Working with Measurement Uncertainty

Start with the repeated measurement statistics tool to calculate a mean and its standard error from readings. Check the independence and measurement-condition assumptions before using that standard error as an uncertainty component.

The two-input uncertainty calculator takes absolute standard uncertainties in the units of the corresponding estimates. It does not take a tolerance, percentage or confidence interval in those fields.

Under an explicitly assumed uniform distribution over ±a, standard uncertainty is a/√3. For the mean of n independent repeated observations, it is s/√n, where s is their sample standard deviation. These are different input models; neither follows from a number labeled “accuracy” alone. See NIST’s definitions and assumptions.

Original input examples: a ±0.3 V bound under a uniform model gives 0.173205 V standard uncertainty. A set of 25 independent observations with sample standard deviation 0.5 V gives 0.1 V standard uncertainty of the mean. The latter is not the uncertainty of an individual observation, and repeating readings does not automatically remove other uncertainty components.

Original calculated example: use voltage A = 10 V and current B = 2 A, with standard uncertainties uA = 0.1 V and uB = 0.02 A. Choose multiplication, correlation r = 0 and coverage factor k = 2. The estimate is P = AB = 20 W. The calculator opens with these numbers; interpret its generic output unit as watts.

For this product, ∂P/∂A = B = 2 A and ∂P/∂B = A = 10 V. Each sensitivity times its input uncertainty contributes 0.2 W. With zero covariance, the first-order combined uncertainty is √(0.2² + 0.2²) = 0.282843 W. Adding the two contributions directly would instead give 0.4 W and would represent a different correlation assumption here.

The relative standard uncertainty is 1.4142%. Expanded uncertainty U = k uc = 0.565685 W. The displayed numerical interval is 19.434315 to 20.565685 W. These extra digits help reproduce the calculation; they are not a prescription for reporting precision.

The two-input first-order variance is uc² = (cA uA)² + (cB uB)² + 2r cA cB uA uB. Sensitivities cA and cB retain their signs. These comparisons change only the assumed correlation in the power example.

For a difference A−B, the second sensitivity is negative. Equal standard uncertainties with perfect positive correlation then cancel in the linear model. Do not select correlation to obtain a preferred answer: it must describe the measurements and their shared uncertainty components.

Products and ratios use a first-order approximation. A zero first-order result can miss higher-order terms; a quotient becomes especially problematic when its denominator is uncertain near zero. The tool does not model distributions or perform Monte Carlo propagation.

Multiplying by k scales uncertainty. It does not, by itself, determine a confidence level. Interpreting coverage requires more information about the distribution and reliability of the uncertainty estimate; see NIST’s expanded-uncertainty guidance. Record the measurement equation, input uncertainty basis, correlation, units and coverage-factor choice alongside your result.