Percent error and uncertainty propagation calculator

How to calculate percent error

Divide the gap between the measured and accepted values by the accepted value: % error = |measured − accepted| ÷ |accepted| × 100. Measuring g as 9.72 m/s² against the accepted 9.81 m/s² gives 0.09 ÷ 9.81 × 100 = 0.917%. Use it when a trusted reference exists.

Signed percent error and edge cases

Drop the absolute value to keep the direction: (9.72 − 9.81) ÷ 9.81 × 100 = −0.917%, which says the measurement came out low.

The denominator always uses the size of the accepted value. Measuring −9 against an accepted −10 gives a signed error of +10%, because the reading is numerically above the reference.

An accepted value of zero makes percent error undefined. Report the absolute error, |measured − accepted|, in the quantity's own units instead.

Percent error is not uncertainty

Percent error compares one result with a reference. Uncertainty, written ±, is your estimate of how much the measurement itself can vary, and it exists even when no accepted value does.

Use the two together to judge agreement. 9.72 ± 0.12 m/s² is consistent with 9.81, since the 0.09 gap lies inside the uncertainty; 9.72 ± 0.05 m/s² is not.

How to propagate uncertainty

Absolute uncertainties combine for sums and differences. Relative uncertainties, δa/|a|, combine for products, quotients and powers. For independent random errors, both combine in quadrature:

• Sum or difference: δq = √(δa² + δb²). (10.0 ± 0.3) + (5.0 ± 0.4) = 15.0 ± 0.5.
• Product or quotient: δq/|q| = √((δa/a)² + (δb/b)²). In (2.0 ± 0.1) × (3.0 ± 0.2), 5% and 6.7% combine to 8.3% of 6.0, so q = 6.0 ± 0.5.
• Power: δq/|q| = |n| × δa/|a|. (2.0 ± 0.1)³ = 8.0 ± 1.2, and a square root halves the relative uncertainty: √(4.0 ± 0.2) = 2.00 ± 0.05.

Subtracting close values is where uncertainty grows. (10.0 ± 0.3) − (9.5 ± 0.4) = 0.5 ± 0.5, a 100% relative uncertainty, although the inputs were uncertain by only 3% and 4.2%.

For a longer formula, apply the rules one operation at a time, as long as each input appears only once. When an input appears twice, its errors are correlated with themselves and the general method in NIST's combination guide applies.

Quadrature or worst case?

Add in quadrature when the input errors are independent and random. That is the assumption behind the combined standard uncertainty in NIST Technical Note 1297, and the result never exceeds the linear sum.

Add linearly, δa + δb, for a bound that holds even when the errors are correlated, such as two lengths measured with the same faulty ruler. The sum example becomes 15.0 ± 0.7, and the product 6.0 ± 0.7.

How to round the reported result

Round the uncertainty to one significant figure, or two when its first digit is 1, then round the value to the same decimal place. This is the convention in Taylor's An Introduction to Error Analysis, section 2.2.

An uncertainty of 1.2 starts with 1 and keeps two figures, giving 8.0 ± 1.2. An uncertainty of 0.05 keeps one, so the value is written to two decimals: 2.00 ± 0.05. Carry unrounded numbers through intermediate steps and round once at the end.

How the propagation examples are reported

Sum quadrature and sum worst case are both (10.0 ± 0.3) + (5.0 ± 0.4). Product is (2.0 ± 0.1) × (3.0 ± 0.2), and cube is (2.0 ± 0.1)³.

Common mistakes

• Entering a percentage as the ± value. For 2.0 ± 5%, the uncertainty is 0.1, not 5.
• Combining absolute uncertainties for a product. Products and quotients use relative uncertainties.
• Reporting percent error as the uncertainty. One compares with a reference; the other describes the measurement.
• Using quadrature for correlated errors. It can understate the combined uncertainty; the worst case gives a safe bound.
• Applying the rules to large uncertainties. They are first-order formulas that assume each uncertainty is small next to its value.
• Comparing values in different units. Convert 981 cm to 9.81 m before comparing it with 9.72 m.

Value and uncertainty conventions

• Uncertainties are absolute, in the same unit as their value.
• The exponent of a power is exact; use 0.5 for a square root.
• A negative base needs a whole-number exponent.
• Products, quotients and powers need nonzero inputs, because a relative uncertainty divides by the value.

To round a result to a chosen number of significant figures, use the rounding calculator. The Ohm's law calculator is a common source of products and quotients to propagate, such as P = V × I.