M10 · optional contextual support
Compare paired experiments and interactions
Keep task identity when comparing methods and enumerate a small bootstrap.
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
On the same tasks, baseline errors [4,6] and candidate errors [3,5] give paired improvements [1,1]. Pairing removes task difficulty from the comparison; shuffling identities answers a different question.
2. Predict before changing a value
Change average improvement and between-task spread. Enumerate all four-task bootstrap samples and inspect the interval. Change AB’s factorial score while keeping separate A and B scores fixed. 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
- Paired mean improvement
- 1.00000
- Exact empirical bootstrap central ~95% interval
- 0.50000 to 1.50000
- Factorial interaction (AB−A−B+base)
- 0.00000
Vertical axis: Paired improvement. 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.
| Task | Baseline error | Candidate error | Paired improvement |
|---|---|---|---|
| 0 | 4 | 2.5 | 1.5 |
| 1 | 6 | 5.5 | 0.5 |
| 2 | 3 | 1.5 | 1.5 |
| 3 | 7 | 6.5 | 0.5 |
All 4⁴=256 bootstrap resamples of four task differences are enumerated. This empirical interval does not establish population coverage with four authored tasks. Factorial scores base=0,A=1,B=2,AB=n must use matched resources; interaction differs from either main effect.
Change average improvement and between-task spread. Enumerate all four-task bootstrap samples and inspect the interval. Change AB’s factorial score while keeping separate A and B scores fixed.
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
On the same tasks, baseline errors [4,6] and candidate errors [3,5] give paired improvements [1,1]. Pairing removes task difficulty from the comparison; shuffling identities answers a different question.
Keep task identity when comparing methods and enumerate a small bootstrap. 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
differencei=baselinei−candidatei; mean difference=Σdifference/n; interaction=AB−A−B+baseline
Positive difference means lower candidate error. n counts independent tasks here, not repeated seed measurements or rows. A and B label interventions, and AB applies both under a matched resource protocol.
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
Change average improvement and between-task spread. Enumerate all four-task bootstrap samples and inspect the interval. Change AB’s factorial score while keeping separate A and B scores fixed. 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 C14, 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
Resample the actual independent unit. If tasks have several seeds, a task-cluster bootstrap retains their dependence. An empirical interval from four authored tasks is illustrative, not a population guarantee. Factorial interaction tests whether the combined effect differs from adding the separate effects; extra compute is a competing explanation unless controlled.
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
Resample the actual independent unit. If tasks have several seeds, a task-cluster bootstrap retains their dependence. An empirical interval from four authored tasks is illustrative, not a population guarantee. Factorial interaction tests whether the combined effect differs from adding the separate effects; extra compute is a competing explanation unless controlled.
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 C14, C15. Browse every support topic and course connection.