Concept Architecture
Concept
Theoretically, Rounding Error is the numerical error introduced when an exact value is replaced by a finite-precision approximation. It is an inherent consequence of finite-precision arithmetic used by digital computers and numerical algorithms. Rounding Error is a fundamental concept in numerical analysis because virtually all computational methods involve representing real numbers with a limited number of digits, leading to unavoidable approximation.
Mathematically, Rounding Error is defined as the difference between the exact mathematical value and its rounded representation. Although each individual rounding error is typically small, repeated arithmetic operations can cause errors to accumulate or propagate throughout numerical computations. The magnitude and behaviour of Rounding Error depend on the precision of the numerical representation, the rounding rule employed and the stability of the underlying algorithm.
In practice, Rounding Error is important in health economics because economic models, simulation studies, optimisation algorithms and statistical analyses rely on finite-precision computation. Understanding and controlling rounding error improves numerical stability, enhances reproducibility and ensures that calculated costs, utilities and incremental outcomes remain sufficiently accurate for healthcare decision-making.
Purpose
Used to quantify numerical approximation introduced by finite-precision computation, assess computational accuracy, evaluate numerical stability and support reliable implementation of health economic models.
Mathematical Formulae
Primary Formula
Rounding Error = Rounded Value ? Exact Value
or, in magnitude,
|Rounding Error| = |Rounded Value ? Exact Value|
Supporting Formulae
Relative Rounding Error
Relative Rounding Error = |Rounded Value ? Exact Value| � |Exact Value|
Machine Epsilon
� = smallest positive number such that
1 + � > 1
Related Mathematical Methods
- Absolute Error
- Relative Error
- Floating-point arithmetic
- Machine precision
- Numerical stability
- Error propagation
Example
Suppose the exact value is:
� = 3.141592654
Rounded to four decimal places:
3.1416
Rounding Error
= 3.1416 ? 3.141592654
= 0.000007346
The numerical approximation therefore introduces a Rounding Error of approximately 7.35 ? 10??, which may accumulate if repeated throughout large-scale computations.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| ROUND | =ROUND(A2,4) | Round calculated values to a specified precision. |
| ABS | =ABS(ROUND(A2,4)-A2) | Calculate the magnitude of the Rounding Error. |
| Formula | =ABS((ROUND(A2,4)-A2)/A2) | Calculate Relative Rounding Error. |
| SUM | =SUM(B2:B101) | Assess accumulated rounding error across calculations. |
| IF | =IF(ABS(ROUND(A2,4)-A2)<0.0001,"Acceptable","Review Precision") | Evaluate whether rounding precision is acceptable. |
VBA (Optional)
Automate identification of rounding errors within health economic models, evaluate cumulative numerical precision and generate reports highlighting calculations susceptible to excessive rounding error.
Sources
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Wilkinson JH. Rounding Errors in Algebraic Processes.
- Burden RL, Faires JD. Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
- Goldberg D. What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (7)
Library
Publications
1
Accuracy and Stability of Numerical Algorithms — Nicholas J. Higham, 2nd Edition ed., 2002 (Society for Industrial and Applied Mathematics (SIAM))
The standard reference on floating-point error, conditioning, stability and the numerical reliability of matrix, polynomial and linear-algebra algorithms.
BookView source →
Frequently Asked Questions (6)
What is rounding error?
The difference introduced when a numerical value is rounded to fit a limited number of digits.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What causes rounding error?
Rounding error is caused by rounding a numerical value to fit a limited number of digits, so a value needing more digits than are available is replaced by a nearby representable one. The difference between the true value and this rounded version is the rounding error. This adjustment to fit a limited digit budget is what produces rounding error Because each individual rounding is tiny yet unavoidable in finite-precision arithmetic, the concern is less any single one than how many of them may accumulate over a long calculation.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What is rounding error the difference between?
Rounding error is the difference between a numerical value and the rounded version used to fit a limited number of digits. When the exact value cannot be held, it is replaced by the closest representable value, and the gap between them is the rounding error. This difference introduced by rounding is what the term names Because each individual rounding is tiny yet unavoidable in finite-precision arithmetic, the concern is less any single one than how many of them may accumulate over a long calculation.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Why does rounding error arise in computation?
Rounding error arises in computation because numbers are stored with a limited number of digits, so values that need more must be rounded to fit, introducing a small difference. Finite-precision arithmetic makes such rounding unavoidable for many values. This limitation on available digits is why rounding error appears Because each individual rounding is tiny yet unavoidable in finite-precision arithmetic, the concern is less any single one than how many of them may accumulate over a long calculation.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Is rounding error usually large?
Rounding error is usually small for a single rounding, since it is only the difference introduced when a value is rounded to fit a limited number of digits, and that limit is typically fine. However, many such errors can accumulate through a long calculation. The individual rounding error being small is characteristic, though its effects can build up Because each individual rounding is tiny yet unavoidable in finite-precision arithmetic, the concern is less any single one than how many of them may accumulate over a long calculation.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does rounding error relate to truncation error?
Rounding error is the difference introduced when a value is rounded to fit a limited number of digits, while truncation error is the error from approximating a mathematical process using a finite number of terms or steps. Rounding error comes from representing numbers, whereas truncation error comes from cutting a process short. The two are connected as distinct sources of numerical error in a computation Because each individual rounding is tiny yet unavoidable in finite-precision arithmetic, the concern is less any single one than how many of them may accumulate over a long calculation.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Jun 2026
Content version: 1.0.0
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- Persistent URI
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- Term code
- CS-NA-EA-003
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