Run a blank — reagents, no analyte — and the instrument does not read zero. It reads a small, wandering number that changes every time you measure it: electronic noise, a trace of contamination, the baseline moving. Add a little analyte and its signal has to climb out of that wandering blank to be seen; near the bottom of the curve the two are the same size. There is a level below which a real signal cannot be told from a lucky-high blank — you cannot honestly say the analyte is there — and a higher one below which you can say it is there but cannot pin down how much with useful precision. Those are the limit of detection and the limit of quantitation, and getting them right is the difference between an honest trace result and a confident guess.
Two ingredients: how much the blank wanders, how steeply you respond
A limit in concentration is a multiple of the blank's noise divided by the calibration slope — so both matter. The noise is the variability of the blank (or low-level) signal, its standard deviation σblank (not its mean, which you subtract off). The slope S is the response per unit concentration. A quieter blank or a steeper slope both push the floor lower; a restless blank raises it, and a sensitive instrument (large S) partly offsets that. The whole subject is one ratio asked twice — how far above the noise a signal must be before you believe it, and how far before you can measure it.
Detection is a decision — with two ways to be wrong
"Is the analyte present?" is a yes/no call made on a noisy signal, so it can fail two independent ways: call a blank "detected" when nothing is there (a false positive), or miss a real low sample as "not detected" (a false negative). Currie's framework — followed by IUPAC and ISO 11843, though not by every standard — separates these into two different kinds of number, and conflating them is the most common mistake in the whole topic.1
The critical value (LC) is a threshold you apply to one result's signal: read above it, declare "detected." It is set so a true blank rarely crosses it — false positives at rate α. For a normal blank of known σ, one-sided, α = 0.05, that is LC ≈ 1.65·σblank in signal (≈ 1.65·σ/S once divided through the slope). This is the line each individual result is judged against.
The detection limit (LD, the "LOD") is a different thing: not a per-result cutoff but the method's capability — the lowest true concentration whose signal will clear LC reliably (false negatives at rate β). A true level sitting right at LC would be missed half the time, so LD is set higher. With α = β = 0.05 and constant known σ, LD ≈ (1.65 + 1.65)·σ = 3.3·σblank. So you decide each result against LC and you quote LD as the method's LOD — and the "3.3" is two error rates added, not a bare 3σ.
The working formulas — and which convention you are in
Divide those signal thresholds by the slope S to get concentrations. Here the conventions genuinely diverge, so name the one you mean:
ICH collapses the two-threshold picture into one number, LOD = 3.3σ/S, without stating α and β; EPA's method detection limit is a third construction (below); the S/N "3 and 10" are a separate instrument convention, not the 3.3. They are not interchangeable, and a limit is only comparable to another when both name the same recipe. The LOQ uses 10 because measuring is a higher bar than detecting: a signal ten times the noise gives a relative standard deviation near 10% — provided the scatter at that level is still about the blank's. Where variance grows with concentration, 10σblank is optimistic, and the honest LOQ is set from a directly measured precision at a low spike.
Those factors hold only for a normal, constant-variance blank with known σ, using the normal-curve z-values (1.65, 3.29). Two things break them in practice. If σ is estimated from a few replicates — as it always is — the honest factor uses Student's t (larger, and larger still for small n), not z; ICH's habit of plugging an estimated s into the known-σ factor is a simplification. And residual scatter from a wide-range regression is not the blank's σ, and the SD of an intercept depends on the design of the standards — so a limit built from a curve fitted far above the floor can misstate it. A subtracted paired blank — rather than a well-characterised blank mean — adds a factor of √2 to the effective noise. And for counting detectors the noise itself grows as √signal, so none of the fixed factors apply cleanly. When it matters, work from ISO 11843 (or the Hubaux–Vos treatment), not a rule of thumb.4
How σ is actually estimated — and why numbers disagree
The formulas are simple; the honesty is in the σ, and there are several accepted ways to get it that do not agree. From repeated blanks or low spikes: measure many and take their standard deviation directly (the basis of the regulatory limit below). From the calibration: ICH allows the residual standard deviation or the SD of the intercept. A third, separate route does not estimate σ at all: signal-to-noise, where instrument software reports the limit directly at S/N ≈ 3 or 10 — quick, but meaningful only when the baseline noise is genuinely characterised (RMS versus peak-to-peak changes the number, and smoothing or a hand-picked noise window can quietly game it). Each approach is legitimate; each can give a different limit for the same method — so state which you used. And an s from a handful of replicates is itself uncertain (its own relative uncertainty is about 1/√(2·df)) — which is why a limit is an estimate with scatter, quoted to two significant figures, never as a sharp edge.
Instrument, or method?
A limit measured on a clean standard in solvent — an instrument detection limit — flatters the method, because it never saw the extraction, the dilution, or the sample's matrix. The number that means something is the method detection limit (MDL): low-level spikes carried through the entire procedure in the real matrix, so its noise is the noise you actually fight. The U.S. regulatory MDL (40 CFR Part 136, Appendix B) is a specific recipe — replicate spikes (n ≥ 7), MDL = t(n−1, 0.99) · s, at 99% confidence, and since 2016 also computed from method blanks with the larger of the two taken. Note what that is: a false-positive-controlled, LC-like threshold at α = 0.01, not a Currie LD at α = β = 0.05 — a different construction wearing the word "limit."3 A dirtier matrix raises the noise floor (and often changes the slope too), so the in-matrix limit sits well above a reagent-blank one — report the limit that belongs to the sample.
Reporting honestly across the floor
The thresholds carve the low end into honest zones — but judge each result against the decision threshold, not the method's quoted LOD. A single result whose signal is below the critical value LC is not detected (report "< LC," or the framework's reporting limit). A result above LC is a detection — including one that falls between LC and the method's quoted LOD, which is a real detection sitting below the level the method is guaranteed to catch, not a non-detect. If it is still below the LOQ it is a detected trace — report it with a qualifier (a "J" flag or "< LOQ"), an estimated number, not one to be trusted like a value above the LOQ. At or above the LOQ, report the value plainly. (Regulatory frameworks that set a single reporting limit at the MDL are, in effect, using an LC-type threshold for the decision — consistent with this, once you see the MDL for what it is.) Two quiet sins: quoting a below-LOQ number as if it were quantitative, and — worse — substituting zero, the LOD, or LOD⁄2 for non-detects and letting that fiction average into a result. A non-detect is censored, not zero; where such data must be summarised, it needs statistics built for censoring, not a made-up fill value.
Reagent blanks carried through the method give a signal standard deviation of about σ = 2.0 (arbitrary units); the calibration slope is S = 0.65 units per ppb. Taking the Currie known-σ convention for teaching:
| Quantity | Value | Meaning |
|---|---|---|
| LC = 1.65·σ/S | ≈ 5 ppb | decide each result against this |
| LOD = 3.3·σ/S | ≈ 10 ppb | the method capability you quote |
| LOQ = 10·σ/S | ≈ 31 ppb | lowest you report a number |
So a result under ~5 ppb is not detected; ~5–31 ppb is a detected trace, reported with a qualifier; ~31 ppb and up is reported as a value. Caveats the numbers hide: σ here was estimated, so rigorous factors run a little larger than 1.65 and 3.3 — Student's t for the critical value, and a non-central-t / ISO 11843 treatment for the detection limit, not simply a doubled t. An EPA method detection limit is a different construction again: at least seven spiked replicates, MDL = t(6,0.99)·s ≈ 3.14·s at 99%, an LC-type threshold, not the 3.3σ LD. And a matrix that doubles σ doubles the limits only if it leaves the slope alone — which it often does not.
Where it sits in the family
This is the bottom of the very curves the last pieces built. It is where the low-end recovery that an R² hides finally becomes unmeasurable; where a matrix effect bites hardest because the matrix is the noise; where the replicate scatter that an internal standard tightens sets the floor directly. External standard, standard addition, internal standard — every one of them runs out here, at the level where the signal is no longer reliably above the blank. Each calibration has an honest bottom, and naming it — with the right threshold, from the right σ, on the real method — is part of owning the measurement.
The judgement
A detection limit is not a property of the instrument you can look up; it is a claim about your blank, your slope, your matrix, and the error rates you chose — only as good as the σ behind it and the honesty of how it was found. Say which convention and which σ you used, keep the per-result decision separate from the method's quoted capability, tie the number to the real method, quote it modestly, and let it govern what you report. The instrument will always show a number below the floor; the skill is knowing when it means "present," when it means "measured," and when it means nothing at all.
A foundations piece — the reasoning behind detection and quantitation limits, not a validation protocol. For the exact procedure, error rates, and how to determine and verify a limit, work from ICH Q2(R2), the IUPAC/Currie recommendations and ISO 11843, the applicable EPA/ASTM method, and your laboratory's quality system.