Understand · explore · practise
Investigate original paired data with a spreadsheet: preserve observations, compare justified subsets, fit regression and label missing-response estimates separately.
Before you startScatter diagrams, residuals, regression interpretation and prediction limits.
01 / Ask a question before fitting
Keep raw observations, inclusion decisions and predictions separate.
Do not delete an inconvenient observation or treat an estimated value as a measurement.
Our fictional question is whether distance helps predict delivery duration. A recorded road closure may be relevant to all deliveries but outside a specifically defined routine-conditions question. Compare both scopes openly rather than selecting whichever line looks tidiest.
Six routine deliveries and one recorded road-closure delivery are constructed examples, not real customer records. Both fits use observed complete pairs only. The missing duration at 9 km stays missing.
Using all seven recorded deliveries, the predicted duration at 8 km is about 32.14 minutes. This is a fitted estimate, not a replacement for either recorded 8 km duration.
02 / Inspect the paired records
On a narrow screen, scroll the table sideways to read the conditions.
| ID | Distance / km | Recorded duration / min | Conditions |
|---|---|---|---|
| A | 2 | 14 | Routine |
| B | 4 | 16 | Routine |
| C | 6 | 24 | Routine |
| D | 8 | 29 | Routine |
| E | 10 | 31 | Routine |
| F | 12 | 39 | Routine |
| G | 8 | 55 | Road closure recorded |
| H | 9 | Missing | Conditions not recorded |
All values are constructed for this activity. H cannot supply a complete observed pair. D and G are different deliveries at the same distance, so they are not duplicate records merely because x repeats.
How many complete observed pairs are available for an analysis of all recorded deliveries?
H has no recorded duration.
Seven. Keep H in the raw table but exclude it from calculations requiring both observed coordinates.
Why should a spreadsheet sort entire rows rather than the distance and duration columns separately?
Each duration belongs to a particular delivery.
Sorting columns independently breaks the pairing and creates observations that were never recorded.
03 / Build a reproducible spreadsheet
Download the original investigation workbook or download the raw constructed records as CSV.
Why use a scatterplot with a numeric distance axis rather than a category line chart for these pairs?
The numerical spacing of distances matters.
A scatterplot positions each pair using both numeric coordinates. Category spacing can conceal the actual distances between x values.
Should H’s missing duration be replaced by zero to make the regression run?
Zero is an actual numerical claim.
No. A missing value is not a zero-minute delivery. Fit from complete observed pairs and retain a missing marker for H.
04 / Compare two stated scopes
Pause, replay or seek freely. The notes explain the same idea and stay in view.
At 9 km, predicted duration is 30.5 minutes
This describes the routine subset.
Including the closure delivery gives t̂ ≈ 9.52419 + 2.82661d
Use unrounded coefficients for calculations.
At 9 km, that fit predicts about 34.96 minutes
The question and inclusion scope explain the difference.
Is “the closure point makes the line less neat” a defensible reason to delete G?
Distinguish evidence of error from disagreement with a trend.
No. G is documented as a recorded closure delivery. Retain it in raw data. A routine-only analysis can exclude it for an explicitly narrower question, while reporting that decision and comparing sensitivity.
The two recorded durations at 8 km are 29 and 55 minutes. Which should the line replace?
A fitted value is a summary, not a correction.
Neither. Retain both observations. Their differing conditions may explain variation; the fitted value remains a separate model output.
05 / Label an estimate as an estimate
The routine-only line predicts 30.5 minutes at H’s 9 km. What should you record?
H’s conditions are not recorded.
Record 30.5 as a model estimate conditional on routine conditions, in a separate prediction field. Keep the observed-duration field missing; applicability to H is uncertain.
Why should you not fill H’s raw duration with a model prediction and then count it as an eighth observed pair?
The estimate was generated by the same data and model.
That invents evidence and can make fit or sample-size claims misleading. It adds no independent measurement. Keep an audit trail and use the seven genuinely observed pairs.
06 / Audit range and residuals
Both observed scopes have distances 2–12 km. Classify a prediction at 16 km.
Compare the supplied distance with the observed distance range.
It is extrapolation for either scope. The routine fit gives 48 minutes; the all-recorded fit gives 54.75 minutes. Neither extension is validated just by evaluating the formula.
For the routine fit, find the residual of the routine delivery D at 8 km and 29 minutes.
Observed minus predicted.
The fitted value is 28 minutes, so the residual is 1 minute. This does not change the recorded value 29.
07 / Limit the conclusion to the evidence
Can these constructed records prove the same relation applies to every delivery company or city?
Consider data origin, sampling and context.
No. They are an illustrative dataset, not a representative sample of a real population. A real investigation would need suitable sampling, comparable conditions and validation.
Does the fitted gradient measure the causal effect of adding a kilometre to any delivery?
Other conditions can vary with distance.
Not from these observational pairs alone. It describes a fitted association under the chosen scope, with traffic, route and other possible influences.
08 / Preserve, compare and qualify
Preserve complete pairs and missing markers, define the population and scope, inspect a numeric scatterplot, fit the right response, and compare justified inclusion choices. Keep estimates separate and explain the range, residual variation and sampling limitations.
Section 1 of 8 · Ask a question before fitting