M02 · optional contextual support
Describe variation without inventing evidence
Inspect mean, variance, covariance and the assumptions behind uncertainty summaries.
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
Measurements 1, 2 and 3 have mean 2. Squared deviations are 1, 0 and 1, so population variance is 2/3. Repeating each row creates six rows but no new independent measurement.
2. Predict before changing a value
Move the center without changing spread, then change spread. Watch which statistics change. Increase row count and distinguish authored interpolation from new independent sampling. 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
- Mean
- 0.00000
- Population variance (divisor n)
- 0.50000
- Covariance with y
- 0.25000
- Illustrative standard error
- 0.31623
Vertical axis: y measurement. 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.
| Index | x | y |
|---|---|---|
| 0 | -1.00000 | -1.00000 |
| 1 | -0.50000 | 0.25000 |
| 2 | 0.00000 | -0.50000 |
| 3 | 0.50000 | 0.75000 |
| 4 | 1.00000 | 0.00000 |
Authored evenly spaced measurements, not a random sample. Standard error is only an illustration under independence; duplicates do not supply independent units.
Move the center without changing spread, then change spread. Watch which statistics change. Increase row count and distinguish authored interpolation from new independent sampling.
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
Measurements 1, 2 and 3 have mean 2. Squared deviations are 1, 0 and 1, so population variance is 2/3. Repeating each row creates six rows but no new independent measurement.
Inspect mean, variance, covariance and the assumptions behind uncertainty summaries. 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
mean = Σx/n; variance = Σ(x−mean)²/n; covariance = Σ(x−mean x)(y−mean y)/n
Σ means add the displayed terms. n is the number of rows. Variance has squared measurement units; covariance uses the product of both units. This descriptive formula uses divisor n.
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
Move the center without changing spread, then change spread. Watch which statistics change. Increase row count and distinguish authored interpolation from new independent sampling. 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 C01, C02, 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
Sample-variance estimation often uses n−1 under an independent common-distribution model. Standard error σ/√n needs independent units or an appropriate dependence correction. Calibration compares predicted probabilities with observed frequencies; small bins and selected samples make that comparison uncertain.
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
Sample-variance estimation often uses n−1 under an independent common-distribution model. Standard error σ/√n needs independent units or an appropriate dependence correction. Calibration compares predicted probabilities with observed frequencies; small bins and selected samples make that comparison uncertain.
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 C01, C02, C09, C15. Browse every support topic and course connection.