Compatibility of independent lifetime measurements ($Q=({\hat{\tau }}_1-{\hat{\tau }}_2{)}^2/(s_1^2+s_2^2)$)

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 $s_1^2+s_2^2-2\mathrm{Cov}({\hat{\tau }}_1,{\hat{\tau }}_2)$.