Context. Canary is the multi-object adaptive optics (MOAO) on-sky pathfinder developed in the perspective of multi-object spectrograph on extremely large telescopes (ELTs). In 2013, Canary was operated on-sky at the William Herschel telescope (WHT), using three off-axis natural guide stars (NGS) and four off-axis Rayleigh laser guide stars (LGS), in open-loop, with the on-axis compensated turbulence observed with a H-band imaging camera and a Truth wave-front sensor (TS) for diagnostic purposes.Aims. Our purpose is to establish a reliable and accurate wave-front error breakdown for LGS MOAO. This will enable a comprehensive analysis of Canary on-sky results and provide tools for validating simulations of MOAO systems for ELTs.Methods. To evaluate the MOAO performance, we compared the Canary on-sky results running in MOAO, in single conjugated adaptive optics (SCAO) and in ground layer adaptive optics (GLAO) modes, over a large set of data acquired in 2013. We provide a statistical study of the seeing. We also evaluated the wave-front error breakdown from both analytic computations, one based on a MOAO system modelling and the other on the measurements from the Canary TS. We have focussed especially on the tomographic error and we detail its vertical error decomposition. Results. We show that Canary obtained 30.1%, 21.4% and 17.1% H-band Strehl ratios in SCAO, MOAO and GLAO respectively, for median seeing conditions with 0.66′′ of total seeing including 0.59′′ at the ground. Moreover, we get 99% of correlation over 4500 samples, for any AO modes, between two analytic computations of residual phase variance. Based on these variances, we obtain a reasonable Strehl-ratio (SR) estimation when compared to the measured IR image SR. We evaluate the gain in compensation for the altitude turbulence brought by MOAO when compared to GLAO.