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Detection and quantification limits — Examples

1 · Prerequisites

2 · Summary

Worked examples substitute the five-percent false-positive and false-negative targets and ten-percent CV criterion into the known-blank normal formulas. They then change the model by adding estimated-blank variance or concentration-dependent variance. The final example treats a below-threshold report as a censored event and computes its probability under the declared response law.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Known-blank normal limits at five-percent error and ten-percent CV

Example

Assume AC and $σ_{0} > 0$ . In the known-blank model $X \sim N (c, σ_{0}^{2})$ , take $α = β = 0.05$ and $k_{Q} = 0.10$ . Then $L_{C} = z_{0.95} σ_{0}, L_{D} = 2 z_{0.95} σ_{0}, L_{Q} = 10 σ_{0} .$ The NIST SP 958 table reports the corresponding known-blank values rounded to $1.64 σ_{0}$ , $3.29 σ_{0}$ , and $10 σ_{0}$ . These are conditional on the equal-known-variance normal law and the stated error and CV targets.

Facts & Assumptions

Given: AC; $σ_{0} > 0$ ; the known-blank experiment and definitions in Exact limits in the known-blank equal-variance normal model and Critical, detection, and quantification levels, with $α = β = 0.05$ and $k_{Q} = 0.10$ .

[A1]

AC is assumed by the normal-limit theorem and the standard normal quantile lemma. The Axiom of Choice Exact limits in the known-blank equal-variance normal model The standard normal CDF has unique symmetric quantiles

[F1]

Under the known-blank equal-variance normal model, $L_{C} = z_{1 - α} σ_{0}$ , $L_{D} = (z_{1 - α} + z_{1 - β}) σ_{0}$ , and $L_{Q} = σ_{0} / k_{Q}$ . Exact limits in the known-blank equal-variance normal model

[F2]

The standard normal $q$ -quantile is the unique $z_{q}$ with $Φ (z_{q}) = q$ , and $z_{1 - q} = - z_{q}$ . The standard normal CDF has unique symmetric quantiles

Verification

Given: AC; $σ_{0} > 0$ ; $α = β = 0.05 = 1 / 20$ ; $k_{Q} = 0.10 = 1 / 10$ ; the known-blank model; and the suppliers [F1]–[F2].

1.1A1F1F2algebra

Since $1 - α = 1 - β = 0.95$ , [A1] and [F1] give $L_{C} = z_{0.95} σ_{0}$ and $L_{D} = (z_{0.95} + z_{0.95}) σ_{0} = 2 z_{0.95} σ_{0}$ , with $z_{0.95}$ the unique quantile from [F2].

2.1F1step 1.1algebra∎

By [F1], the stated CV criterion gives $L_{Q} = σ_{0} / k_{Q} = σ_{0} / (1 / 10) = 10 σ_{0}$ . Together with step 1.1, this gives all three exact limits.

Source notes

The exact expressions follow from [F1]. The NIST table reports the known-blank row as $1.64$ , $3.29$ , and $10$ times the blank standard deviation for the same targets; those table entries are rounded values, not exact replacements for $z_{0.95}$ .

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

An estimated blank increases the effective normal scale

Example

Assume AC, $σ_{0} > 0$ , and $τ > 0$ . Suppose the blank-corrected response is explicitly modeled by $D_{c} \sim N (c, σ_{0}^{2} + τ^{2})$ for every $c \geq 0$ , where $τ^{2} > 0$ is the declared additional variance contribution from estimating the blank. Put $σ_{e f f} = \sqrt{σ_{0}^{2} + τ^{2}}$ . The exact normal formulas use $σ_{e f f}$ in place of $σ_{0}$ : $L_{C} = z_{1 - α} σ_{e f f}, L_{D} = (z_{1 - α} + z_{1 - β}) σ_{e f f}, L_{Q} = \frac{σ_{e f f}}{k_{Q}} .$ This is a stated comparison model, not a claim that every blank-estimation procedure produces an independent normal error with this variance. The blank-correction uncertainty must be included in the response law used to calculate the limits.

Facts & Assumptions

Given: AC; $σ_{0} > 0$ , $τ > 0$ , $0 < α, β < 1 / 2$ , $k_{Q} > 0$ , and the corrected-response law in the Example.

[A1]

AC is assumed by the known-normal limit theorem used below. The Axiom of Choice Exact limits in the known-blank equal-variance normal model

[F1]

For a known-blank equal-variance normal family with scale $σ > 0$ , the three limits are $z_{1 - α} σ$ , $(z_{1 - α} + z_{1 - β}) σ$ , and $σ / k_{Q}$ . Exact limits in the known-blank equal-variance normal model

Verification

Given: AC; $σ_{0} > 0$ , $τ > 0$ , $0 < α, β < 1 / 2$ , $k_{Q} > 0$ ; and the corrected-response model in the Example.

1.1givenA1F1algebra

Set $σ_{e f f} = \sqrt{σ_{0}^{2} + τ^{2}}$ . Since $σ_{0} > 0$ and $τ > 0$ , $σ_{e f f} > σ_{0} > 0$ . Thus the corrected-response family has exactly the form required by [F1], with $σ = σ_{e f f}$ .

2.1A1F1step 1.1algebra∎

Substitution into [F1] yields the three displayed limits. Since $0 < α, β < 1 / 2$ , both $z_{1 - α}$ and $z_{1 - β}$ are positive; since $k_{Q} > 0$ , each multiplier of $σ_{e f f}$ is positive. Together with $σ_{e f f} > σ_{0}$ , each corresponding limit is strictly larger than its known-blank value.

Source notes

Eurachem §5.3.2 and Example 4 show that routine averaging and blank correction change the standard deviation used for LOD/LOQ; §5.3.5 notes that day-to-day blank variation can require intermediate-precision data. The example's exact normal law is an explicit assumption, and the limit formulas are proved locally from [F1].

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Detection-limit assumptions and left-censored readings

Example

Consider the heteroscedastic response law $Y_{c} \sim N (c, σ_{0}^{2} (1 + ρ c)^{2})$ for $c \geq 0$ , with $σ_{0} > 0$ and $ρ \geq 0$ . The blank variance is $σ_{0}^{2}$ , but the standard deviation increases with concentration when $ρ > 0$ . For $0 < α, β < 1 / 2$ , write $a = z_{1 - β}$ . The critical level is $L_{C} = z_{1 - α} σ_{0}$ . A finite detection limit for false-negative target $β$ exists exactly when $a ρ σ_{0} < 1$ , and then $L_{D} = \frac{L_{C} + a σ_{0}}{1 - a ρ σ_{0}} .$ For $k_{Q} > 0$ , the coefficient-of-variation criterion has a finite quantification limit exactly when $k_{Q} > ρ σ_{0}$ , and then $L_{Q} = \frac{σ_{0}}{k_{Q} - ρ σ_{0}} .$ When a reporting procedure records only the category $Y \leq L_{C}$ , its likelihood contribution at concentration $c$ is the probability $P_{c} (Y \leq L_{C}) = Φ ⁣ (\frac{L_{C} - c}{σ_{0} (1 + ρ c)}),$ not a density value at $L_{C}$ .

Facts & Assumptions

Given: AC; the heteroscedastic family and parameter restrictions in the Example; and the positive-decision rule is $Y > L_{C}$ .

[A1]

AC is required by the standard normal probability-law construction. The Axiom of Choice Standard normal and normal laws

[F1]

The standard normal CDF is strictly increasing, has no atoms, and its quantiles obey $z_{q} = - z_{1 - q}$ for every $q \in (0, 1)$ . The standard normal CDF has unique symmetric quantiles

[F2]

A normal response with mean $m$ and scale $σ$ has expectation $m$ and variance $σ^{2}$ . Location-scale normal laws have their stated mean and variance

[F3]

$N (c, σ^{2})$ is the law of $c + σ Z$ for standard normal $Z$ . Standard normal and normal laws

[F4]

A real random variable's CDF is $F (x) = P (Y \leq x)$ . Cumulative distribution function of a real random variable

[F5]

The exact critical level and the false-negative target are defined relative to the specified blank and signal laws. Critical, detection, and quantification levels

[F6]

For a probability measure, $P (A^{c}) = 1 - P (A)$ . Basic identities for a probability measure

Verification

Given: The response model, positive-decision rule, and targets in the Example.

1.1A1F1F3F5F6givenalgebra

At $c = 0$ , the model is $N (0, σ_{0}^{2})$ , so [F3] represents $Y_{0} = σ_{0} Z$ in law. Set $L_{C} = z_{1 - α} σ_{0}$ ; by [F1], [F3], and complement identity [F6], $P_{0} (Y_{0} > L_{C}) = 1 - Φ (z_{1 - α}) = α$ . Also [F1] gives $z_{1 / 2} = - z_{1 / 2}$ , hence $z_{1 / 2} = 0$ and $Φ (0) = 1 / 2$ . Since $1 - α > 1 / 2$ , strict increase gives $z_{1 - α} > 0$ , and [F5] makes this an exact critical level.

1.2F2givenalgebra

By [F2], $CV (c) = σ (c) / c = σ_{0} / c + ρ σ_{0}$ for $c > 0$ . If $k_{Q} > ρ σ_{0}$ , the inequality $CV (c) \leq k_{Q}$ is equivalent to $c \geq σ_{0} / (k_{Q} - ρ σ_{0})$ , giving the displayed least value. If $k_{Q} \leq ρ σ_{0}$ , then $CV (c) > ρ σ_{0} \geq k_{Q}$ for every finite $c > 0$ , so no finite quantification limit exists.

2.1A1F1F3F5givenstep 1.1algebra

For $c > 0$ , the scale is $σ (c) = σ_{0} (1 + ρ c) > 0$ . By [F3], $P_{c} (Y \leq L_{C}) = Φ ((L_{C} - c) / σ (c))$ . Since $a = z_{1 - β} > 0$ , strict increase in [F1] makes this equal to $β$ exactly when $c - L_{C} = a σ_{0} (1 + ρ c), a = z_{1 - β} > 0.$ Rearranging gives $c (1 - a ρ σ_{0}) = L_{C} + a σ_{0}$ , whose right-hand side is positive. If $a ρ σ_{0} < 1$ , the unique solution is the positive displayed $L_{D}$ . If $a ρ σ_{0} = 1$ , the equation would equate zero to a positive number; if $a ρ σ_{0} > 1$ , its only algebraic solution is negative. Thus a nonnegative detection limit exists exactly under the stated condition.

3.1A1F1F3F4step 2.1algebra∎

If the recorded observation is the category $Y \leq L_{C}$ , its probability under the stated law is the CDF at $L_{C}$ , namely the displayed $Φ ((L_{C} - c) / σ (c))$ by [F4]. The affine normal law has no atom at $L_{C}$ by [F1] and [F3], while the CDF gives the probability of the reported half-line. Substituting $L_{C}$ as an observed point is therefore not the likelihood of this censored report.

Source notes

Eurachem §5.3.2 explains that blank correction changes the standard deviation used for LOD/LOQ; §5.3.5 discusses changing blanks across days. The heteroscedastic law and censoring rule here are explicit model assumptions. Their consequences are derived above rather than imported as universal measurement claims.

Detection and quantification limits — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Known-blank normal limits at five-percent error and ten-percent CV

Example

Facts & Assumptions

Verification

Source notes

An estimated blank increases the effective normal scale

Example

Facts & Assumptions

Verification

Source notes

Detection-limit assumptions and left-censored readings

Example

Facts & Assumptions

Verification

Source notes

Sources