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Error Propagation Calculator

Combine uncertainties when measured quantities are added, multiplied, divided, or raised to powers

Parameters

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Show Trail

Controls

xⓘ

Calculated Values

Result f:
50.00;50.00;
Uncertainty σ_f:
1.41;1.41;
Relative uncertainty:
2.83;2.83;%
Reported value:
50.00;±1.450.00;± 1.4

Examples

Resistance R = V / I

V = 12.0 ± 0.1 V, I = 0.50 ± 0.02 A (quotient).

  • R (Ω): 24.0024.00
  • σ_R: 1.001.00

Area A = L × W

L = 2.50 ± 0.02 m, W = 1.20 ± 0.01 m (product).

  • A (m²): 3.003.00
  • σ_A: 0.030.03

Total mass m1 + m2

Independent weighings combined by sum rule.

  • m_total (g): 150.00150.00
  • s: 0.580.58

Visualization

Propagating Experimental Uncertainty

Every physics measurement carries uncertainty. When you plug measured values into a formula, the uncertainty of the result must be calculated rigorously from the uncertainties of the inputs — not inferred from the last digit you write down.

For a function f(x, y, …) of independent variables, the general formula is σ_f² = (∂f/∂x)²σ_x² + (∂f/∂y)²σ_y² + … . This is the first-order Taylor expansion and is standard in undergraduate lab courses.

When variables are added or subtracted (f = x ± y), absolute uncertainties combine in quadrature: σ_f = √(σ_x² + σ_y²). This applies to total length from two rulers, or net force from perpendicular components.

When variables are multiplied, divided, or raised to powers, it is easier to use relative uncertainties: σ_f/|f| = √((∂ln f/∂x · σ_x)² + …). For f = xᵐ yⁿ, this gives σ_f/|f| = √((mσ_x/x)² + (nσ_y/y)²).

Always check that uncertainties are independent. Correlated errors (same instrument, same calibration) require the full covariance form, which is beyond this calculator but essential in research.

After propagation, report as (value ± σ) unit and round using the Significant Figures calculator. Validate your model with the Chi-Square Fit tool when comparing theory to multiple data points.

Key Concepts

  • General: σ_f² = Σ (∂f/∂xᵢ)² σ_xᵢ² for independent errors
  • Sum/difference: σ_f = √(σ_x² + σ_y²)
  • Product f = xᵐ yⁿ: σ_f/|f| = √((mσ_x/x)² + (nσ_y/y)²)
  • Quotient x/y: same relative formula as products
  • Relative uncertainty: σ_f/|f| often quoted as a percentage
  • Independent vs correlated errors — correlation matters in advanced labs

Real-World Applications

  • Ohm's law: R = V/I from voltmeter and ammeter readings
  • Density ρ = m/V from balance and graduated cylinder
  • Kinetic energy from measured mass and speed
  • Refractive index from measured angles in optics labs
  • Gravitational acceleration from pendulum period and length

Explore Further

More measurement uncertainty tools

  • Significant Figures

    Round measurements and calculated results to the correct precision for lab reports.

  • Chi-Square Fit

    Compute χ² and reduced χ² to test whether data agree with a physics model within uncertainty.

Physics Equations

General (independent):
σf2=(∂f∂x)2σx2+(∂f∂y)2σy2\sigma_f^2 = \left(\frac{\partial f}{\partial x}\right)^2\sigma_x^2 + \left(\frac{\partial f}{\partial y}\right)^2\sigma_y^2
Sum:
σf=σx2+σy2\sigma_f = \sqrt{\sigma_x^2 + \sigma_y^2}
Product / quotient (relative):
σf∣f∣=(mσxx)2+(nσyy)2\frac{\sigma_f}{|f|} = \sqrt{\left(m\frac{\sigma_x}{x}\right)^2 + \left(n\frac{\sigma_y}{y}\right)^2}
Power f = x^m:
σf∣f∣=∣mσxx∣\frac{\sigma_f}{|f|} = \left|m\frac{\sigma_x}{x}\right|

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: List measured quantities and uncertainties

Write each input with its standard uncertainty (same units as the quantity).

Result:

x=10±0.2,y=5±0.1x = 10 ± 0.2, y = 5 ± 0.1

Explanation:

Independent random errors are assumed. Systematic errors would need a separate analysis.

2

Step 2: Identify the combining operation

Selected rule: Product x^m · y^n.

Equation:

f=x1y1f = x^{1} y^{1}

Explanation:

The propagation formula depends on whether variables add, multiply, divide, or appear in powers.

3

Step 3: Evaluate the central value f

Substitute best estimates (not rounded values) into the formula.

Calculation:

f=50.000000f = 50.000000

Result:

f=50.0000f = 50.0000

Explanation:

Use full calculator precision here; round only in the final reported answer.

4

Step 4: Apply relative uncertainty formula

For products, quotients, and powers, work with fractional errors first.

Equation:

σf∣f∣=(1σxx)2+(1σyy)2\frac{\sigma_f}{|f|} = \sqrt{\left(1\frac{\sigma_x}{x}\right)^2 + \left(1\frac{\sigma_y}{y}\right)^2}

Calculation:

σf∣f∣=0.000400+0.000400=0.028284\frac{\sigma_f}{|f|} = \sqrt{0.000400 + 0.000400} = 0.028284

Result:

σf/∣f∣=0.0283\sigma_f / |f| = 0.0283

Explanation:

Partial derivatives of ln f give the squared terms inside the square root.

5

Step 5: Convert to absolute uncertainty

Multiply relative uncertainty by |f|.

Equation:

σf=∣f∣⋅σf∣f∣\sigma_f = |f| \cdot \frac{\sigma_f}{|f|}

Calculation:

σf=50.0000×0.028284=1.4142\sigma_f = 50.0000 \times 0.028284 = 1.4142

Result:

σf=1.4142\sigma_f = 1.4142
6

Step 6: Report the final result

State value ± uncertainty with correct units; round σ to 1–2 significant figures.

Result:

50.000 ± 1.4

Explanation:

Match the decimal place of f to the last significant digit of σ (e.g. 24.0 ± 1.0 Ω).

Frequently Asked Questions (FAQ)

When do I add errors in quadrature?

For independent random uncertainties, always combine in quadrature (square, sum, square root). Never add σ_x + σ_y directly unless errors are perfectly correlated and identical sign.

Can fractional uncertainty be larger than 100%?

Yes, when σ_x is a large fraction of x. That signals a poor measurement or that x is near zero — reconsider your method.

Do I propagate before or after unit conversion?

Convert to SI first, propagate in those units, then present final results. Changing units does not change relative uncertainty.

What about systematic errors?

Treat known offsets separately. A calibration error common to all points is not reduced by averaging and may not enter the same s formula.

Practice MCQs

  1. If f = x + y with independent σ_x and σ_y, σ_f equals:
  2. For f = x × y, the relative uncertainty satisfies:
  3. Doubling σ_x while keeping x fixed will:
  4. Which operation uses ONLY absolute quadrature (no relative terms)?
  5. If σ_x/x = 1%, σ_y/y = 2%, and f = x × y, then σ_f/f is approximately:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

Significant FiguresRound calculated results before reporting experimental answers.Round your final reported value after propagating uncertainty.Chi-Square FitTest whether data agree with a physics model within uncertainty.Check whether propagated errors match scatter in data.Ohm's LawCombine measured V, I, R with propagated uncertainties.Projectile MotionKinematic trajectories under gravity — compare simulated vs measured positions.Monte Carlo IntroStochastic estimation of integrals and constants.Monte Carlo standard error (~1/√N) combines with instrument limits.Gradient DescentIteratively minimize f(x) by following the negative gradient with animated path visualization.Related Computational Physics topic.