= Correlated spectral endpoint error
\b[Correlated spectral endpoint error] means that the errors in two estimated frequencies have nonzero <covariance>. Their effect on a frequency difference depends on that covariance as well as on the uncertainty of each endpoint. Shared errors can cancel in a difference without making the absolute frequencies accurately known.
Let endpoint errors be $\epsilon_0,\epsilon_T$, with <standard uncertainties> $u_0,u_T$ and correlation coefficient $\rho$. The error in the increment is $\epsilon_T-\epsilon_0$, so
$$
u_\Delta^2=u_T^2+u_0^2-2\rho u_Tu_0.
$$
For equal uncertainties $u$, the result is $2u^2(1-\rho)$. Independent errors give $2u^2$, perfectly shared errors give zero, and perfect anticorrelation gives $4u^2$. This is correlated-input <uncertainty propagation> applied to a difference.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, sensitivity coefficients and covariance]{ref}
An original common-offset model writes $\epsilon_0=b+\eta_0$ and $\epsilon_T=b+\eta_T$. Suppose $b$ has variance $s_b^2$ and the two independent residual errors each have variance $s^2$, all mutually independent. Each endpoint then has variance $s_b^2+s^2$ and their covariance is $s_b^2$, but
$$
\operatorname{Var}(\epsilon_T-\epsilon_0)=2s^2.
$$
With $s_b=0.3$ and $s=0.1$ in compatible frequency units, each endpoint has standard uncertainty about $0.316$, while the increment error has standard uncertainty about $0.141$. This <spectral common-mode offset rejection> follows from subtraction, rather than from knowing the common offset's value.
A shared gain error behaves differently. If $Y_t=(1+g)\omega_t+b$, the constant offset cancels but $Y_T-Y_0=(1+g)(\omega_T-\omega_0)$. A shared $1\%$ scale error therefore remains a $1\%$ increment error. Calling an error common does not establish which part cancels.
To estimate covariance, repeated measurements should preserve the reference and data-processing relationships relevant to the two endpoints. Filtering or overlapping fitting windows can introduce correlations. The <spectral frequency-increment measurement noise> contribution also depends on covariance between physical fluctuations and estimation errors; the endpoint formula alone does not justify dropping that separate term. A <two-dimensional infrared spectroscopy> statistic should be interpreted through its actual estimator and response model, not by automatically adding independent noise to two abstract frequency labels.
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