= Effective repetition count for detector intervals with shared drift
{title2=$N_{\mathrm{eff}}$}
The <effective repetition count for detector intervals with shared drift> is the number of independent repetitions that would give the same <variance> for an average indicator as the actual dependent repetitions. In <overdispersion of detector-interval counts under shared drift>, let $X_i$ indicate an empty interval, with common probability $\overline p$ satisfying $0<\overline p<1$. If the <covariance> between distinct indicators is $v=\operatorname{Var}(p(H))$, their <correlation coefficient> is
$$
\rho=\frac{v}{\overline p(1-\overline p)}.
$$
The parent's <law of total variance> calculation therefore gives
$$
\operatorname{Var}(\overline X)
=\frac{\overline p(1-\overline p)}N\{1+(N-1)\rho\}
=\frac{\overline p(1-\overline p)}{N_{\rm eff}},
\qquad N_{\rm eff}=\frac{N}{1+(N-1)\rho}.
$$
This is an observable-specific variance equivalence, not a literal count of independent traces or a guarantee of <confidence interval> coverage. A different statistic or a different dependence pattern may require a different equivalence.
For an original feasible mixture, take $\overline p=0.1$ and $\rho=0.1$. These moments can arise when the block probability is $0$ with probability $9/19$ and $0.19$ with probability $10/19$: its <variance> is $0.009$. With $N=50$, the variance-inflation multiplier is $5.9$ and
$$
N_{\rm eff}=50/5.9\approx8.47,\qquad
\operatorname{Var}(\overline X)=\frac{0.09}{50}(5.9)=0.01062.
$$
Doubling the repetitions to $100$ only raises $N_{\rm eff}$ to about $9.17$. For a fixed positive $\rho$, its large-$N$ limit is $1/\rho$, here $10$.
The experiment must therefore record which traces share a reference or drift state. Acquiring genuinely independent calibration blocks addresses a different limitation from oversampling one block; <effective repetition count across independent calibration blocks> depends on that grouping. A <cluster bootstrap> can preserve the observed within-block dependence when blocks are suitable independent resampling units. NIST's uncertainty law explicitly retains shared-input <covariance>, supporting the need to model reference reuse rather than discard it.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
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