SCIENCE · INVESTIGATE AND EXPLAIN

Inspect what a model misses

Use residuals to identify a systematic model mismatch.

Learning outcome: Compare observed and predicted positions with consistent units.

Starting recap

A residual is observation minus prediction. Its pattern can reveal a missing relationship.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

A linear position model is compared with a synthetic cart whose motion includes acceleration.

Open the experiment

Predict

Will changing only the linear coefficient remove all residuals?

Your prediction and reason

Experiment

Tune the linear coefficient with quadratic coefficient zero. Then compare a quadratic model.

Compare a model with observations

Worked teaching model · synthetic values · Four synthetic positions follow y = 2t + 0.5t² exactly. They contain no measurement noise. Matching this curve is not evidence that a real cart follows it.

Linear coefficient: 2 m/s Quadratic coefficient: 0 m/s²

Reset keeps your prediction, captures and practice history.

Calculated values on the same fixed scale
Row	Quantity	Value (m)
1	Observed at 1 s	2.5
2	Predicted at 1 s	2
3	Observed at 2 s	6
4	Predicted at 2 s	4
5	Observed at 3 s	10.5
6	Predicted at 3 s	6
7	Observed at 4 s	16
8	Predicted at 4 s	8

Root mean squared residual: 4.704 m.

Prediction = linear coefficient × t + quadratic coefficient × t². Residual = observation − prediction. Four tuning points are development evidence; reserve another condition to test transfer.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

A residual tells you what the prediction missed

The synthetic observations follow position y = 2t + 0.5t², where t is seconds and y is metres. The coefficient 2 has units m/s; 0.5 has units m/s². In a constant-acceleration interpretation, the quadratic coefficient is half the acceleration.

At t = 2 seconds the observation is 4 + 2 = 6 metres. A linear model y = 2t predicts 4 metres. Its residual, observation minus prediction, is +2 metres.

At times 1, 2, 3 and 4, this linear model’s residuals are 0.5, 2, 4.5 and 8 metres. Their increasing pattern suggests that one constant slope misses a relationship.

A summary does not replace inspection

Root mean squared residual squares each residual, averages the squares and takes the square root. It retains metres as its unit and penalizes larger discrepancies. For the four residuals above it is about 4.704 metres.

Changing the linear coefficient may reduce the summary error, but it cannot reproduce all four quadratic values exactly. Add the correct quadratic coefficient and every residual becomes zero in this noise-free teaching case.

The plotted values use a shared fixed scale. Inspect the pairs, explain the mechanism of mismatch and keep the tuned observations distinct from a later test.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Identify the tuning data and state why fitting them is not an independent test.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate random variation from systematic error Next: Test a model beyond its tuning cases

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.