  Minimal Composite Higgs Models (MCHM) have long provided a solution to the
hierarchy problem of the Standard Model, yet suffer from various sources of
fine tuning that are becoming increasingly problematic with the lack of new
physics observations at the LHC. We develop a new fine tuning measure that
accurately counts each contribution to fine tuning (single, double, triple,
etc) that can occur in a theory with $n_p$ parameters, that must reproduce
$n_o$ observables. We then use a novel scanning procedure to perform a
comprehensive study of three different two-site, 4D, $SO(5)\rightarrow SO(4)$
MCHMs with all third generation fermions included, distinguished by the choice
of the lepton embeddings. These are the
MCHM$^{\textbf{5-5-5}}_{\textbf{5-5-5}}$,
MCHM$^{\textbf{5-5-5}}_{\textbf{14-14-10}}$ and
MCHM$^{\textbf{5-5-5}}_{\textbf{14-1-10}}$, where MCHM$^{q-t-b}_{l-\tau-\nu}$
has the lepton doublet partner in representation $l$, tau partner in
representation $\tau$, and so on. We find that embedding at least one massive
lepton in the symmetric $\textbf{14}$ of $SO(5)$ reduces the tuning for the
case of low top partner masses (in line with previous results), but that this
is balanced against the increased complexity of the model when one properly
accounts for all sources of fine tuning. We study both the current relative
fine-tuning of each scenario, and the future prospects. Noting that the
different scenarios behave differently with respect to future improvements in
collider measurements, we find that the
MCHM$^{\textbf{5-5-5}}_{\textbf{14-1-10}}$ enjoys a relatively low increase in
fine tuning even for a future lower bound on the top partner masses of 3.4 TeV
(or equivalently a maximum Higgs-fermion or Higgs-gluon coupling deviation of
2%).
