propagation of errors calculator

What Is Propagation of Errors?

Propagation of errors (or uncertainty propagation) is the process of determining how measurement uncertainty in your inputs affects uncertainty in your final result. If you measure quantities like length, mass, voltage, concentration, or time, each value contains some uncertainty. When those values are added, multiplied, divided, or transformed, the uncertainty changes.

This calculator helps you estimate that change quickly using standard formulas taught in physics, chemistry, engineering, and lab science.

How to Use This Calculator

• Choose the operation that matches your equation.
• Enter your measured values (x, and optionally y).
• Enter the standard uncertainties (σx and σy).
• Click Calculate Uncertainty.
• Read the result as value ± uncertainty and check the relative uncertainty percentage.

Formulas Used

Addition and Subtraction

For z = x ± y with independent uncertainties:

σz = √(σx² + σy²)

Notice that uncertainties add in quadrature, not by direct sign. Even in subtraction, uncertainty still increases.

Multiplication

For z = x·y:

σz = √((yσx)² + (xσy)²)

This is equivalent to combining relative uncertainties when both variables are nonzero.

Division

For z = x / y:

σz = √((σx/y)² + (xσy/y²)²)

This expression comes from partial derivatives and is robust for typical lab calculations.

Power Function

For z = xⁿ:

σz = |n·xⁿ⁻¹|σx

This is especially useful for area (x²), volume-like terms (x³), and inverse powers.

Worked Examples

Example 1: Rectangle Area

If length is 10.0 ± 0.2 cm and width is 4.0 ± 0.1 cm, then area is A = L×W = 40.0 cm². The propagated uncertainty from multiplication gives approximately ±1.0 cm² (depending on rounding).

Example 2: Ratio Measurement

If x = 5.0 ± 0.2 and y = 2.0 ± 0.1, then z = x/y = 2.5. Division propagation gives the uncertainty in the ratio directly, which is often needed in calibration and normalization workflows.

Assumptions and Limitations

• Inputs are treated as independent (uncorrelated).
• Uncertainties are interpreted as 1σ standard uncertainties.
• Method uses first-order Taylor approximation (good for small uncertainties).
• For strongly nonlinear models or large uncertainties, Monte Carlo methods may be better.

Practical Tips for Better Results

• Keep units consistent before calculating.
• Use realistic uncertainty estimates from instrument specs or repeated trials.
• Report both absolute uncertainty (±) and relative uncertainty (%).
• Round uncertainty to one or two significant digits, then round the value accordingly.

Why This Matters

Good uncertainty reporting improves scientific transparency, engineering safety, and decision quality. Whether you are writing a lab report, validating a sensor, or presenting model outputs, propagated uncertainty communicates confidence in your result—not just the result itself.