We present a measurement of the Lyman α flux probability distribution function (PDF) obtained from a set of eight high-resolution quasar spectra with emission redshifts in the range 3.3 ≤z≤ 3.8. We carefully study the effect of metal absorption lines on the shape of the PDF. Metals have a larger impact on the PDF measurements at lower redshift, where there are relatively fewer Lyman α absorption lines. This may be explained by an increase in the number of metal lines that are blended with Lyman α absorption lines towards higher redshift, but may also be due to the presence of fewer metals in the intergalactic medium (IGM) at earlier times. We also provide a new measurement of the redshift evolution of the effective optical depth, τeff, at 2.8 ≤z≤ 3.6, and find no evidence for a deviation from a power-law evolution in the log (τeff)–log (1 +z) plane. The flux PDF measurements are furthermore of interest for studies of the thermal state of the IGM at z≃ 3. By comparing the PDF to state-of-the-art cosmological hydrodynamical simulations, we place constraints on the temperature of the IGM and compare our results with previous measurements of the PDF at lower redshift. At redshift z= 3, our new PDF measurements are consistent with an isothermal temperature–density relation, T=T0Δγ− 1, with a temperature at the mean density of T0= 19 250 ± 4800 K and a slope γ= 0.90 ± 0.21 (1σ uncertainties). In comparison, joint constraints with existing lower redshift PDF measurements at z < 3 favour an inverted temperature–density relation with T0= 17 900 ± 3500 K and γ= 0.70 ± 0.12, in broad agreement with previous analyses.