Singapore School Mathematics Operating Manual · Chapter 36
A multi-stage calculation can be locally accurate and still end with a poor final answer if small errors accumulate.
Measurements may be rounded. Intermediate values may be approximated. Calculator displays may be truncated. A graph may be read only to limited resolution. Several estimated quantities may be combined. An early approximation may be reused many times downstream.
This chapter develops the idea of an error budget: the total amount of uncertainty or approximation the final answer can tolerate, and how that allowance is distributed across the stages that create the answer.
The aim is not advanced uncertainty calculus. It is a practical operating discipline: know where error enters, know how it can grow, and keep enough precision so the final decision remains trustworthy.
1. Error can enter before calculation begins
If a length is measured as 10.0 cm to the nearest 0.1 cm, the input is already uncertain.
Exact arithmetic performed on 10.0 does not make the physical length exact.
2. Error can also be introduced by the solver
Replacing √2 with 1.41 creates approximation error.
Rounding 2/3 to 0.67 creates approximation error.
These errors are optional in the sense that exact form could have been preserved longer.
3. Input uncertainty and computational rounding are different sources
A measured value may be uncertain because reality was observed imperfectly.
A calculated decimal may be rounded because the solver chose finite representation.
Both affect the final answer, but they have different provenance.
4. Absolute error measures raw deviation
If true value is T and approximation is A, absolute error is |A−T|.
If π is approximated by 3.14, the absolute error is about 0.00159.
5. Relative error measures error against scale
Relative error is |A−T|/|T| when T≠0.
A 0.1-unit error matters much more for a value near 1 than for a value near 10,000.
6. Percentage error is relative error expressed in percent
If a measurement 50 is off by 1, relative error is 1/50=0.02, or 2%.
This helps compare error significance across different scales.
7. Addition combines absolute uncertainties naturally
Suppose A is known within ±0.2 and B within ±0.3.
A conservative worst-case bound for A+B is ±0.5.
The errors can reinforce each other.
8. Subtraction can be especially delicate
If A=100.1±0.2 and B=100.0±0.2, the difference of central values is 0.1.
But worst-case uncertainty in the difference is ±0.4.
The sign of the true difference is not guaranteed.
Subtracting nearly equal uncertain numbers can destroy relative precision.
9. Multiplication propagates relative uncertainty
For small independent percentage uncertainties, relative errors in a product roughly add in worst-case first-order reasoning.
If length is uncertain by about 1% and width by about 2%, area may carry roughly 3% relative uncertainty as a simple conservative estimate.
This is an approximation rule, not a replacement for exact bounds where those are available.
10. Powers amplify relative uncertainty
If A=πr², a small percentage error in r produces roughly twice that percentage error in area.
If V∝r³, it produces roughly three times the percentage effect in volume.
This links error budgets to scale sensitivity.
11. Denominators near zero can consume the entire error budget
For y=1/x, uncertainty in x becomes dangerous near x=0.
A small denominator perturbation can produce a very large output change.
The Conditioning chapter explains this structural sensitivity.
12. Reusing one uncertain value creates correlated downstream error
If the same measured radius is used to calculate circumference and area, the two output errors are not independent.
Both inherit the same radius error.
A common source can make several results move together.
13. An error budget should focus on high-leverage quantities
If one intermediate value feeds five later calculations, preserve more precision and check it carefully.
A low-leverage value used once near the end may deserve less attention.
Result lineage helps allocate checking effort.
14. Early rounding can consume budget unnecessarily
Suppose x=√3 is used in several later products.
Keeping √3 exact preserves information.
Rounding to 1.73 at the first step injects error into every descendant calculation.
15. Guard digits protect intermediate work
If a final answer is required to 3 decimal places, intermediate calculations should generally retain more than 3 decimals.
The exact number of guard digits depends on the calculation’s sensitivity.
The principle is to round once, late, rather than repeatedly.
16. Repeated rounding can drift
A sequence of individually tiny rounding decisions can accumulate into a noticeable difference from a single final rounding.
This is why calculators should carry full internal precision where practical.
17. Rounding direction matters in one-sided decisions
If a minimum capacity is required, rounding a required resource count downward can violate feasibility.
If a maximum permitted quantity is involved, upward rounding may violate the bound.
Error budgeting must respect the decision direction.
18. Capacity problems have discrete tolerance
If a calculation gives 2.01 buses, the practical answer is still 3 when every passenger must fit.
The discrete decision may be insensitive to small numerical changes unless those changes cross an integer threshold.
19. Threshold distance is a useful robustness margin
If a result is 8.0±0.1 and threshold is 10, the uncertainty is far from the boundary.
If result is 9.98±0.1, the interval crosses the threshold.
The second decision has exhausted its robustness margin.
20. Bounds provide guaranteed error budgets
If x lies in [a,b], propagate the full interval through a monotone formula where possible.
The resulting output interval gives a guaranteed uncertainty range rather than an approximate percentage rule.
21. Interval width is an error certificate
If a root is bracketed between 1.4142 and 1.4143, the interval width is 0.0001.
The midpoint has maximum absolute error 0.00005.
This is a direct error budget.
22. Graph-reading uncertainty should match graph resolution
If grid spacing is 1 unit, reporting an intersection to six decimal places is unjustified.
Precision should reflect the observation scale.
23. Grouped-data estimates carry structural approximation
Using class midpoints to estimate a mean assumes observations can be represented by those midpoints.
The final statistic inherits that approximation.
24. Sampling uncertainty is separate from measurement uncertainty
A sample mean can be measured perfectly for the sample but still differ from the population mean because only some members were sampled.
Error budgets can have multiple layers.
25. Model uncertainty can dominate numerical error
A travel-time result may be calculated to the nearest second while the constant-speed assumption is unrealistic to several minutes.
Improving arithmetic precision does not repair a much larger modelling uncertainty.
26. Spend precision where it matters
Do not allocate twelve-digit precision to a minor arithmetic step if the model input is known only roughly.
Conversely, do not round a sensitive intermediate value aggressively when the final threshold is tight.
An error budget is a resource-allocation problem.
27. Backward tolerance allocation starts from the final requirement
If final absolute error must be below 0.1, ask how much error each contributing stage can safely introduce.
A simple conservative plan may allocate smaller tolerances to high-sensitivity stages and larger tolerances to stable ones.
28. Not every stage needs an equal share
If one operation multiplies error strongly and another barely affects it, equal tolerance allocation is inefficient.
Allocate by influence, not symmetry.
29. Independent error sources can partially cancel—but do not rely on cancellation without justification
One approximation may be high and another low.
They can cancel by chance.
A guaranteed error budget should normally use conservative bounds unless a probabilistic error model is explicitly justified.
30. Worst-case and typical error are different ideas
Worst-case analysis asks what could happen within allowed bounds.
Statistical error models ask what is likely under a probability distribution.
School problems should not mix these without stating the model.
31. Exact symbolic simplification can eliminate error
√3×√3=3 exactly.
If both values were rounded first, the result would only be approximate.
Exact structure can sometimes remove the need for an error budget altogether.
32. Verification should test whether the final tolerance was actually met
For a numerical root, inspect the bracket or residual.
For a measurement-based result, inspect propagated bounds.
For a threshold decision, inspect whether the entire plausible interval lies on one side.
33. Error budgets belong to communication too
Use “approximately”, interval notation, significant figures or tolerance statements where appropriate.
A result should not look more certain than its provenance supports.
34. A practical error-budget audit
Ask:
Where does uncertainty enter? Which sources are measured, rounded, sampled or modelled? Which quantities are reused downstream? Which operations amplify error? How close is the final result to a threshold? What precision is required? Can exact form or bounds preserve more information? Have I rounded only as much as the task permits?
35. Independent practice
1. A=5.0±0.1 and B=3.0±0.2. Give a conservative bound on A+B.
2. Give a conservative uncertainty for A−B using the same data.
3. Why is subtracting nearly equal uncertain numbers dangerous?
4. A radius has about 2% relative uncertainty. Roughly what relative uncertainty may area πr² inherit?
5. Why should guard digits be retained?
6. A result is 9.95±0.1 with threshold 10. Is the decision guaranteed?
7. A result is 8.0±0.1 with threshold 10. What is guaranteed?
8. Why do twelve calculator digits not repair uncertain measurements?
9. What kind of intermediate value deserves extra precision?
10. Why is exact √3 often better than 1.73 in intermediate work?
36. Worked answers
1. Central sum 8.0 with conservative uncertainty ±0.3.
2. Central difference 2.0 with conservative uncertainty ±0.3.
3. The result can be small relative to the uncertainty in the inputs, making relative error very large.
4. Roughly 4% for small errors, because area depends on r².
5. To prevent repeated intermediate rounding from consuming the final tolerance.
6. No. The interval [9.85,10.05] crosses the threshold.
7. The result is guaranteed below 10 because the upper bound is 8.1.
8. Computation precision cannot create information absent from the original data.
9. A high-leverage or highly sensitive quantity used by many downstream calculations.
10. It preserves exact information and may simplify exactly later.
37. Batch 09 as one information-control layer
Observability and Identifiability asks whether the available information determines the hidden state.
Compression and Sufficient Summaries asks what information has been retained or discarded.
Mathematical Provenance and Result Lineage asks where each conclusion came from.
This chapter asks how much uncertainty that lineage can carry before the final result becomes unreliable.
Return to the BTT Mathematics Hub for Batch 09.