Question 6 of 7: Sensitivity, Selectivity, and Detection Reliability

Question 6: Sensitivity, Selectivity, and Detection Reliability (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

a) Sensitivity is only meaningful relative to the instrument's own noise floor: a small true input produces a small output change, and that change is only detectable if it is large enough to be distinguished from the random fluctuations already present in the baseline (noise) response. However steep the instrument's nominal calibration slope, the practical smallest signal it can actually resolve is set by where the expected signal change becomes comparable to the noise — so the signal-to-noise ratio, not the calibration slope alone, is what ultimately determines achievable sensitivity.

b) The lowest detectable signal (limit of detection) is defined as the smallest input that produces a response reliably distinguishable from the response to a blank (zero-input) sample — conventionally the input whose response exceeds the mean blank response by some fixed multiple (commonly three times) the standard deviation of the blank/noise measurement.

c) Given. Repeated measurements of a blank (zero concentration/zero-input) sample, and the calibration curve's sensitivity (slope) near zero.

Find. The lowest detectable signal, in input units.

Approach. Characterize the blank's noise statistically, then convert the detection threshold from output units to input units through the calibration slope.

1. Blank statistics. Measure the blank repeatedly and compute its mean response $\bar y_{blank}$ and standard deviation $\sigma_{blank}$.
2. Detection threshold in output units. Take the response threshold as $$y_{LOD}=\bar y_{blank}+3\,\sigma_{blank}$$ (the conventional $3\sigma$ criterion for reliable detection, as distinct from the stricter $10\sigma$ criterion sometimes used for reliable quantification).
3. Convert to input units. $$\boxed{\text{LOD}=\dfrac{y_{LOD}-\bar y_{blank}}{\text{sensitivity}}=\dfrac{3\sigma_{blank}}{dy/dx}}$$ — the LOD improves (gets smaller/better) with either a quieter blank ($\sigma_{blank}\downarrow$) or a steeper calibration slope ($dy/dx\uparrow$).

d) A selectivity ratio expresses how strongly the instrument responds to the target measurand relative to a specific interfering species, typically as the ratio of the sensor's sensitivity (response per unit concentration/level) to the target divided by its sensitivity to the interferent, $$\text{selectivity ratio}=\dfrac{\text{sensitivity}_{target}}{\text{sensitivity}_{interferent}}.$$ A large ratio means the instrument responds much more strongly to the intended measurand than to that interferent.

e) Low sensitivity leaves the true signal close to (or below) the noise floor, so a genuinely present target is missed — increasing the rate of false negatives. Low selectivity means an interferent produces a response large enough to be mistaken for the target, which raises the rate of false positives. A reliable instrument needs both properties together: high sensitivity alone (with poor selectivity) trades false negatives for false positives, and high selectivity alone (with poor sensitivity) does the reverse — neither error type is controlled by only one of the two properties.

f) A false negative on a toxic-gas detector means workers are told an atmosphere is safe when it is not, exposing them to potential injury or death; this is a direct duty-of-care and occupational health and safety failure (under Canadian provincial OH&S legislation, e.g. WorkSafeBC or CCOHS-aligned regulations, the employer has a positive obligation to provide instruments that reliably detect the hazards present), and a documented failure to select or maintain a sufficiently sensitive/selective instrument is a foreseeable-harm finding in any subsequent incident investigation, exposing the employer and instrument supplier to liability. A false positive is less immediately dangerous but is not free of risk: repeated unnecessary alarms/evacuations erode confidence in the system, and a workforce that has learned to distrust "cry wolf" alarms is more likely to ignore or delay response to a subsequent genuine alarm — converting a nuisance cost into a delayed-response liability of its own, in addition to the direct costs of lost production and unnecessary shutdowns.