  Bell nonlocality and uncertainty relations are distinct features of quantum
theory from classical physics. Bell nonlocality concerns the correlation
strength among local observables on different quantum particles, whereas the
uncertainty relations set the lower bound of the sum or product of the variance
square of observables. Here we establish the statistical link between these two
quantum characters using the Aharonov-Vaidman identity. Therein, the upper
bounds of Bell-type inequalities are expressed in terms of the product of the
local sum of the variance square. On the other hand, instead of evaluating
local uncertainty relations, the uncertainty relations on two or more quantum
systems are upper-bounded by the amount of Bell nonlocality therein.
