New bounds are developed for the Čebyšev functional utilising an identity involving a Riemann-Stieltjes integral. A refinement of the classical Čebyšev inequality is produced for f monotonic non-decreasing, g continuous
and M(g; t, b) −M(g; a, t) ≥ 0, for t ∈ [a, b] where M(g; c, d) is the integral mean over [c, d].