Coverage Factor⚠ unverified

Understanding the Parameters

• Confidence level $C$ — the fraction of the probability the interval should contain (e.g. $0.95$ for $95\%$). Higher confidence demands a wider interval and hence a larger $k$. The relationship is nonlinear: pushing from $95\%$ to $99\%$ raises $k$ from $1.96$ to $2.58$, and chasing $99.99\%$ needs $k \approx 3.9$ — the tails of the normal thin out fast, so extreme confidence is expensive.

• The two-sided argument $(1+C)/2$ — because the interval is symmetric ($\pm k u$), the confidence $C$ is split between the two tails. To leave $(1-C)/2$ in each tail, the upper edge must sit at the $\Phi^{-1}((1+C)/2)$ quantile. For $C = 0.95$: $(1+0.95)/2 = 0.975$, and $\Phi^{-1}(0.975) = 1.96$ — the origin of the ubiquitous $k=2$ rounding.

• The output $k$ — fed directly into $U = k u_c$. It is dimensionless: a pure multiplier on the standard uncertainty. The default $k = 2$ (from $C \approx 0.95$) is the metrology convention.

• Normal assumption. This $k$ assumes the measurement result is normally distributed with well-known standard deviation (many degrees of freedom). With few measurements, the heavier-tailed Student's $t$ distribution gives a larger $k$ for the same confidence.

Derivation (Approaching a Proof)

We want the interval $[\mu - k\sigma,\ \mu + k\sigma]$ about a normally-distributed result to contain probability $C$. By symmetry of the normal, this leaves $(1-C)/2$ of the probability in each tail. The upper limit $\mu + k\sigma$ must therefore correspond to the cumulative probability

$$\Phi(k) = 1 - \frac{1-C}{2} = \frac{1+C}{2},$$

where $\Phi$ is the standard normal CDF. Inverting,

$$k = \Phi^{-1}\!\left(\frac{1+C}{2}\right). \qquad\blacksquare$$

Evaluating at the standard confidences reproduces the tabulated coverage factors: $C = 0.90 \Rightarrow \Phi^{-1}(0.95) = 1.645$; $C = 0.95 \Rightarrow \Phi^{-1}(0.975) = 1.960$; $C = 0.99 \Rightarrow \Phi^{-1}(0.995) = 2.576$. The registry's lookup returns exactly these (rounded) values at and above each threshold — correct at the common levels, but a step approximation to the smooth curve.

Dimensional check. $C$ and $(1+C)/2$ are probabilities (dimensionless), and $\Phi^{-1}$ of a probability is a dimensionless quantile, so $k$ is dimensionless. $\checkmark$

History and Development

• The GUM framework. The coverage factor is a defined quantity in the Guide to the Expression of Uncertainty in Measurement, the bridge between the standard uncertainty (one sigma) and the reported expanded uncertainty (Expanded Uncertainty). It standardised how confidence is attached to measurement results.

• The $t$-distribution refinement. When the standard uncertainty is estimated from few observations, the GUM prescribes using the Student's $t$ coverage factor with the effective degrees of freedom (Welch–Satterthwaite), which exceeds the normal $k$ — a correction this normal-only formula omits.

• Rounding conventions. The near-universal "$k = 2$ for $95\%$" is a deliberate, convenient rounding of the exact $1.96$; likewise "$k = 3$" for $\approx 99.7\%$. These conventions appear on calibration certificates worldwide.

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

Notes: Registry calculator coverage-factor (unverified). Exact form $k = \Phi^{-1}((1+C)/2)$; the code is a 3-level step lookup ($2.58$/$1.96$/$1.645$ for $C\ge0.99$/$0.95$/else) — correct at the common levels, snapped elsewhere. Two-sided interval splits $C$ across both tails. Normal assumption; use Student-$t$ for few degrees of freedom. Flagged in Known Issues.