  This paper presents a measurement of the polarisation of $W$ bosons from
$t\bar{t}$ decays, reconstructed in events with one high-$p_{\mbox{T}}$ lepton
and at least four jets. Data from $pp$ collisions at the LHC were collected at
$\sqrt{s}$ = 8 TeV and correspond to an integrated luminosity of 20.2
fb$^{-1}$. The angle $\theta^{*}$ between the $b$-quark from the top quark
decay and a direct $W$ boson decay product in the $W$ boson rest frame is
sensitive to the $W$ boson polarisation. Two different $W$ decay products are
used as polarisation analysers: the charged lepton and the down-type quark for
the leptonically and hadronically decaying $W$ boson, respectively. The most
precise measurement of the $W$ boson polarisation via the distribution of
$\cos{\theta^{*}}$ is obtained using the leptonic analyser and events in which
at least two of the jets are tagged as $b$-quark jets. The fitted fractions of
longitudinal, left- and right-handed polarisation states are $F_{\mathrm{0}}=
0.709 \pm {0.019}$, $F_{\mathrm{L}}= 0.299 \pm {0.015}$ and $F_{\mathrm{R}}=
-0.008 \pm {0.014}$, and are the most precisely measured $W$ boson polarisation
fractions to date. Limits on anomalous couplings of the $Wtb$ vertex are set.
