Grade by Grade
D. Data · Data, Statistics and Mathematical Modelling

Reporting a Model: Answers, Fit, Limitations and Predictions

D2.5 A1About 70 minutes
Learning goals
  • I can report how my model answers the question of interest.
  • I can describe how well a model fits using statistics, residuals and tests with new data.
  • I can describe the limitations of a model.
  • I can state predictions with suitable uncertainty and identify interpolations and extrapolations.

Words to know

modelling report
A written or oral account of a model: the question, data, model, answer, fit, limitations and predictions.
validate
To test a model with new data that was not used to build it.
percent error
The size of the error divided by the actual value, times 100%.
uncertainty
How unsure we are about a value or prediction.
recommendation
Advice for a decision, based on the model.

The model is done. Now what?

A model that stays in your notebook does not help anyone. The last part of the modelling process is to report it: explain how the model answers the question of interest, how well it fits, what its limitations are, and what predictions it can make. The people reading the report (a coach, a town council, your family) need to understand the answer and how much to trust it.

1. Questionwhat we wanted to know2. Datasource, plan, sample3. Modeldisplay, equation, why chosen4. Answerthe answer to the question5. FitR², residuals, tests6. Limitationsassumptions, range, errors7. Predictionswhat we can predict, and how sure+ recommendations andnext steps (optional)
The parts of a modelling report. Parts 4 to 7 match what the curriculum asks you to report: the answer, the fit, the limitations and the predictions.

Describing how well a model fits

Use several kinds of evidence:

  • Statistics: r (for a linear model) and R². "R² = 0.98 means the model explains about 98% of the variation in y."
  • The graph: does the model pass through the middle of the points across the whole range?
  • Residuals: are they small and randomly scattered? Give the largest one in context ("every prediction was within 0.5 cm of the measured mean").
  • Testing with new data: collect a few new data points that were not used to build the model and see how close the predictions are. This is the strongest test.

Describing limitations

Every model has limitations. Common ones:

  • Assumptions that may not always be true (same ball, same floor, same weather).
  • Range: the model was built from x-values in a certain range; predictions far outside it are extrapolations.
  • Sample: the size of the sample, and whether it represents the population.
  • Measurement error in the data.
  • Missing variables that also affect the result.
  • Impossible values: where would the model give answers that make no sense (negative heights, more than 100%)?

Describing predictions

State predictions with suitable uncertainty words: "about", "approximately", "if the trend continues", "for drops between 50 and 200 cm". Separate interpolations (more reliable) from extrapolations (less reliable), and say why.

Worked example

Example 1: A complete report for the bounce project

Write a report for the question: How high will a tennis ball bounce when dropped from different heights, and from what height must it be dropped to bounce back up 1 m?

  1. Question: How does a tennis ball's bounce height depend on its drop height, and what drop height gives a 1 m bounce?
  2. Data: primary data from an experiment. One tennis ball was dropped onto a hard floor from 7 heights (50 to 200 cm), 3 trials each. Heights were read at eye level from slow-motion video. Mean bounce heights: 27, 40, 53, 66, 79, 91 and 104 cm.
  3. Model: the scatter plot showed a strong linear pattern. Linear, quadratic and exponential regressions were tested. The linear model was chosen because it fits almost perfectly (r² ≈ 0.9999), the quadratic model was practically the same line, and a constant bounce fraction makes sense physically. Model: bounce height ≈ 0.513 × drop height + 1.61 (in cm).
  4. Answer: a drop height of about 192 cm gives a 1 m bounce. (A direct-variation model, bounce = 0.528 × drop, gives about 189 cm, so the answer is about 1.9 m either way.)
  5. Fit: all residuals were within 0.5 cm, scattered above and below zero. Test: three new drops from 192 cm gave bounces of 99, 101 and 100 cm (mean 100 cm), matching the prediction.
  6. Limitations: the model applies only to this ball on this floor; a different ball, a carpeted floor, or a ball that has lost pressure would bounce differently. Heights were read by eye, so each value may be off by about 1 cm. Only 7 heights were tested.
  7. Predictions: for drops between 50 and 200 cm, predictions should be within about 1 cm (interpolation). For higher drops, the model is an extrapolation. A test drop at 250 cm bounced 127 cm, about 3 cm less than the model's 129.9 cm, so the model may over-predict for high drops (air resistance has more effect on longer falls).

A good report answers the question in one clear sentence, then gives the evidence (fit), the limits (limitations) and what else the model can and cannot predict.

Testing the bounce model with new drops040801201602002402800306090120150drop height (cm)bounce height (cm)test: 100test: 127(model: 129.9)extrapolation
New test drops (larger open circles). At 192 cm the test bounce (100 cm) matched the model's 100.1 cm very closely. At 250 cm (shaded extrapolation region), the ball bounced 127 cm, a little below the model's 129.9 cm.
Worked example

Example 2: Improve a weak report

A student's report says: "My model is y = 0.098x + 0.21. It is really good. The rink should make 16 L." What is missing? Rewrite it.

  1. Missing: what x and y stand for, the question, where the data came from, how well the model fits (evidence, not "really good"), limitations and the range of skater numbers where it applies.
  2. Rewrite: We wanted to know how much hot chocolate to prepare for a public skate. Using the rink's records for 10 skates (60 to 240 skaters, all on days between −15 °C and −5 °C), we found a strong linear relationship between the number of skaters (x) and litres sold (y): y ≈ 0.098x + 0.21, with r ≈ 0.995. That is about 0.1 L per skater.
  3. For an expected 160 skaters, the model predicts about 16 L. We recommend preparing 17 to 18 L so the rink does not run out.
  4. Limitations: the model is based on only 10 sessions in one temperature range. On mild days, or for crowds larger than 240, it may not be accurate. Holiday skates might also be different. Recording temperature and sales over the rest of the season would improve the model.

The rewrite names the variables, gives evidence of fit, answers the question, and states limitations and conditions.

Sentence starters for reports

  • "The question we investigated was..."
  • "We chose a ___ model because..."
  • "The model fits well/poorly because r² = ___ and..."
  • "The model predicts that... if..."
  • "This prediction is reliable/less reliable because it is an interpolation/extrapolation."
  • "One limitation is... This could be improved by..."
Just for fun

A weather forecaster's report: "Tomorrow: 100% chance of a forecast." At least the model's confidence in that is well justified.

Hands-on activity

Report to your family

You need: your model from the last lesson (or the bounce or hot chocolate example), paper or a slide app, and your family as the audience.

  1. Write a report with the seven headings: Question, Data, Model, Answer, Fit, Limitations, Predictions. Include your graph.
  2. If you can, test your model: collect 2 or 3 new data points and compare them with your predictions.
  3. Present your report to your family in under 3 minutes.
  4. Ask your audience: "Do you trust my answer? What would make you trust it more?" Write down their responses and one improvement you would make.

What to look for: a one-sentence answer to the question, evidence for fit (R², residuals or a test), at least two specific limitations, and predictions labelled as interpolation or extrapolation.

Your turn

1

List the seven parts of a modelling report.

2

A model has R² = 0.94. Write a sentence about its fit.

3

Give two limitations of the hot-chocolate model.

4

Using the bounce model y = 0.513x + 1.61, predict the bounce for a 30 cm drop. Is this interpolation or extrapolation?

5

Why is testing a model with new data stronger evidence than R²?

6

Rewrite this prediction with suitable uncertainty: "The ball will bounce 155.5 cm if dropped from 300 cm."

More practice: warm-up

1

Which part of a report tells the reader how much to trust the model?

2

Which is usually more reliable: interpolation or extrapolation?

3

Give one uncertainty word or phrase for predictions.

4

What is a limitation?

5

Should a report include the original question?

More practice: on your own

1

A model predicts a test value of 48 and the actual value is 45. Find the residual and say whether the model over- or under-predicted.

2

A report says, "Our model proves that more practice causes more free throws." Critique this sentence.

3

A town's model for road-salt use (S = 2.5d) was built with 5 winters of data, each with 40 to 60 snow days. Write one limitation and one prediction with appropriate uncertainty.

4

Explain why a report should state the range of the data.

5

Write a one-sentence answer to the bounce question that a Grade 5 student could understand.

Challenge

1

The bounce model predicted 129.9 cm for a 250 cm drop, but the test gave 127 cm. Calculate the percent error and explain what it suggests about using the model for extrapolation.

2

Write the "Limitations" and "Predictions" sections of a report for the video-views model y = 119.67(1.601)ˣ (days 0 to 7).

3

Explain why communicating limitations honestly makes a model more useful to decision-makers, not less.

Check your understanding

Quiz

1.

Which is the best statement of fit?

2.

Which is a limitation?

3.

A model built from x = 10 to x = 50 is used at x = 80. In the report, this prediction should be described as...

4.

What is the strongest test of a model's predictions?

5.

A model predicts 64 and the actual value is 60. What is the percent error, relative to the actual value?

6.

Which sentence best answers a question of interest in a report?

For parents and teachers

Big idea: D2.5 asks students to report how the model can be used to answer the question of interest, how well it fits the context, its potential limitations and what predictions can be made. Communicating clearly and honestly is as important as the mathematics.

How to help: Be a real audience for your teen's report. Ask the questions a decision-maker would ask: "How sure are you? What if...? Where does this not work?"

Common difficulties:

  • Reporting numbers (r, equations) without saying what they mean in context.
  • Calling a model "perfect" or saying it "proves" something.
  • Listing vague limitations ("human error"). Push for specific ones: which measurement, which assumption, which range.