To test a model, put its predictions beside the supplied data and look at the differences. Then write one sentence that says where the model fits, where it does not, and how you know.
This lesson belongs to SPM Mathematics mathematical modelling. It follows choosing a simple model for a real situation.
What does it mean to test a model?
A model predicts a value for each input. Testing means comparing each prediction with the supplied value and asking whether the gap is small enough for the purpose.
Write the gap as observed − predicted. A positive gap means the model predicts too low, and a negative gap means it predicts too high.
Worked example: a phone battery
A student records a phone battery level every hour. The readings (hour, percent) are (0, 100), (1, 92), (2, 83), (3, 77), (4, 68) and (5, 60).
The student builds a rule from the first and last readings. The drop is 100 − 60 = 40 over 5 hours, which is 8 per hour, so B = 100 − 8t.
| Hour t | Observed | Predicted | Observed − predicted |
|---|---|---|---|
| 0 | 100 | 100 | 0 |
| 1 | 92 | 92 | 0 |
| 2 | 83 | 84 | −1 |
| 3 | 77 | 76 | 1 |
| 4 | 68 | 68 | 0 |
| 5 | 60 | 60 | 0 |
The largest difference is 1 percentage point. The differences are small and change sign, so there is no drift in one direction.
Conclusion: the model B = 100 − 8t fits the readings for 0 ≤ t ≤ 5. After 12.5 hours it would predict a negative battery level, so it should not be used there.
The mistake that costs marks
The common slip is to say “the model passes through (0, 100) and (5, 60), so it fits.” Those two points built the rule, so they must match.
The fair test uses the points that did not build it. Here, that means hours 1 to 4, where the differences are 0, −1, 1 and 0.
| Wrong | Right | |
|---|---|---|
| Points tested | The two used to build the rule | The remaining points |
| Evidence | “It passes through them” | A difference table |
| Conclusion | “Fits perfectly” | “Fits within 1 point for hours 0 to 5” |
A method you can reuse
- Write the model with the variables named.
- Calculate the predicted value at each supplied input.
- Find observed − predicted for each row.
- Look for size, sign and trend in the differences.
- Write a conclusion with the range, the largest difference and any limit.
If you want to see differences on a plot, the graph evidence and fair-comparison lab lets you compare a line with a set of points.
Check yourself
A model is y = 2x + 1. The data are (1, 3.1), (2, 5.0), (3, 7.4) and (4, 8.9). Does the model fit?
Answer
Predicted values: 3, 5, 7, 9.
Differences (observed − predicted): 0.1, 0, 0.4 and −0.1.
The largest difference is 0.4 at x = 3, and the differences are small and change sign. The model fits for x from 1 to 4. Note that the question gives no evidence beyond x = 4.
What to study next
Once you can judge a fit, learn to say why a model breaks down in explaining limitations and refining a model. Then try the mathematical modelling practice set. Record any slips in the mistake log and paper-error review.
If you want a teacher to check your written conclusions, see online one-to-one Mathematics tuition.