Error Propagation & Measurement Uncertainty Calculator — Absolute and Relative Uncertainty
Calculate result uncertainty using value, input uncertainty, equation type, exponent, and confidence factor.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Equation type
Variables
Exponent
Analysis settings
Results
Engineering Pro Save, document and continue this calculation
Turn this result into a reusable engineering record with saving, PDF export and reporting workflows.
Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
result, absolute uncertainty, and relative uncertainty formula:
Result y:
- For a · b: y = a · b
- For a · b · c: y = a · b · c
- For a / b: y = a / b
- For aⁿ: y = aⁿ
- For (a · b) / c: y = (a · b) / c
Relative input uncertainties:
- ra = Δa / a
- rb = Δb / b
- rc = Δc / c
Combined relative uncertainty:
- For a · b or a · b · c: rtotal = √(ra² + rb² + rc²)
- For a / b: rtotal = √(ra² + rb²)
- For aⁿ: rtotal = |n| · ra
- For (a · b) / c: rtotal = √(ra² + rb² + rc²)
absolute uncertainty formula:
Δy = |y| · rtotal · k
relative uncertainty formula:
ε = rtotal · 100
where:
- y — result (-)
- Δy — absolute uncertainty (-)
- ε — relative uncertainty (%)
- a — value (-)
- Δa — uncertainty of a (-)
- b — value (-)
- Δb — uncertainty of b (-)
- c — value (-)
- Δc — uncertainty of c (-)
- n — exponent (-)
- k — confidence factor (-)
- rtotal — combined relative uncertainty (-)
When to use this calculator
result, absolute uncertainty, and relative uncertainty formula:
Result y:
- For a · b: y = a · b
- For a · b · c: y = a · b · c
- For a / b: y = a / b
- For aⁿ: y = aⁿ
- For (a · b) / c: y = (a · b) / c
Relative input uncertainties:
- ra = Δa / a
- rb = Δb / b
- rc = Δc / c
Combined relative uncertainty:
- For a · b or a · b · c: rtotal = √(ra² + rb² + rc²)
- For a / b: rtotal = √(ra² + rb²)
- For aⁿ: rtotal = |n| · ra
- For (a · b) / c: rtotal = √(ra² + rb² + rc²)
absolute uncertainty formula:
Δy = |y| · rtotal · k
relative uncertainty formula:
ε = rtotal · 100
where:
- y — result (-)
- Δy — absolute uncertainty (-)
- ε — relative uncertainty (%)
- a — value (-)
- Δa — uncertainty of a (-)
- b — value (-)
- Δb — uncertainty of b (-)
- c — value (-)
- Δc — uncertainty of c (-)
- n — exponent (-)
- k — confidence factor (-)
- rtotal — combined relative uncertainty (-)
How to interpret the result
The result is defined as the calculated output of the selected equation type: multiplication, division, power, or mixed multiplication and division.
Absolute uncertainty is defined as the result range width controlled by the combined relative uncertainty and the confidence factor. Increasing the confidence factor increases absolute uncertainty.
Relative uncertainty depends on the relative uncertainties of the active input values. Increasing any active input uncertainty increases relative uncertainty.
- Safe — used when every active relative input uncertainty is below 10% and relative uncertainty is not greater than 5%.
- Warning — used when at least one active relative input uncertainty is 10% or higher, or relative uncertainty is greater than 5%.
- Unsafe — used when relative uncertainty is greater than 30%.
- Invalid — used when the selected equation, input values, input uncertainties, exponent, confidence factor, or computed uncertainty cannot satisfy the validation limits.
The result is used to evaluate whether the computed value has a narrow or wide uncertainty interval and which measured input contributes the dominant uncertainty share.
Calculation example
Example:
A user calculates a result from two measured values using the product equation a · b and wants to estimate the propagated result uncertainty.
- Equation type: a · b
- a = 10
- Δa = 0.2
- b = 5
- Δb = 0.1
- k = 2
Result: y = 50, Δy ≈ 2.83, ε ≈ 2.83%.
Assumptions and limitations
- The selected equation type is limited to a · b, a / b, aⁿ, or (a · b) / c.
- Every active input value used in multiplication, division, or mixed equations must be non-zero.
- For the power equation, the base value must be greater than 0.
- Every active input uncertainty must be greater than or equal to 0.
- Every active relative input uncertainty must be lower than 1.
- The confidence factor must be greater than 0 and not greater than 10.
- The exponent must be finite and its absolute value must not be greater than 20.
- The propagated uncertainty model uses relative uncertainty combination based on squared relative uncertainty terms.