  The difference between individual solar cycles in the magnetic butterfly
diagram can mostly be ascribed to the stochasticity of the emergence process.
We aim to obtain the expectation value of the butterfly diagram from
observations of four cycles. This allows us to further determine the generation
rate of the surface radial magnetic field. We use data from Wilcox Solar
Observatory to generate time-latitude diagrams spanning cycles 21 to 24 of the
surface radial and toroidal magnetic fields, symmetrize them across the equator
and cycle-average them. From the mean butterfly diagram and surface toroidal
field we then infer the mean poloidal field generation rate at the surface of
the Sun. The averaging procedure removes realization noise from individual
cycles. The amount of emerging flux required to account for the evolution of
the surface radial field is found to match that provided by the observed
surface toroidal field and Joy's law. Cycle-averaging butterfly diagrams
removes realization noise and artefacts due to imperfect scale separation, and
corresponds to an ensemble average that can be interpreted in the mean-field
framework. The result can then be directly compared to $\alpha\Omega$-type
dynamo models. The Babcock-Leighton $\alpha$-effect is consistent with
observations, a result that can be appreciated only if the observational data
is averaged in some way.
