  Measurements of the Hubble constant H(z) are increasingly being used to test
the expansion rate predicted by various cosmological models. But the recent
application of 2-point diagnostics, such as Om(z_i,z_j) and Omh^2(z_i,z_j), has
produced considerable tension between LCDM's predictions and several
observations, with other models faring even worse. Part of this problem is
attributable to the continued mixing of truly model-independent measurements
using the cosmic-chronomter approach, and model-dependent data extracted from
BAOs. In this paper, we advance the use of 2-point diagnostics beyond their
current status, and introduce new variations, which we call Delta h(z_i,z_j),
that are more useful for model comparisons. But we restrict our analysis
exclusively to cosmic-chronometer data, which are truly model independent. Even
for these measurements, however, we confirm the conclusions drawn by earlier
workers that the data have strongly non-Gaussian uncertainties, requiring the
use of both "median" and "mean" statistical approaches. Our results reveal that
previous analyses using 2-point diagnostics greatly underestimated the errors,
thereby misinterpreting the level of tension between theoretical predictions
and H(z) data. Instead, we demonstrate that as of today, only Einstein-de
Sitter is ruled out by the 2-point diagnostics at a level of significance
exceeding ~ 3 sigma. The R_h=ct universe is slightly favoured over the
remaining models, including LCDM and Chevalier-Polarski-Linder, though all of
them (other than Einstein-de Sitter) are consistent to within 1 sigma with the
measured mean of the Delta h(z_i,z_j) diagnostics.
