This paper presents a measurement of the polarisation of W bosons from t (t) over bar t decays, reconstructed in events with one high-p(T) lepton and at least four jets. Data from pp collisions at the LHC were collected at root 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-0 = 0.709 +/- 0.019, F-L = 0.299 +/- 0.015 and F-R = - 0.008 +/- 0.014, and are the most precisely measured W boson polarisation fractions to date. Limits on anomalous couplings of the W tb vertex are set.