Error Propagation Calculator

Combine two measured values and their uncertainties through a sum, difference, product, quotient, or power to get the result and its propagated error, added in quadrature.

Quick Facts

Method
Quadrature (root-sum-of-squares) propagation for independent errors
Sums/differences combine absolute errors; products/quotients/powers combine relative errors.

Your Results

Calculated
Result q
-
Value of the combined quantity
Propagated uncertainty δq
-
Absolute error, added in quadrature
Relative uncertainty
-
δq ÷ |q|, as a percent
Reported result
-
q ± δq

Ready

Enter your values and uncertainties, then calculate.

How error propagation works

Every measurement carries uncertainty. When you combine measured quantities in a calculation, that uncertainty carries through to the answer. Error propagation is the set of rules that tells you how large the uncertainty in the final result is, given the uncertainties in each input. This calculator implements the standard rules for independent, random errors, where uncertainties combine "in quadrature" — the square root of the sum of squares — because independent errors partly cancel rather than always reinforcing each other.

The propagation rules

Let x and y be measured values with absolute uncertainties δx and δy. For the common operations:

  • Sum or difference (q = x + y or q = x − y): the absolute uncertainties add in quadrature. δq = √(δx² + δy²). Note that the uncertainty of a difference is the same as the uncertainty of the sum — subtraction does not cancel error.
  • Product or quotient (q = x·y or q = x/y): the relative uncertainties add in quadrature. δq/|q| = √((δx/x)² + (δy/y)²), then δq = |q| · (δq/|q|).
  • Power (q = xⁿ): the relative uncertainty scales by the magnitude of the exponent. δq/|q| = |n| · (δx/|x|). For example, squaring a value (n = 2) doubles its fractional uncertainty; taking a square root (n = 0.5) halves it.

These rules come from a first-order Taylor expansion of the function around the measured values, keeping only the linear terms — which is why they are sometimes called the "first-order" or "linearized" propagation formulas (Taylor, An Introduction to Error Analysis).

Why quadrature and not a simple sum

If you simply added the errors (δq = δx + δy for a sum), you would be assuming both measurements are wrong in the same direction at the same time — the worst possible case. In reality, independent random errors are just as likely to push in opposite directions, so on average they partially cancel. The root-sum-of-squares captures this: it always gives a smaller, more realistic uncertainty than the linear sum. A linear sum is only appropriate when errors are perfectly correlated or when you deliberately want a conservative worst-case bound.

A worked example

Suppose you measure a length x = 10.0 ± 0.5 cm and a width y = 4.0 ± 0.2 cm and want the area A = x·y. The result is 10.0 × 4.0 = 40.0 cm². The relative uncertainty is √((0.5/10)² + (0.2/4)²) = √(0.0025 + 0.0025) = √0.005 ≈ 0.0707, or about 7.07%. So δA = 40.0 × 0.0707 ≈ 2.83 cm², and you report A = 40.0 ± 2.8 cm².

Frequently Asked Questions

Why are uncertainties added in quadrature instead of just summed?
When the errors in x and y are independent and random, they are equally likely to push the result in either direction, so they partly cancel on average. Adding them in quadrature — the square root of the sum of squares — reflects this statistical cancellation. A plain sum assumes both errors always conspire in the same worst-case direction, which overstates the true uncertainty.
What is the difference between absolute and relative uncertainty?
Absolute uncertainty δx has the same units as x (for example ±0.5 cm). Relative (fractional) uncertainty is δx divided by x, a dimensionless ratio often written as a percent. For sums and differences you combine absolute uncertainties in quadrature; for products, quotients, and powers you combine relative uncertainties in quadrature, then multiply by the result to get its absolute uncertainty.
Does this calculator assume the errors are independent?
Yes. The quadrature rules apply when the uncertainties in x and y are uncorrelated. If x and y share a common source of error — for instance both are read from the same mis-calibrated instrument — the true uncertainty can be larger, and a full treatment must include a covariance term. When in doubt for correlated inputs, a linear sum of errors gives a safe upper bound.
Why doesn't subtracting two numbers reduce the uncertainty?
The propagated uncertainty depends on the square of each error, not its sign, so a difference has the same absolute uncertainty as a sum: √(δx² + δy²). What subtraction can hurt is the relative uncertainty: when two nearly equal numbers are subtracted the result q is small while δq stays fixed, so δq/|q| can become huge. This is the well-known "catastrophic cancellation" problem.

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