= Compatibility of independent lifetime measurements
{title2=$Q=(\widehat\tau_1-\widehat\tau_2)^2/(s_1^2+s_2^2)$}
If two independent approximately Gaussian <mean lifetime> estimates have known <standard deviations> $s_1,s_2$, equality of their true lifetimes implies that their difference has mean zero and <variance> $s_1^2+s_2^2$. Its squared standardized difference therefore has a one-degree-of-freedom <chi-squared distribution>, giving the statistic in the title. A chosen significance threshold converts its upper-tail <P-value> into a compatibility decision.
Testing a fitted decay curve against a fixed numerical lifetime is a different question: that test does not include the external measurement uncertainty. Comparing the two experiments requires both uncertainties or a joint likelihood with a common lifetime. Correlated estimates replace the denominator by $s_1^2+s_2^2-2\operatorname{Cov}(\widehat\tau_1,\widehat\tau_2)$.
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