If you measure quantities with uncertainty, the Error Propagation Calculator computes the resulting uncertainty of a derived value. It uses standard rules for addition, subtraction, multiplication, division, and powers, so you can report results with confidence.
This guide explains the formulas, shows how each input affects the final uncertainty, and helps you avoid common mistakes like mixing units or forgetting correlated errors.
What Is Error Propagation?
Error propagation is the process of converting uncertainties in measured inputs into an uncertainty for a calculated output. If you have uncertainties on the inputs, you can’t just add them to the final number—you must combine them using the correct rule for the math operation.
Most lab and engineering calculations assume the uncertainties are small and approximately independent. Under that assumption, the combined standard uncertainty is computed using derivatives (the “sensitivity” of the output to each input).
Core Formulas (Independent Uncertainties)
Let your measured inputs be x, y, etc., with uncertainties u(x), u(y). The calculator below uses these common rules for independent errors.
1) Addition and Subtraction
For z = x + y or z = x − y, the uncertainties add in quadrature:
u(z) = √(u(x)² + u(y)²)
2) Multiplication and Division
For z = x·y or z = x/y, use relative (fractional) uncertainties:
u(z) = |z| · √((u(x)/x)² + (u(y)/y)²)
This rule works because products and quotients scale with the percentage uncertainty of each factor.
3) Powers
For z = x^n, where n is a known exponent, the uncertainty is:
u(z) = |z| · |n| · u(x)/|x|
Here the exponent is treated as exact (no uncertainty in n). If your exponent is uncertain, you need a more general method.
Units and Unit Conversions
Uncertainty must be in the same unit as the quantity being reported. The calculator supports unit selection for each input so you can enter values in common units (like meters vs. centimeters, or seconds vs. milliseconds) and still get a consistent output.
Internally, the calculator converts inputs and uncertainties into a consistent base unit, applies the propagation formulas, then reports the result in the unit you choose.
How to Use the Error Propagation Calculator
Choose the operation type, enter each measured value and its uncertainty, then read the output value and propagated uncertainty. The calculator also returns a relative uncertainty percentage, which is often the most useful way to compare results.
- Operation: Pick addition/subtraction, multiplication/division, or power.
- Inputs: Enter each value and its uncertainty (absolute uncertainty).
- Units: Select units for values and uncertainties so they match.
- Output: The calculator computes z and u(z).
Practical Examples (Real-World Use Cases)
Example 1: Combining Length Measurements
Suppose you measure two lengths and want the total. Let x = 1.20 m with u(x)=0.03 m, and y = 0.80 m with u(y)=0.02 m. For z = x + y:
z = 2.00 m
u(z) = √(0.03² + 0.02²) = 0.036 m
You would report: (2.00 ± 0.04) m (rounded).
Example 2: Uncertainty in a Derived Area
If you compute area from two perpendicular lengths, you multiply them. Let x = 0.50 m with u(x)=0.01 m, and y = 0.30 m with u(y)=0.005 m. For z = x·y:
z = 0.15 m²
u(z) = |z| · √((0.01/0.50)² + (0.005/0.30)²)
Compute the square-root term to get the final uncertainty, then report (0.15 ± u) m².
Common Mistakes to Avoid
- Forgetting units: If you enter centimeters as meters, your uncertainty will be wrong.
- Mixing absolute and relative uncertainty: The calculator expects absolute uncertainty (same unit as the value).
- Using the wrong rule: Addition/subtraction uses absolute uncertainties in quadrature; multiplication/division uses relative uncertainties.
- Zero values in relative formulas: For products/quotients, the relative uncertainty needs x ≠ 0 and y ≠ 0. The calculator flags invalid inputs.
Frequently Asked Questions
What does “independent uncertainties” mean in error propagation?
Independent uncertainties assume the errors in different measurements do not move together. In that case, the combined uncertainty uses quadrature (square root of summed squares). If errors are correlated, you need additional covariance terms, and the quadrature-only results can be too small or too large.
Should I use absolute or percentage uncertainty in the calculator?
This calculator uses absolute uncertainty (same unit as the measured value). If you only know a percentage uncertainty, convert it first: absolute uncertainty = value × percentage/100. Then enter both the value and its absolute uncertainty in the selected units.
How do I round the final uncertainty and result?
Report uncertainty with one or two significant figures, then round the final value to the same decimal place as the uncertainty. For example, if u(z) = 0.036, you might report 0.04. The value should be rounded consistently with that.
Why does multiplication/division use relative uncertainty?
For z = x·y or z = x/y, the output scales with the inputs. A 2% error in x produces about a 2% error in the result, regardless of the units. That is why the formulas use u(x)/x and u(y)/y, then multiply by |z|.
Can I propagate uncertainty for more complex formulas than these?
Yes, but you must use a general derivative-based approach: u(z) = √(Σ (∂z/∂xi)² u(xi)²). The calculator here covers common operations (sum, product, quotient, and power). For other expressions, compute partial derivatives or use a more general tool.
Next Step: Use the Calculator for Your Measurement Workflow
When you collect data, uncertainties are not optional—they are part of the result. Use the Error Propagation Calculator to compute uncertainty quickly, keep units consistent, and produce results you can defend in a report, paper, or design review.