This paper presents a measurement of the polarisation of W bosons from tt¯tt¯ decays, reconstructed in events with one high- p T p T lepton and at least four jets. Data from pp collisions at the LHC were collected at s√s = 8 TeV and correspond to an integrated luminosity of 20.2 fb −1−1 . The angle θ∗θ∗ 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 cosθ∗cosθ∗ 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 F0= 0.709 ± 0.019F0= 0.709 ± 0.019 , FL= 0.299 ± 0.015FL= 0.299 ± 0.015 and FR= −0.008 ± 0.014FR= −0.008 ± 0.014 , and are the most precisely measured W boson polarisation fractions to date. Limits on anomalous couplings of the Wtb vertex are set.