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Error Propagation Calculator

Free error propagation calculator. GUM, NIST, Taylor 1997. Propagate uncertainties through add, sub,

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

σ
GUM · NIST · TAYLOR 1997Uncertainty Propagation

Error Propagation — Uncertainties Through Operations

Add, sub, mul, div, power, √(a²+b²). Partial derivatives + Monte Carlo. 7 real-world examples.

Absolute Uncertainty →Confidence Intervals →

Real-World Scenarios — Click to Load

Operation & Inputs

error_propagation.sh
CALCULATED
$ propagate --op="mul" --x=5 --σx=0.1 --y=3 --σy=0.05
Result
15.000000
Abs. uncertainty (σf)
0.390512
Rel. uncertainty
2.60%
95% CI
[14.235, 15.765]
Share:
Error Propagation Result
Multiplication (x × y)
f = 15.0000 ± 0.3905
Rel: 2.60%95% CI: [14.235, 15.765]
numbervibe.com/calculators/statistics/error-propagation-calculator

Result Value

f = 15.0000 ± 0.3905

Uncertainty Contribution (%)

Calculation Breakdown

COMPUTATION
Result f
15.000000
f = x × y = 5 × 3
Relative σx/x
0.020000
0.1/5
Relative σy/y
0.016667
0.05/3
(σf/f)²
0.000678
(\text{sigma} x/x)^{2} + (\text{sigma} y/y)^{2}
σf
0.390512
|f| × √((σx/x)² + (σy/y)²)
REPORTING
Relative uncertainty
2.60%
(\text{sigma} f/f) imes 100
95% CI (1.96σ)
[14.234596, 15.765404]
f ± 1.96σf

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • Error propagation combines uncertainties through mathematical operations using partial derivatives (GUM law of propagation)
  • • For independent variables: σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² + ...
  • • Add/Sub: σf = √(σx² + σy²) — uncertainties add in quadrature
  • • Mul/Div: (σf/f)² = (σx/x)² + (σy/y)² — relative uncertainties add in quadrature
  • • Power: σf/f = |n| × (σx/x)
  • • Monte Carlo: Sample from distributions, compute f, take SD — works for any function

Did You Know?

📐The quadrature rule assumes uncorrelated errors. Correlated errors require covariance terms in the propagation formula.Source: GUM JCGM 100
🔬NIST and ISO guides (GUM) standardize uncertainty propagation for metrology and lab measurements worldwide.Source: NIST
📊Monte Carlo propagation doesn't assume normality — it works for any distribution and nonlinear functions.Source: Taylor 1997
⚡In physics, error propagation is essential: "R = 6.0 ± 0.3 Ω" means uncertainty propagated from V and I.Source: Lab reporting
📏For area A = L×W, (σA/A)² = (σL/L)² + (σW/W)². If both 1%, area uncertainty ≈ 1.4%.Source: Taylor 1997
🌡️F = 1.8C + 32: the 32 has no uncertainty; only the 1.8×C term propagates error.Source: Temperature conversion

Formulas (GUM-compliant)

General: σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² + ...

Assumes independent variables

f = x ± y: σf = √(σx² + σy²)

Add and subtract: same formula

f = x × y or x/y: (σf/f)² = (σx/x)² + (σy/y)²

Relative uncertainties add in quadrature

f = x^n: σf/f = |n| × σx/x

Power rule

f = √(a²+b²): σf² = (a/f)²σa² + (b/f)²σb²

Pythagoras / hypotenuse

Monte Carlo Propagation

Sample x and y from their uncertainty distributions, compute f(x,y) for each sample, then take the standard deviation. Works for any function, including highly nonlinear ones, and doesn't assume small uncertainties (Taylor 1997).

Frequently Asked Questions

When do I use analytic vs Monte Carlo propagation?

Analytic formulas are exact for the listed operations and assume small uncertainties. Use Monte Carlo for custom/complex functions, large uncertainties, or non-normal distributions.

What if my variables are correlated?

The standard formula assumes independence. For correlated x,y, add covariance terms: σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² + 2(∂f/∂x)(∂f/∂y)ρσxσy.

Why "add in quadrature"?

Variances add: σf² = σx² + σy². So σf = √(σx² + σy²). Errors don't simply add; they combine as the square root of the sum of squares.

How do I report the result?

Report as f ± σf with appropriate significant figures. E.g., "A = 15.0 ± 0.3 m²" or "R = 6.00 ± 0.15 Ω".

What about division by zero?

If y = 0 in f = x/y, the result is undefined. Ensure y has a non-zero value.

Limitations

  • • Analytic formulas assume small relative uncertainties (linear approximation)
  • • Assumes independent (uncorrelated) variables
  • • Often assumes normal distributions; Monte Carlo can use other distributions
  • • For very nonlinear functions, Monte Carlo is more accurate

Applications

Physics & Engineering

Force, resistance, area, velocity, tolerance stack-up.

Chemistry & Lab

Dilution factors, concentrations, titration, density.

Metrology

NIST GUM: Guide to the Expression of Uncertainty in Measurement.

Data Analysis

Propagate measurement errors through derived quantities.

Dominant Error Source

When one uncertainty is much larger than the others, it dominates. For f=x×y with σx/x=1% and σy/y=10%, σf/f ≈ 10%. Improve the dominant measurement first to reduce overall uncertainty.

Disclaimer: This calculator uses standard error propagation formulas for independent variables (GUM). For correlated variables or complex functions, consult the GUM or use Monte Carlo simulation.

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