Worked problem

Measurement Uncertainty: Combining Independent Inputs

Combine independent uncertainties in P = V × I by root-sum-square and compare the result with a worst-case bound.

Controls & Instrumentation · Last reviewed

Given

The power of a load is calculated as P = V × I. The voltage is measured as 230.0 V with a standard uncertainty of 0.5 V. The current is measured as 10.00 A with a standard uncertainty of 0.05 A. The two measurements are independent.

Find

The power, its combined standard uncertainty, the expanded uncertainty for a coverage factor k = 2, and the worst-case bound for comparison.

Solution

u(P)² = (∂P/∂V × u(V))² + (∂P/∂I × u(I))²
  1. Power: P = 230.0 × 10.00 = 2,300 W.
  2. Sensitivity coefficients: ∂P/∂V = I = 10.00 A and ∂P/∂I = V = 230.0 V.
  3. Voltage contribution: 10.00 × 0.5 = 5.0 W. Current contribution: 230.0 × 0.05 = 11.5 W.
  4. Combined: u(P) = √(5.0² + 11.5²) = √(25 + 132.25) = 12.5 W, which is 0.55% of the reading.
  5. Expanded uncertainty with k = 2 (about 95% coverage): 2 × 12.5 = 25 W.
  6. Worst case (all errors add in the same direction): 5.0 + 11.5 = 16.5 W, or 0.72%.

Answer

P = 2,300 W ± 12.5 W (standard) | ± 25 W (k = 2) | worst case ± 16.5 W

Why root-sum-square

Independent errors are unlikely to reach their extremes together in the same direction, so they partly cancel. The root-sum-square combination reflects that and gives a smaller result than the straight sum. The worst-case sum is an upper bound that is rarely approached.

  • The current dominates: its relative uncertainty (0.5%) is larger than the voltage one (0.22%), and its contribution to the power is 11.5 W against 5.0 W.
  • Improving the voltage meter would barely help. Halving the current uncertainty would cut u(P) to 7.9 W.
  • For a product or quotient, the relative uncertainties combine: √(0.00217² + 0.005²) = 0.55%.

Common mistakes

  • Adding standard uncertainties directly when the inputs are independent.
  • Using root-sum-square when the errors are correlated, for example two readings from the same instrument.
  • Mixing standard uncertainty with an expanded one, or with a quoted accuracy limit.
  • Quoting the result with more digits than the uncertainty supports.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

More worked problems

Questions

What is a standard uncertainty?

An uncertainty expressed as one standard deviation. Manufacturers often give limits instead, and these must be converted before combining them.

What does k = 2 mean?

A coverage factor of 2 gives an interval that contains the true value with about 95% probability when the distribution is roughly normal.

Does this cover calibration and drift?

Only if those are included in the standard uncertainties of the inputs. A full uncertainty budget lists each source.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

← All worked problems