  We study the fluctuations of the spin per site around the thermodynamic
magnetization in the mean-field Blume-Capel model. Our main theorem generalizes
the main result in a previous paper (Ellis, Machta, and Otto) in which the
first rigorous confirmation of the statistical mechanical theory of finite-size
scaling for a mean-field model is given. In that paper our goal is to determine
whether the thermodynamic magnetization is a physically relevant estimator of
the finite-size magnetization. This is done by comparing the asymptotic
behaviors of these two quantities along parameter sequences converging to
either a second-order point or the tricritical point in the mean-field
Blume-Capel model. The main result is that the thermodynamic magnetization and
the finite-size magnetization are asymptotic when the parameter $\alpha$
governing the speed at which the sequence approaches criticality is below a
certain threshold $\alpha_0$. Our main theorem in the present paper on the
fluctuations of the spin per site around the thermodynamic magnetization is
based on a new conditional limit theorem for the spin, which is closely related
to a new conditional central limit theorem for the spin.
