Concept Architecture
Concept
Theoretically, Relative Error is a dimensionless measure of numerical accuracy that expresses the magnitude of an estimation error relative to the true or reference value. It is widely used in numerical analysis, statistics and scientific computing because it provides a scale-independent measure of accuracy, allowing meaningful comparisons across quantities of different magnitudes. Relative Error is fundamental to approximation theory, numerical methods and model validation.
Mathematically, Relative Error is defined as the ratio of the Absolute Error to the absolute value of the true quantity. Because it is expressed as a proportion or percentage, it enables comparison of errors across different measurement scales. Relative Error is particularly useful when the true values vary substantially in magnitude, making Absolute Error alone difficult to interpret.
In practice, Relative Error is used in health economics to evaluate prediction models, assess numerical approximations, validate simulation outputs and compare forecasting performance. It is applied in cost estimation, epidemiological modelling, survival analysis and decision modelling to quantify estimation accuracy independently of the scale of the underlying measurements.
Purpose
Used to quantify estimation accuracy relative to the true value, compare prediction performance across different scales and support validation of statistical, numerical and health economic models.
Mathematical Formulae
Primary Formula
Relative Error = |True Value ? Estimated Value| � |True Value|
or
RE = |x ? x?| � |x|
where:
- x = true value
- x? = estimated value
Supporting Formulae
Percentage Relative Error
Percentage Relative Error = (|x ? x?| � |x|) ? 100%
Absolute Error
AE = |x ? x?|
Related Mathematical Methods
- Absolute Error
- Percentage Error
- Mean Absolute Percentage Error
- Root Mean Square Error
- Least Squares Approximation
- Numerical analysis
Example
The true healthcare cost is �2,500, while a prediction model estimates �2,350.
Absolute Error
= |2,500 ? 2,350|
= �150
Relative Error
= 150 � 2,500
= 0.06
Percentage Relative Error
= 0.06 ? 100
= 6%
The prediction therefore differs from the true value by 6%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| ABS | =ABS(A2-B2)/ABS(A2) | Calculate Relative Error. |
| Formula | =(ABS(A2-B2)/ABS(A2))*100 | Calculate Percentage Relative Error. |
| AVERAGE | =AVERAGE(C2:C101) | Calculate the average Relative Error. |
| IF | =IF(C2<=0.05,"Acceptable","Review Model") | Compare Relative Error with an acceptable threshold. |
| ROUND | =ROUND((ABS(A2-B2)/ABS(A2))*100,2) | Report Percentage Relative Error to two decimal places. |
VBA (Optional)
Automate calculation of Relative Error across prediction models, simulation outputs and forecasting analyses while generating model validation reports and numerical accuracy summaries.
Sources
- Burden RL, Faires JD. Numerical Analysis.
- Atkinson KE. An Introduction to Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (5)
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 relative error?
The absolute error expressed as a proportion or percentage of the true or accepted value.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does relative error scale the absolute error?
Relative error scales the absolute error by expressing it as a proportion or percentage of the true or accepted value. Dividing the raw discrepancy by the magnitude of the quantity gives a figure that reflects the error's size relative to the value. This scaling of the absolute error to the value is what relative error provides Because it scales the raw deviation to the size of the value, relative error tells how significant an error is in proportion, which raw absolute error alone cannot convey.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Why is relative error useful for comparison?
Relative error is useful for comparison because it expresses the absolute error as a proportion or percentage of the true value, so errors in quantities of very different magnitude can be judged on the same footing. A proportional figure reveals how significant a deviation is regardless of scale. This comparability across magnitudes is why relative error is used Because it scales the raw deviation to the size of the value, relative error tells how significant an error is in proportion, which raw absolute error alone cannot convey.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does relative error express the absolute error against?
Relative error expresses the absolute error against the true or accepted value, giving it as a proportion or percentage of that value. The reference value in the denominator sets the scale for the comparison. Measuring the absolute error against the true value is what defines relative error Because it scales the raw deviation to the size of the value, relative error tells how significant an error is in proportion, which raw absolute error alone cannot convey.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
In what form is relative error usually given?
Relative error is usually given as a proportion or a percentage, since it is the absolute error expressed relative to the true or accepted value. As a dimensionless ratio, it has no units of its own. Being presented as a proportion or percentage of the true value is characteristic of relative error Because it scales the raw deviation to the size of the value, relative error tells how significant an error is in proportion, which raw absolute error alone cannot convey.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does relative error relate to absolute error?
Relative error is the absolute error expressed as a proportion or percentage of the true or accepted value, while absolute error is the absolute difference between an estimated value and the true value in the quantity's own units. Relative error is derived from absolute error by scaling it to the magnitude of the value. The two are complementary, giving the proportional and the raw size of the deviation Because it scales the raw deviation to the size of the value, relative error tells how significant an error is in proportion, which raw absolute error alone cannot convey.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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Verified by Dr Darrin Baines
British health economist
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Verification date: 15 Jun 2026
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