  Here we report that the observed braking indices of the 366 pulsars in the
sample of Hobbs et al. range from about $-10^8$ to about $+10^8$ and are
significantly correlated with their characteristic ages. Using the model of
magnetic field evolution we developed previously based on the same data, we
derived an analytical expression for the braking index, which agrees with all
the observed statistical properties of the braking indices of the pulsars in
the sample of Hobbs et al. Our model is, however, incompatible with the
previous interpretation that magnetic field growth is responsible for the small
values of braking indices ($<3$) observed for "baby" pulsars with
characteristic ages of less than $2\times 10^3$ yr. We find that the
"instantaneous" braking index of a pulsar may be different from the "averaged"
braking index obtained from fitting the data over a certain time span. The
close match between our model-predicted "instantaneous" braking indices and the
observed "averaged" braking indices suggests that the time spans used
previously are usually smaller than or comparable to their magnetic field
oscillation periods. Our model can be tested with the existing data, by
calculating the braking index as a function of the time span for each pulsar.
In doing so, one can obtain for each pulsar all the parameters in our magnetic
field evolution model, and may be able to improve the sensitivity of using
pulsars to detect gravitational waves.
