= Overdispersion of detector-interval counts under shared drift
{title2=$\operatorname{Var}(M)=N\overline p(1-\overline p)+N(N-1)\operatorname{Var}(p)$}
\b[Overdispersion of detector-interval counts under shared drift] is excess count variability caused by a common experimental condition changing the chance of an interval outcome. Let $M$ count empty physical detector-width intervals among $N$ readings. Conditional on a block's calibration state $H$, suppose the readings are independent with empty probability $p(H)$, so $M\mid H\sim\operatorname{Binomial}(N,p(H))$.
Writing $\overline p=\mathbb E[p(H)]$, the <law of total variance> gives
$$
\operatorname{Var}(M)=N\overline p(1-\overline p)+N(N-1)\operatorname{Var}(p(H)).
$$
The first term is the binomial <variance> using the mean probability. The second comes from sharing one fluctuating condition across the block. Individual indicators have <covariance> $\operatorname{Var}(p(H))$ across different readings, even though they are independent after conditioning on $H$.
For an original mixture example, suppose equally likely block conditions have empty probabilities $0.01$ and $0.09$. Then $\overline p=0.05$ and $\operatorname{Var}(p(H))=0.0016$. With $N=100$, the expected count is five, but its <variance> is $4.75+15.84=20.59$, with <standard deviation> about $4.54$. An independent binomial model with probability $0.05$ would give <standard deviation> only about $2.18$.
The <effective repetition count for detector intervals with shared drift> can consequently be much smaller than the number of stored traces. Repeating measurements within one unchanged calibration block does not supply the same information about drift as acquiring independent blocks. Record acquisition grouping and reference reuse before pooling interval categories.
The <negative covariance of empty and positive detector-interval counts> applies to a fixed-probability multinomial model. Shared drift adds cross-repetition dependence and can change that count <covariance> as well as each count's <variance>. In the example, the two probability states are stipulated; a physical model would relate them to the receiver's drift, <variance>-difference estimator and uncertainty calibration.
NIST retains shared influences in uncertainty propagation and explains that confidence interpretations depend on the assumed distributions.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3; §§6.2–6.3, Expanded Uncertainty]{ref} Observed overdispersion can motivate checking the block model, but does not by itself identify which physical influence caused it.
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