All eddy-covariance flux measurements are associated with random uncertainties which are a combination of sampling error due to natural variability in turbulence and sensor noise. The former is the principal error for systems where the signal-to-noise ratio of the analyser is high, as is usually the case when measuring fluxes of heat, CO2 or H2O. Where signal is limited, which is often the case for measurements of other trace gases and aerosols, instrument uncertainties dominate. Here, we are applying a consistent approach based on auto- and cross-covariance functions to quantify the total random flux error and the random error due to instrument noise separately. As with previous approaches, the random error quantification assumes that the time lag between wind and concentration measurement is known. However, if combined with commonly used automated methods that identify the individual time lag by looking for the maximum in the cross-covariance function of the two entities, analyser noise additionally leads to a systematic bias in the fluxes. Combining data sets from several analysers and using simulations, we show that the method of time-lag determination becomes increasingly important as the magnitude of the instrument error approaches that of the sampling error. The flux bias can be particularly significant for disjunct data, whereas using a prescribed time lag eliminates these effects (provided the time lag does not fluctuate unduly over time). We also demonstrate that when sampling at higher elevations, where low frequency turbulence dominates and covariance peaks are broader, both the probability and magnitude of bias are magnified. We show that the statistical significance of noisy flux data can be increased (limit of detection can be decreased) by appropriate averaging of individual fluxes, but only if systematic biases are avoided by using a prescribed time lag. Finally, we make recommendations for the analysis and reporting of data with low signal-to-noise and their associated errors.