Error Propagation
Print PageAn Error Propagation Calculator (also known as a Propagation of Uncertainty Calculator, Experimental Measurement Error Utility, Partial Derivative Uncertainty Generator, or Quadrature Sum Analyzer) computes total propagated standard uncertainties (σf = √[ ∑ ( (∂f ÷ ∂xi) · σxi )2 ]), relative percentage uncertainties ([ σf ÷ f ] · 100%), fractional error contributions, and expanded confidence intervals across physics experiments, chemical titration assays, engineering designs, and ISO 17025 metrology audits.
When multi-variable scientific measurements are combined into a calculated result, individual measurement errors propagate and compound. **Error propagation analysis** uses Taylor series partial derivative expansions to quantify how uncertainties in independent inputs impact final computed outcomes.
Our free online Error Propagation Calculator provides instant calculations across all standard uncertainty propagation functions:
- General Partial Derivative Propagation Formula (Uncorrelated Variables):
σf = √[ ( (∂f ÷ ∂x) · σx )2 + ( (∂f ÷ ∂y) · σy )2 + (…) ]. - Addition / Subtraction Rule (f = x ± y):
σf = √[ σx2 + σy2 ]. - Multiplication / Division Rule (f = x · y ÷ z):
σf ÷ |f| = √[ ( σx ÷ x )2 + ( σy ÷ y )2 + ( σz ÷ z )2 ]. - Power Function Law (f = xn):
σf ÷ |f| = |n| · ( σx ÷ x ). - Natural Logarithm Rule (f = ln[x]):
σf = σx ÷ x. - Exponential Function Rule (f = ex):
σf = ex · σx. - Kinetic Energy Physics Equation (Ek = ½ m v2):
σEk ÷ Ek = √[ ( σm ÷ m )2 + ( 2 · [ σv ÷ v ] )2 ].
Master Error Propagation Reference Table (Physics Experiment: Kinetic Energy Ek = ½ m v2)
The table below displays nominal measurements, absolute input uncertainties, partial derivative sensitivity factors, component error contributions, and total propagated uncertainty for a kinetic energy experiment (Mass m = 2.000 ± 0.040 kg [2.0%], Velocity v = 10.000 ± 0.300 m/s [3.0%]):
| Physical Variable / Parameter | Nominal Measurement Value | Input Absolute Uncertainty (σ) | Partial Derivative Sensitivity (∂f / ∂x) | Absolute Component Error Contribution | Variance Share of Total Error (%) |
|---|---|---|---|---|---|
| Mass Input (m) | m = 2.000 kg | σm = ±0.040 kg (2.0%) | ∂Ek / ∂m = ½ v2 = 50.00 J/kg | 50.00 · 0.040 = 2.000 Joules | 10.000% (4.00 / 40.00) |
| Velocity Input (v – Squared Impact) | v = 10.000 m/s | σv = ±0.300 m/s (3.0%) | ∂Ek / ∂v = m v = 20.00 J/(m/s) | 20.00 · 0.300 = 6.000 Joules | 90.000% (36.00 / 40.00) |
| Calculated Nominal Kinetic Energy (Ek) | Ek = ½ (2.0) (10.0)2 | 100.000 Joules | Nominal Output | Quadrature Sum | 100.0% Total Error |
| Total Propagated Absolute Error (σEk) | √[ (2.0)2 + (6.0)2 ] | σEk = ±6.3246 Joules (±6.325 J) | Expanded Measurement Error | Final Experimental Result | 100.00 ± 6.32 Joules |
| Relative Propagated Uncertainty (%) | ( 6.3246 ÷ 100.00 ) · 100% | ±6.3246% (±6.32%) | √[ 2.0%2 + (2·3.0%)2 ] | √[ 4% + 36% ] = √40% | 6.32% Relative Error |
Step-by-Step Kinetic Energy Uncertainty Calculation
To calculate propagated uncertainty for kinetic energy Ek = ½ m v2 given m = 2.000 ± 0.040 kg and v = 10.000 ± 0.300 m/s:
Step 1 (Calculate Nominal Value): E_k = 0.5 · 2.000 · (10.000)^2 = 0.5 · 2.000 · 100.00 = 100.000 Joules
Step 2 (Partial Derivative wrt Mass): ∂E_k / ∂m = 0.5 · v^2 = 0.5 · 100.00 = 50.00 J/kg &implies; Component_m = 50.00 · 0.040 = 2.000 J
Step 3 (Partial Derivative wrt Velocity): ∂E_k / ∂v = m · v = 2.000 · 10.000 = 20.00 J/(m/s) &implies; Component_v = 20.00 · 0.300 = 6.000 J
Step 4 (Quadrature Summation): σ_E_k = √[ (2.000)^2 + (6.000)^2 ] = √[ 4.000 + 36.000 ] = √40.000 = 6.32455 ≈ ±6.325 Joules
Step 5 (Relative Uncertainty): Relative Error = (6.32455 ÷ 100.000) · 100% = 6.3246% ≈ 6.32%
Thus, the final experimental result is reported as 100.00 ± 6.32 Joules (6.32% relative uncertainty), with velocity contributing 90% of the total measurement variance.
Uncertainty Methodologies Comparison: Partial Derivatives vs. Quadrature Sums vs. Monte Carlo
Below is a comparative reference chart detailing when to use partial derivative propagation versus alternative uncertainty estimation methods:
| Uncertainty Propagation Method | Mathematical Complexity Basis | Small Error Assumption Requirement | Primary Practical Application |
|---|---|---|---|
| Taylor Series Partial Derivatives | First-order derivative matrix expansions | YES (σx / x << 1 for linear approximation) | Standard ISO GUM lab reports & physics experiments. |
| Quadrature Power Laws | Relative percentage sum of squares | YES (Pure multiplication / power terms) | Quick hand calculations for physics product functions. |
| Monte Carlo Simulation | Repeated random sampling (100,000 runs) | NO (Handles highly non-linear large errors) | Complex non-linear models & asymmetric errors. |
History & Mathematics: 1795 Carl Friedrich Gauss to 1993 ISO GUM
1795 Carl Friedrich Gauss & Least Squares
In 1795, legendary German mathematician Carl Friedrich Gauss formulated the method of least squares and established error combination rules to calculate astronomical planetary orbits.
1993 ISO / BIPM GUM Guide
In 1993, the International Organization for Standardization (ISO) and BIPM published the Guide to the Expression of Uncertainty in Measurement (GUM), establishing partial derivative Taylor expansion as the universal standard for international scientific measurement.
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Frequently Asked Questions (FAQ)
What is the general formula for error propagation?
The general formula for uncorrelated variables is σf = √[ ∑ ( (∂f ÷ ∂xi) · σxi )2 ].
How do you propagate errors when adding or subtracting numbers?
For f = x ± y, add absolute variances in quadrature: σf = √[ σx2 + σy2 ].
How do you propagate errors when multiplying or dividing?
For f = x · y or f = x ÷ y, add relative fractional variances in quadrature: σf ÷ |f| = √[ ( σx ÷ x )2 + ( σy ÷ y )2 ].