  We investigate the uncertainty in the value of the strong coupling
$\alpha_S(M_Z^2)$ when allowing it to be a free parameter in the recent MMHT
global analyses of deep-inelastic and related hard scattering data that was
undertaken to determine the parton distribution functions (PDFs) of the proton.
The analysis uses the standard framework of leading twist fixed--order
collinear factorisation in the ${\overline {\rm MS}}$ scheme. We study the
constraints on the value of $\alpha_S(M_Z^2)$ coming from the individual data
sets by repeating the NNLO and NLO global analyses at various fixed values of
$\alpha_S(M_Z^2)$ about its optimum values, spanning the range
$\alpha_S(M_Z^2)=0.108$ to $0.128$ in units of $0.001$, and making all PDFs
sets available. The inclusion of the measurements of the cross section for
inclusive $t\bar{t}$ production in the global fit allows us to explore the
correlation between the values taken for the mass $m_t$ of the top quark and
$\alpha_S(M_Z^2)$. We find that the best fit values are
$\alpha_S(M_Z^2)=0.1201\pm0.0015$ and $0.1172\pm 0.0013$ at NLO and NNLO
respectively, with the central values changing to $\alpha_S(M_Z^2)=0.1195$ and
$0.1178$ when the world average of $\alpha_S(M_Z^2)$ is used as a data point.
We investigate the interplay between the uncertainties on $\alpha_S(M_Z^2)$ and
the uncertainties on the PDFs. In particular we calculate the cross sections
for key processes at the LHC and show how the uncertainties coming from the
standard MMHT2014 PDFs and from $\alpha_S(M_Z^2)$ can be provided independently
and can be combined.
