LOD and LOQ calculator

Detection and quantitation limits by ICH Q2(R2), from every σ your data supports at once — because the same method gives different limits depending on which σ you choose, and that is worth seeing.

The formula

LOD = 3.3 σ / S LOQ = 10 σ / S S/N route: LOD = C × 3 / (S/N) LOQ = C × 10 / (S/N)

σ: standard deviation of the response — the SD of the y-intercept, the residual SD S(y/x), or the SD of the blank
S: slope of the calibration line, in response units per concentration unit
C: concentration of the low-level standard whose peak you measured
S/N: peak height divided by baseline noise, both as heights

Worked example

Five standards, three σ, three answers

The same five standards as the calibration page — 2.00, 5.00, 10.00, 20.00 and 50.00 µg/mL giving peak-area ratios of 0.1140, 0.2695, 0.5301, 1.0462 and 2.6105. The fit has a slope of 0.05201, an intercept standard error of 0.001271 and S(y/x) = 0.002010.

3.3σ/S on the intercept standard error gives LOD = 0.08064 µg/mL and LOQ = 0.2444 µg/mL. The slope and intercept SE above are the full-precision fit rounded for display — not the operands the calculator used. From S(y/x), the same curve gives LOD = 0.1275 µg/mL and LOQ = 0.3865 µg/mL. Ten blank injections with an illustrative standard deviation of 0.0015 — a figure chosen to show the third route, not measured on an instrument — would give a third pair, 0.09518 and 0.2884 µg/mL.

One method, one data set, and the detection limit ranges from 0.08064 to 0.1275 µg/mL depending only on which σ was chosen — a spread of 58%. Nothing about the method changed.

The chromatographic route works from a trace instead. A 0.20 µg/mL standard giving a peak 45 units high on baseline noise of 3 has S/N = 15.00, so the concentration at S/N 3 is 0.20 × 3 / 15 = 0.04000 µg/mL and at S/N 10 it is 0.1333 µg/mL.

Common pitfalls

Your LOD sits below your lowest standard, so it is an extrapolation
The worked example proposes 0.08064 µg/mL from standards that start at 2.00 µg/mL — a level the method was never tested at. ICH is explicit that the limit must be verified by analysing samples at or near it. The arithmetic proposes a detection limit; only the experiment establishes one. If the proposed limit is far below your lowest standard, the honest next step is to prepare standards down there and find out.
Three defensible σ, three different answers
The SD of the y-intercept, the residual standard deviation and the SD of the blank are all sanctioned by the same guideline, and on the worked example they span 0.08064 to 0.1275 µg/mL. Which one you report belongs in the protocol, written before the data exists. Choosing afterwards, once you can see which gives the prettiest number, is choosing your own detection limit — and an auditor who asks why this σ will not be satisfied by the answer that it was the smallest.
The intercept standard error depends on where your standards sit
It is a function of the spread of the concentrations and their distance from zero, not only of the noise. Add a standard near the bottom of the range and the intercept standard error falls, taking the LOD with it, while the method's actual noise is unchanged. S(y/x) does not behave that way — it is the scatter about the line wherever the standards happen to be. That difference is the strongest argument for reading both rows rather than picking one.
A perfect fit gives a limit of zero, and this page refuses to print it
Standards that fall exactly on a line leave the curve nothing to measure noise with, usually because there were too few of them. What they do not leave is a residual standard deviation of exactly zero: in floating-point arithmetic the residuals collapse to the rounding floor instead, about one part in 10¹⁶ of the size of the responses, and 3.3σ/S turns that into a detection limit near 10⁻¹⁵ rather than 0.000. Either number is arithmetic, not chemistry. This page treats any σ that is negligible against the spread of the responses as no measurement of noise at all: the row is dropped with a note, and the other rows are unaffected.
Signal-to-noise uses peak height, and assumes the noise stays put
The noise is a height, so the ratio must be built from peak height — a ratio of area to noise is not a number. Scaling a measured peak down to S/N 3 also assumes response stays proportional to concentration and that the baseline noise is the same down there as it was at the concentration you measured. Near the baseline both assumptions flatter the method, which is why the σ/S routes and this one rarely agree.
Decimal commas and thousands separators
Values in the concentration and response columns are separated by commas, newlines, tabs and spaces, so a decimal comma cannot also be a decimal point: typing “0,5301” into the responses column reads as two values, not one. Use a dot. Numbers grouped like “1,234,567” — the way a spreadsheet formats a peak area — are rejected outright and listed as unread rather than guessed at.

Where 3.3 comes from

A detection limit has to control two errors: calling a blank a detection, and missing a real one. Allowing 5% for each, one-sided, costs 1.645 standard deviations apiece, and 2 × 1.645 = 3.29 — which is where the conventional 3.3 comes from. That derivation belongs to the detection-limit literature the convention grew out of; ICH states the factor rather than arguing for it. The factor of 10 for the quantitation limit is a convention rather than a derivation: it corresponds to a relative standard deviation of about 10% at the limit, which is the point at which a result is usually considered quantitative. Neither factor is adjustable here, deliberately.

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