Error Propagation Calculator

Calculate propagated values, absolute uncertainty, and percentage uncertainty based on measurements, errors, operations, and correlation. Error Propagation Calculator helps students, engineers, scientists, and analysts assess reliability in experimental and calculated results.

Highlights

  • Calculate propagated values for addition, subtraction, multiplication, and division.
  • Combine measurement uncertainties from two measured quantities.
  • Account for correlation between measurements from -1 to +1.
  • View absolute and percentage uncertainty results instantly.
  • Useful for students, engineers, scientists, and analysts.

Table of Contents

  1. How Error Propagation Calculator Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How Error Propagation Calculator Works?

The Error Propagation Calculator applies first-order uncertainty propagation to two measured quantities, A and B. Enter each measurement, its absolute uncertainty, the mathematical operation, and a correlation coefficient r between -1 and 1. The calculator returns the propagated value, combined absolute uncertainty, and percentage uncertainty.

The correlation coefficient accounts for shared or dependent sources of error. If the measurements are independent, use r=0. Positive correlation generally increases the combined uncertainty for addition and multiplication, while negative correlation can reduce it.


OperationPropagated valueAbsolute uncertainty
Additionz=A+Bσz=σA2+σB2+2rσAσB
Subtractionz=A−Bσz=σA2+σB2−2rσAσB
Multiplicationz=ABσz=B2σA2+A2σB2+2rABσAσB
Divisionz=A/Bσz=σA2B2+A2σB2B4−2rAσAσBB3

Here, σA and σB are the entered absolute uncertainties. The percentage uncertainty is calculated from the magnitude of the result:

Percentage uncertainty=σz|z|×100%

The measurement and uncertainty fields accept the calculator's supported labels for length, mass, time, volume, pressure, and electrical or mechanical quantities, including millimeter, centimeter, meter, kilometer, inch, foot, yard, mile, milligram, gram, kilogram, ounce, pound, millisecond, second, minute, hour, milliliter, liter, gallon, pascal, kilopascal, bar, psi, volt, ampere, watt, joule, and newton. Values combined by addition or subtraction should represent compatible quantities, and division requires a nonzero Measurement B.

Practical Use Cases

Students and laboratory learners can combine measured lengths, masses, times, or electrical readings with their stated uncertainties to report experimental results with both absolute and percentage uncertainty.

Engineers can estimate how tolerance in two input measurements affects a calculated force, power, pressure, dimension, or performance value, while using the correlation coefficient for measurements influenced by a common sensor or process.

Scientists and researchers can propagate uncertainty through addition, subtraction, multiplication, or division when preparing results for experiments, calibration studies, and technical publications.

Quality-control analysts can compare the relative reliability of calculated results by examining percentage uncertainty and identifying whether correlated measurements materially increase or decrease the combined error.

Data and measurement analysts can evaluate the effect of uncertainty in paired observations, such as a numerator and denominator in a ratio, without treating dependent measurements as statistically independent.

Accuracy and Limitations

The calculator uses standard first-order, or linearized, error propagation with absolute input uncertainties and a correlation coefficient from -1 to 1. It is most reliable when uncertainties are relatively small compared with the measurements and the operation is not close to a singularity. Division requires Measurement B to be nonzero, and percentage uncertainty is undefined when the propagated result is zero. The calculation assumes the entered uncertainties appropriately describe the measurements; it does not estimate uncertainty distributions, detect outliers, include systematic effects not entered by the user, or validate whether the selected quantities are dimensionally compatible.

Quick Examples

Example #1: Adding Length Measurements

Estimate the combined length and uncertainty when adding two measured segments with a positive correlation.

Example #2: Subtracting Mass Measurements

Calculate the remaining mass and propagated uncertainty after subtracting a measured sample from a container.

Example #3: Dividing Voltage by Current

Determine electrical resistance and its uncertainty by dividing voltage by current measurements.

Frequently Asked Questions

Measurement uncertainty describes the range of doubt associated with a measured value. It can arise from instrument limits, environmental conditions, calibration, and repeatability.

Absolute uncertainty is expressed in the same units as the measured quantity. Percentage uncertainty expresses the same uncertainty relative to the measured or calculated value.

Correlation indicates whether uncertainties in two measurements tend to vary together. Positive correlation means they generally increase or decrease together, while negative correlation indicates opposing changes.

Measurements should use compatible units before they are combined. For multiplication or division, the resulting units follow the corresponding algebraic combination of the input units.

Report the result together with its uncertainty, using consistent decimal places and appropriate significant figures. Avoid presenting more precision than the uncertainty supports.

Cite This Calculator