Detection and quantification limits

Let $C=[0,\mathrm{\infty })$ be a concentration parameter set and let $\mathsf{E}=((\mathbb{R},\mathcal{B}(\mathbb{R})),C,(P_c{)}_{c\in C})$ be a statistical experiment (Statistical experiments). The response $X(x)=x$ is a real random variable (Random elements and real random variables) with law $P_c$ at concentration $c$; $P_0$ is the blank law. Since each $(t,\mathrm{\infty })$ is open, it is Borel (The Borel sigma-algebra of a topological space), so the indicator rule below is measurable. For a threshold $t\in \mathbb{R}$, the positive-decision rule is ${\phi }_t(x)={\mathbf{1}}_{\{x>t\}}$. Its blank false-positive probability is $P_0(X>t)$, and its false-negative probability at $c>0$ is $P_c(X\le t)$. These are the null rejection probability and one minus the power for the simple null $\{0\}$ and point alternative $\{c\}$ (Randomized tests, Test size and power, The expectation of an indicator is the probability of the event, Basic identities for a probability measure).

For a target $\alpha \in (0,1)$, an exact critical level $L_C$ is a threshold satisfying $P_0(X>L_C)=\alpha \mathrm{.}$ A threshold with probability at most $\alpha$ is an $\alpha$-level threshold; equality need not be attainable for a discrete response. For a target $\beta \in (0,1)$, a detection limit $L_D$ is the least $c>0$ satisfying $P_c(X\le L_C)=\beta ,$ when such a least concentration exists. Thus $L_C$ is measured in response units, whereas $L_D$ is measured in concentration units.

When $0<E_c[X]<\mathrm{\infty }$ and ${\mathrm{Var}}_c(X)<\mathrm{\infty }$, the coefficient of variation is $\mathrm{CV}(c):=\frac{\sqrt{{\mathrm{Var}}_c(X)}}{E_c[X]}\mathrm{.}$ Expectation under $P_c$ is as in Expectation of a nonnegative or integrable random variable, and variance has the meaning in Moments, variance, and covariance on a probability space. For a declared tolerance $k_Q>0$, a quantification limit $L_Q$ is the least $c>0$ satisfying $\mathrm{CV}(c)\le k_Q$, when such a least concentration exists. This definition does not make the three limits universal: each is tied to the response law and the stated error or precision criterion.

The measurement procedure and its observation law must include any blank correction used in routine reporting. A below-threshold report is an event such as $X\le L_C$; its likelihood contribution is determined by the declared measurement law, not by substituting $L_C$ as though it were an observed response.

Assume the Axiom of Choice. Let $Z$ have the standard normal law and let $\mathrm{\Phi }(x)=P(Z\le x)$ be its cumulative distribution function. Then $\mathrm{\Phi }$ is continuous and strictly increasing, with limits $0$ and $1$ at $-\mathrm{\infty }$ and $+\mathrm{\infty }$, and $P(Z=x)=0$ for every $x\in \mathbb{R}$. For each $q\in (0,1)$ there is a unique real $z_q$ with $\mathrm{\Phi }(z_q)=q$, and $z_{1-q}=-z_q\mathrm{.}$

The quantile existence, uniqueness, and symmetry are proved locally from the published standard normal density and measure definitions; the source references document the normal-law convention and its use in detection-limit calibration.

Assume AC. For $m\in \mathbb{R}$, $\sigma \ge 0$, and a real random variable $X$ with law $N(m,{\sigma }^2)$, $E[X]=m,\mathrm{Var}(X)={\sigma }^2\mathrm{.}$

The moment calculation is supplied locally from the library's proved Gaussian even-moment result and the normal-law push-forward definition. The cited sources document the scale-as-standard-deviation convention used in the detection-limit model.

Assume AC, ${\sigma }_0>0$, $0<\alpha ,\beta <1\mathrm{/}2$, and $k_Q>0$. Consider the response-only experiment with concentration $c\in [0,\mathrm{\infty })$ and $X\sim N(c,{\sigma }_0^2)$ under each $c$. Reject the blank when $X>L_C$. For the coefficient-of-variation criterion $\mathrm{CV}(c)\le k_Q$, the exact critical level, detection limit for false-negative target $\beta$, and quantification limit are $L_C=z_{1-\alpha }{\sigma }_0,L_D=(z_{1-\alpha }+z_{1-\beta }){\sigma }_0,L_Q=\frac{{\sigma }_0}{k_Q}\mathrm{.}$ where $z_q$ is the unique standard normal $q$-quantile from The standard normal CDF has unique symmetric quantiles. These formulas are conditional on the known-blank, equal-variance normal model.

The exact formulas are derived from the stated normal law and the local quantile and moment lemmas. NIST SP 958 reports rounded known-blank values under $\alpha =\beta =0.05$, constant ${\sigma }_0$, and 10% relative standard deviation; Eurachem's normal-tail discussion uses the corresponding approximate $1.65+1.65$ factor. Neither source is used in place of the local derivation.