
- 0.40000
Vertical axis: Success fraction. Fixed scales; 0 points clipped for display. Lines join sampled values to aid reading; they do not create extra observations. Exact numerical values are below.
| Group | Treated success rate | Control success rate |
|---|---|---|
| Easy | 0.8 | 0.7 |
| Hard | 0.2 | 0.1 |
Treatment improves each authored subgroup by 0.1, but different group mixtures can reverse the observed aggregate. Standardization answers a chosen mixture question; causal interpretation still needs consistency, overlap and no unmeasured confounding. Randomization is a design choice, not a claim inferred from this table.
Give treatment fewer easy cases than control, then equalize the mixtures. Explain the reversal without changing any within-group rate.
Controls save locally as practice. No timers, remote services or GPU are required. Maximum numerical work is a bounded small table or 256 bootstrap resamples.
4. Understand the calculation
Start with a calculation you can inspect
Treatment succeeds in 80% of easy cases and 20% of hard cases; control succeeds in 70% and 10%. Treatment helps within both groups, but giving it mostly hard cases can make its overall rate look worse.
Change cohort composition and diagnose an aggregate reversal. The experiment changes one named quantity at a time and shows the intermediate values, rather than asking you to accept a final score. Before changing a control, write down which value you expect to move and why.
Read the formula in words and symbols
aggregate rate = easy fraction×easy rate + hard fraction×hard rate
Fractions sum to one inside each cohort. A standardized rate uses the same declared mixture for both methods. Association compares observed groups; an intervention claim needs additional design assumptions.
A subscript names a coordinate, case or time index; it is not multiplication. A summation sign means add the terms for the stated indices. Parentheses identify which calculation happens first. Carry units through arithmetic: adding incompatible quantities or confusing a rate with a count changes the question.
Connect the visual and the numbers
Give treatment fewer easy cases than control, then equalize the mixtures. Explain the reversal without changing any within-group rate. The plot uses declared fixed axes, so a new setting does not silently rescale the apparent change. Any clipped values are counted and remain available in the numerical table. Do not infer an exact curve or a new observation from a line drawn between points.
Use it in an AI experiment
These ideas support C07, C09, C15. Use the return link when you entered from an AI lesson; its original section and experiment remain separate. A mathematical support calculation is a way to inspect an assumption, not evidence that a trained model generalizes.
Optional derivation and implementation depth
A causal contrast needs consistency of treatment definition, overlap and exchangeability or an appropriate identification argument. Standardization can adjust observed group differences but does not remove an unmeasured confounder automatically. Randomization changes the assignment process; it does not guarantee perfect balance in every finite sample.
Recompute the worked example with a separate piece of arithmetic or a short program. State the domain and test a boundary case before using the formula elsewhere. Guided numerical practice, independent implementation and human review are different outcomes.
Optional implementation and research depth
A causal contrast needs consistency of treatment definition, overlap and exchangeability or an appropriate identification argument. Standardization can adjust observed group differences but does not remove an unmeasured confounder automatically. Randomization changes the assignment process; it does not guarantee perfect balance in every finite sample.
Recompute one row independently, then test a boundary case and state which assumption it probes. Passing the supplied practice case does not establish independent implementation proficiency.
5. Apply it to a different case
Practice not yet passed; independent mastery and human review are not claimed.
6. Continue when useful
Completion not recorded. Skipping retains settings and practice; it never awards completion.
Used in C07, C09, C15. Browse every support topic and course connection.