Error Propagation Calculator

Enter your X and Y values along with their uncertainties (±dX, ±dY), choose an operator (+, −, ×, ÷, ln, log, power), and the Error Propagation Calculator applies Gaussian error propagation to give you the result value and its propagated uncertainty (±dF). Perfect for physics, chemistry, and engineering labs where you need to track how measurement errors combine through calculations. Also try the calculate Process Capability Index Cpk (Process Capability Index).

Results

Result f(X, Y)

Propagated Uncertainty ±dF

Relative Uncertainty (%)

Result ± Uncertainty

What is error propagation and why does it matter?

Error propagation (or propagation of uncertainty) is the process of determining how the uncertainty in measured input variables affects the uncertainty of a calculated result. In any real experiment, every measurement has some degree of uncertainty. When you combine measurements mathematically, those uncertainties combine too — error propagation tells you by how much, giving you a scientifically rigorous result with a meaningful error bound. See also our Median Calculator.

What formula does this calculator use?

This calculator uses the Gaussian (quadrature) error propagation formula: δf = √[(∂f/∂x · δx)² + (∂f/∂y · δy)²]. It takes the partial derivatives of your expression with respect to each variable, multiplies each by the corresponding uncertainty, and adds the squared terms under a square root. This assumes measurements are independent and normally distributed.

What is this calculator good for?

It is ideal for students and scientists who need to track how standard deviations or measurement errors propagate through common mathematical operations. For example, if you measure two lengths each with their own uncertainty and compute their product, this tool tells you the uncertainty in that product — a key requirement in physics, chemistry, and engineering lab reports.

How do I enter scientific notation values?

You can enter values in standard decimal format (e.g. 0.001) or use scientific notation such as 1e-3 or 1E-3. Make sure to always precede decimal points with a zero — use '0.01' rather than '.01' — and use a period as the decimal separator, not a comma. You might also find our 68% of Data Falls Between — Empirical Rule useful.

Does this calculator handle single-variable operations like ln(X) or √X?

Yes. For single-variable operations such as ln(X), log₁₀(X), e^X, X^a, and √X, only X and its uncertainty ±dX are needed. The Y fields are ignored for these operations. The propagated uncertainty is computed from the exact partial derivative of each function with respect to X.

What is the difference between absolute and relative uncertainty?

Absolute uncertainty (±dF) is the propagated error in the same units as your result — it tells you the size of the uncertainty directly. Relative uncertainty is that absolute uncertainty divided by the result value, often expressed as a percentage. Relative uncertainty is useful for comparing precision across measurements of different magnitudes.

Why does addition/subtraction add uncertainties in quadrature instead of directly?

When measurements are independent and randomly distributed, the worst-case (direct addition) rarely occurs. Gaussian error propagation adds uncertainties in quadrature — i.e. as the square root of the sum of squares — because this correctly reflects the statistical likelihood that errors partially cancel. Direct addition of uncertainties is only appropriate if you have strong reason to believe errors are fully correlated.

What are the limitations of this calculator?

This calculator assumes that all input variables are independent (uncorrelated) and that uncertainties are small relative to the values (so that a first-order Taylor expansion is valid). It does not account for systematic errors, correlations between variables, or higher-order terms. For highly nonlinear functions or large uncertainties, Monte Carlo methods may provide more accurate results.