  Quantum metrology is a general term for methods to precisely estimate the
value of an unknown parameter by actively using quantum resources. In
particular, some classes of entangled states can be used to significantly
suppress the estimation error. Here, we derive a formula for rigorously
evaluating an upper bound for the estimation error in a general setting of
quantum metrology with arbitrary finite data sets. Unlike in the standard
approach, where lower bounds for the error are evaluated in an ideal setting
with almost infinite data, our method rigorously guarantees the estimation
precision in realistic settings with finite data. We also prove that our upper
bound shows the Heisenberg limit scaling whenever the linearized uncertainty,
which is a popular benchmark in the standard approach, shows it. As an example,
we apply our result to a Ramsey interferometer, and numerically show that the
upper bound can exhibit the quantum enhancement of precision for finite data.
