To compare a prediction with an observation, work out the residual (observed minus predicted), express it as a percentage of the observed value, then decide in words whether the model is good enough for its purpose. The number alone is not the answer, because the last sentence must say what it means.
This lesson sits after interpreting parameters in International Mathematics modelling, and it prepares you to judge predictions beyond the data.
How do you compare step by step?
- Predict by substituting the input into the model.
- Find the residual: observed − predicted.
- Find the percentage error: |residual| ÷ observed × 100.
- Describe the direction: a positive residual means the model underestimates, and a negative one means it overestimates.
- Judge: say whether the gap is acceptable for the use, and give one possible reason for it.
Worked example
A taxi fare model is F = 3.50 + 1.20d, where F is in RM and d is the distance in km. For a trip of 8.5 km, the real fare was RM15.20. Compare the prediction with the observation.
Step 1, predict: F = 3.50 + 1.20 × 8.5 = 3.50 + 10.20 = RM13.70.
Step 2, residual: 15.20 − 13.70 = +RM1.50.
Step 3, percentage error: 1.50 ÷ 15.20 × 100 = 9.87…, so about 9.9%.
Step 4, direction: the residual is positive, so the model underestimates this fare.
Step 5, judge: a gap of about 10% is acceptable for a rough budget but not for a precise bill. A likely reason is that the model ignores waiting time in traffic.
Check: 1.20 × 8.5 = 10.2 because 1.2 × 8 = 9.6 and 1.2 × 0.5 = 0.6, and 9.6 + 0.6 = 10.2. Also 13.70 + 1.50 = 15.20.
The mistake to watch for
A common slip is to reverse the subtraction and then read the sign the wrong way.
Mistaken working: 13.70 − 15.20 = −1.50, so the model overestimates the fare.
The prediction is lower than the real fare, so it underestimates.
The correction is to fix one rule: residual = observed − predicted, and always say the direction in words. Ask yourself which number is larger. If the real value is larger than the prediction, the model is too low.
Check yourself
Try these on paper, then open each answer.
1. A model predicts 42 for a value that was observed as 45. Find the residual and the percentage error.
Show answer
Residual = 45 − 42 = +3. Percentage error = 3 ÷ 45 × 100 = 6.66…, so about 6.7%. The model underestimates.
2. A model predicts that a container holds 250 mL, but the measured amount is 240 mL. Find the residual, the percentage error and the direction.
Show answer
Residual = 240 − 250 = −10 mL. Percentage error = 10 ÷ 240 × 100 = 4.16…, so about 4.2%. The model overestimates.
3. A model gives residuals of +0.2, +0.3 and +0.4 as x increases. What does the pattern suggest?
Show answer
The residuals are all positive and growing, so the model underestimates more and more as x increases. The gradient of the model is probably too small, or the real relationship is curved. A good model has gaps on both sides of zero with no pattern.
Where this leads next
Predictions inside the data range are one thing, and predictions far outside it need more care: explain why extrapolation may fail. If percentages still feel slippery, the percentage-base explorer shows how the base changes the answer. Mixed questions are in the modelling practice set.
Some students get the arithmetic right and then write a verdict that does not follow from it. Our teachers work on that link in online one-to-one Mathematics tuition.