 Detection decision threshold

A detection decision threshold is a cutoff applied to an observed analytical response to decide whether there is evidence for an analyte contribution beyond background. A decision above the cutoff can be wrong even for an analytical blank: this is a type I error. Missing a genuine contribution is a type II error. The chosen rule and the measurement procedure determine those error probabilities.^[ref]

For an original one-sided example, suppose blank responses have a known normal distribution $Y\sim N(100,5^2)$. Set $y_c=100+1.645(5)\approx 108.225$. Declaring detection when $Y>y_c$ gives about a5percent type I error probability. A real sample whose mean is exactly $108.225$ has only a50percent chance of crossing that same detection decision threshold.

Under the additional assumption of unchanged standard deviation, a mean near $116.45$ gives about95percent probability of crossing it: the mean lies another $1.645(5)$ units above $y_c$. Thus an observed-response cutoff and a limit of detection giving a specified detection probability answer different questions. These numbers are a known-variance teaching model, not a universal laboratory convention. Estimating the background, changing the error criterion or allowing signal-dependent variance changes the calculation. The threshold must be tied to the complete procedure whose detection performance is claimed.^[ref]