= Noise-limited detection threshold
{title2=$T=k\sigma$}
A \b[noise-limited detection threshold] is the decision boundary above which a measured statistic is declared to contain a signal, with the boundary chosen from a specified noise model and decision criterion. It belongs to a <measurement procedure>: filtering, observation time, baseline correction and the definition of a positive detection affect the result. A threshold in observed data is distinct from the true signal level required for a specified <detection probability>.
For an original one-sided Gaussian model, let $Y=A+N$, where $A\ge0$ is the signal and $N\sim\mathcal N(0,\sigma^2)$. Declare a detection when $Y>T$ and choose $T=k\sigma$. If $\Phi$ is the <cumulative distribution function> of the <standard normal distribution>, then
$$
\alpha=\Pr(Y>T\mid A=0)=1-\Phi(k),\qquad
P_{\rm det}(A)=1-\Phi\!\left(k-\frac A\sigma\right).
$$
Here $\alpha$ is the per-observation <false alarm rate>. The model assumes the same additive noise distribution with and without the signal.
With $k=5$, the noise-only crossing probability is approximately $2.87\times10^{-7}$ for one independent observation. Yet a signal of size $A=5\sigma$ is detected only half the time: it puts the distribution's center exactly at the decision boundary. “Five times noise” therefore does not by itself mean reliable detection of a signal that large.
To require both a chosen false-positive probability $\alpha$ and a missed-detection probability $\beta$, this model gives
$$
T=\sigma\Phi^{-1}(1-\alpha),\qquad
A_{\rm required}=\sigma\bigl[\Phi^{-1}(1-\alpha)+\Phi^{-1}(1-\beta)\bigr].
$$
For $\alpha=\beta=0.05$, the two values are approximately $1.645\sigma$ and $3.290\sigma$. This distinction accords with the VIM definition of a <limit of detection>, which specifies both kinds of decision error rather than equating the limit with the observed threshold.<International vocabulary of metrology by JCGM>[VIM3 entry 4.18, definition and note 1]{ref}
The experiment should measure baseline fluctuations under the same <bandwidth>, loading and observation rule used for signal detection, then test known signal levels. Repeated searching also matters: for $M$ independent noise-only observations, the chance of at least one threshold crossing is $1-(1-\alpha)^M$. Correlated samples require a different calculation. Baseline drift, non-Gaussian fluctuations and signal-dependent noise likewise invalidate a threshold inferred from the simple model without further characterization.
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