Expanded Uncertainty⚠ unverified

Understanding the Parameters

• Combined standard uncertainty $u_c$ — the "one standard deviation" uncertainty of the measurement result, obtained by combining all the individual uncertainty contributions (in root-sum-square, weighted by sensitivity coefficients). It carries the units of the measurand (the registry labels it dimensionless, but it inherits whatever is being measured — millimetres, volts, kelvin). It represents about a $68\%$ coverage interval on its own.

• Coverage factor $k$ — the multiplier that sets the confidence level (Coverage Factor). $k = 2$ (the default) gives approximately $95\%$ coverage for a normal distribution; $k = 3$ gives about $99.7\%$; $k = 1$ is just the standard uncertainty itself. The choice of $k$ is a statement of how confident the interval should be.

• The output $U$ — the half-width of the reported interval. A result is stated as $y \pm U$, meaning the true value is expected to lie in $[y - U,\, y + U]$ with the confidence implied by $k$. This is the number a customer, auditor, or downstream calculation actually uses.

• Why "expanded." The standard uncertainty is the natural statistical scale (one sigma), but a $68\%$ interval is too weak for most decisions. Expanding it by $k$ produces a practically useful confidence interval — the point of the exercise.

Derivation (Approaching a Proof)

The expanded uncertainty is a scaling of the standard uncertainty to a chosen confidence level, following the GUM (Guide to the Expression of Uncertainty in Measurement). Model the measurement result $Y$ as approximately normally distributed about the true value with standard deviation equal to the combined standard uncertainty $u_c$. For a normal distribution, the interval $\pm k\,u_c$ about the mean contains a fraction of the probability determined solely by $k$:

$$P\big(|Y - y| \le k\,u_c\big) = 2\Phi(k) - 1,$$

where $\Phi$ is the standard normal CDF. Choosing $k$ to hit the desired confidence $C$ (i.e. $k = \Phi^{-1}((1+C)/2)$, the Coverage Factor) and defining the half-width as

$$U = k\,u_c \qquad\blacksquare$$

gives the interval $y \pm U$ covering probability $C$. For $k = 2$, $2\Phi(2) - 1 \approx 0.954$, hence the conventional "$k=2$ for $95\%$." When the effective degrees of freedom are low (few measurements), the normal is replaced by Student's $t$ and $k$ grows accordingly — a refinement the simple $U = ku_c$ form leaves to the choice of $k$.

Dimensional check. $k$ is dimensionless and $u_c$ carries the measurand's units, so $U$ has the measurand's units — the same units as the reported result. $\checkmark$

History and Development

• The GUM. The Guide to the Expression of Uncertainty in Measurement (1993, by the BIPM, ISO, and other metrology bodies) standardised uncertainty reporting worldwide, defining standard uncertainty, combined standard uncertainty, coverage factor, and expanded uncertainty. It replaced a patchwork of "error" conventions with a coherent probabilistic framework.

• $k = 2$ as convention. The near-universal default $k = 2$ (giving $\approx 95\%$) became the standard way to report calibration and test results, appearing on essentially every accredited calibration certificate.

• Type A and Type B. The combined standard uncertainty $u_c$ that feeds this formula is itself built from Type A (statistical, from repeated measurements) and Type B (from specifications, prior knowledge) contributions, combined by the law of propagation of uncertainty — the machinery behind Relative Uncertainty, Coverage Factor, and the uncertainty budget.

Related Concepts: Coverage Factor, Relative Uncertainty, Gage Rr Percent, Normal Distribution PDF, Process Capability Cpk

Notes: Registry calculator expanded-uncertainty (unverified). $U = k\,u_c$ (GUM) — correct as shipped. $u_c$ carries the measurand's units (registry labels dimensionless). $k=2 \Rightarrow \approx95\%$ for a normal distribution; low degrees of freedom need a Student-$t$ $k$. Reported as $y \pm U$.