This paper presents a measurement of the polarisation of W bosons from $$t\bar{t}$$ decays, reconstructed in events with one high- $$p_{\text{ T }}$$ 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.