Error Propagation
Free online Error Propagation Calculator — compute uncertainty propagation for addition, subtraction, multiplication, division, and power operations. Enter values with uncertainties for instant results, charts, and breakdowns.
About This Calculator
The Error Propagation Calculator helps scientists, engineers, students, and researchers compute how measurement uncertainties propagate through mathematical operations. When you measure physical quantities, each measurement carries some uncertainty, and when you combine these measurements through arithmetic, the uncertainties combine in mathematically predictable ways.
This calculator supports five operations: addition (Z = X + Y), subtraction (Z = X − Y), multiplication (Z = X × Y), division (Z = X ÷ Y), and power functions (Z = Xⁿ). For addition and subtraction, absolute uncertainties add in quadrature: ΔZ = √(ΔX² + ΔY²). For multiplication and division, relative uncertainties add in quadrature: ΔZ/|Z| = √((ΔX/X)² + (ΔY/Y)²). For power functions, the relative uncertainty scales with the exponent: ΔZ/|Z| = |n| × ΔX/|X|.
The underlying principle is the general error propagation formula derived from calculus. For a function f that depends on several variables with uncertainties, the total uncertainty is Δf = √(Σ(∂f/∂xᵢ)²Δxᵢ²), where ∂f/∂xᵢ are partial derivatives. This formula assumes the input uncertainties are independent and random, which is the most common case in experimental science and engineering.
This calculator is ideal for laboratory work, experimental physics and chemistry, engineering tolerance analysis, quality control, and any field where measurement uncertainties must be accounted for in derived results. Results are presented with the calculated value Z, the propagated uncertainty ΔZ, and the relative uncertainty as a percentage, plus charts showing the input values with their uncertainty contributions.
Regional Notes: Error propagation principles are universal and apply worldwide. The mathematical formulas for uncertainty propagation do not depend on geographic region — they follow the same calculus-based rules in all countries. The calculator accepts any numeric values without currency or region-specific formatting, making it suitable for global use in scientific and engineering contexts.
Frequently Asked Questions
What is error propagation?
Error propagation is the process of determining how uncertainties in measured quantities affect the result of a calculation. When you measure quantities X and Y with uncertainties ΔX and ΔY, and then compute a derived quantity Z = f(X, Y), the uncertainties propagate to Z following specific mathematical rules.
What is the error propagation formula for addition and subtraction?
For addition (Z = X + Y) and subtraction (Z = X − Y), the error propagation formula is ΔZ = √(ΔX² + ΔY²). The absolute uncertainties add in quadrature, meaning the total uncertainty is the square root of the sum of the squares of the individual uncertainties.
How is error propagation different for multiplication and division?
For multiplication (Z = X × Y) and division (Z = X ÷ Y), relative uncertainties add in quadrature: ΔZ/|Z| = √((ΔX/X)² + (ΔY/Y)²). This means the percentage error in the result depends on the percentage errors of the inputs, not their absolute values.
How do I use the error propagation calculator?
Select the mathematical operation (addition, subtraction, multiplication, division, or power), enter the values X and Y with their uncertainties ΔX and ΔY, then click Calculate. The calculator computes the result Z, the propagated uncertainty ΔZ, and the relative uncertainty. Results include bar charts of values and doughnut charts showing uncertainty contribution from each input.
What is the error propagation formula for power functions?
For Z = Xⁿ, the relative uncertainty is ΔZ/|Z| = |n| × ΔX/|X|. The absolute uncertainty is ΔZ = |n| × |Z| × ΔX/|X|. Only X and its uncertainty ΔX are needed, along with the exponent n.
Why do uncertainties add in quadrature?
Uncertainties add in quadrature (ΔZ = √(ΔX² + ΔY²)) because the input uncertainties are assumed to be independent and random. This follows from the variance sum law for uncorrelated variables. If uncertainties were correlated, the formula would include a covariance term.
What is the general error propagation formula?
The general formula for error propagation is Δf = √(Σ(∂f/∂xᵢ)²Δxᵢ²), where f is a function of variables xᵢ with uncertainties Δxᵢ. This formula calculates the total uncertainty by taking partial derivatives of the function with respect to each variable and adding them in quadrature.