  Two-particle correlation measurements and analysis are an important component
of the relativistic heavy-ion physics program. In particular, particle
pair-number correlations on two-dimensional transverse momentum ($p_t$) allow
unique access to soft, semi-hard and hard-scattering processes in these
collisions. Precise measurements of this type of correlation are essential for
understanding the dynamics in heavy-ion collisions. However, transverse
momentum correlation measurements are especially vulnerable to statistical and
systematic biases. In this paper the origins of these large bias effects are
explained and mathematical correlation forms are derived from mean-$p_t$
fluctuation quantities in the literature. Monte Carlo simulations are then used
to determine the conditions, e.g. multiplicity and collision centrality bin
widths, where each correlation form is minimally biased. The ranges of
applicability for each correlation quantity are compared. Several are found to
reproduce the assumed input correlations with reasonable fidelity over a wide
range of conditions encountered in practical analysis of data.
