Statistics

Error Propagation Calculator

Combine two measured quantities and their uncertainties to see the propagated error. Choose add, subtract, multiply, or divide, and the tool applies the matching rule: sums and differences add the absolute uncertainties in quadrature, while products and quotients combine the relative uncertainties.

What this calculator does

The Error Propagation Calculator estimates your result instantly from the values you enter, using standard statistics formulas. Combine two measured quantities and their uncertainties to see the propagated error. Choose add, subtract, multiply, or divide, and the tool applies the matching rule: sums and differences add the absolute uncertainties in quadrature, while products and quotients combine the relative uncertainties. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.

Error Propagation Calculator

Result

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Result
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Propagated Uncertainty

About Error Propagation Calculator

This calculator takes two values, A and B, each carrying its own uncertainty (±ΔA and ±ΔB), and combines them under the operation you pick. For A + B and A − B it adds the absolute uncertainties in quadrature, √(ΔA² + ΔB²). For A × B and A ÷ B it works in relative terms, so the result's uncertainty is |result| × √((ΔA/A)² + (ΔB/B)²).

It reports the combined result and its propagated uncertainty side by side, both rounded to five decimals.

Formula & Mathematical Calculation

Result = f(Value A, Uncertainty ±ΔA, Value B, Uncertainty ±ΔB, Operation)

Calculated according to standard stats formulas implemented in the calculator core.

Calculation Example

Set Value A = 10 with ±ΔA = 0.5, Value B = 5 with ±ΔB = 0.3, and Operation to A × B:

  • Result = 10 × 5 = 50
  • Propagated uncertainty = 50 × √((0.5/10)² + (0.3/5)²) = 50 × √(0.0025 + 0.0036) = ± 3.90512

The panel shows a Result of 50 and a Propagated Uncertainty of ± 3.90512.

Key Benefits

  • Switches between four operations (add, subtract, multiply, divide) and automatically uses the correct propagation rule for each
  • Uses quadrature for sums and differences and relative-error combination for products and quotients, so you do not have to remember which formula applies
  • Blocks negative uncertainties and division by zero with a clear inline message before any result is shown
  • Shows both the combined value and its propagated uncertainty to five decimals, free and entirely in the browser

How to Use this Calculator

  1. Enter Value A and its uncertainty
    Type your first measurement into Value A and its error into the Uncertainty +/-ΔA field (it cannot be negative).
  2. Enter Value B and its uncertainty
    Fill in Value B and Uncertainty +/-ΔB the same way.
  3. Pick the operation
    Use the Operation menu to select A + B, A - B, A x B, or A / B.
  4. Calculate
    Press Calculate to read the combined Result and its Propagated Uncertainty.

Assumptions & Limitations

What the calculation assumes

  • Every value you enter is treated as an exact, known amount.
  • Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
  • Results use standard published formulas and account only for the values entered above.
  • Rounding is applied for display; the underlying calculation uses your full precision.
  • Results are returned in the same units you enter, unless the tool states the output unit.

What it does not cover

  • Error Propagation Calculator produces a calculated result for reference and learning, not a certified or official measurement.
  • Real-world outcomes differ when tolerances, rounding rules or local standards apply.
  • Definitions and standards vary by region and change over time; verify the convention that applies to your case.
  • Figures are only as accurate as the inputs you supply — check them before acting on the result.
  • For work where the result carries real consequences, confirm it against an authoritative source.

Frequently Asked Questions

For addition and subtraction the tool combines the two absolute uncertainties in quadrature, taking the square root of the sum of their squares, rather than adding them directly.

For products and quotients the calculator combines relative uncertainties (each uncertainty divided by its value), then multiplies by the absolute result, which is the standard rule for those operations.

The tool stops and shows the message that uncertainties cannot be negative, since an uncertainty represents a magnitude.

No. If the operation is division and Value B is zero, the calculator flags that you cannot divide by zero instead of returning a result.

Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.