Monitoring schemes are typically designed under the assumption of perfect measurements. However, in real-life applications, data tend to be subjected to measurement errors, that is, a difference between the real quantities and the measured ones mostly exist even with highly sophisticated advanced measuring instruments. Thus, in this paper, the negative effect of measurement errors on the performance of the homogenously weighted moving average (HWMA) scheme is studied using the linear covariate error model for constant and linearly increasing variance. Monte Carlo simulations are used to evaluate the performance of the proposed HWMA scheme in terms of the run-length characteristics. It is observed that as the smoothing parameter increases, measurement errors have a higher negative effect on the performance of the HWMA X¯¯¯ scheme. More importantly, it is shown that the negative effect of measurement errors is reduced by using multiple measurements and/or by increasing the slope coefficient of the covariate error model. Moreover, the performance of the HWMA X¯¯¯ scheme is compared with the corresponding exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) X¯¯¯ schemes. An illustrative example is provided to help in implementing this monitoring scheme in a real-life situation.