Many data comes from a process with variables having non-normal distributions. The commonly used Shewhart variable control charts which depend heavily on the normality assumption should not be applied here. Further, measurement error of an instrument or gauge is an important factor in industry that influences the outcome of a process. Incapability of the instrument or gauge would cause the observation measure deviates from the true value and consequently lead to a false discrimination between good and bad service quality and manufacturing parts. In this study, we propose a new EWMA control chart to monitor the exponentially distributed service time between consecutive events with the measurement error instead of monitoring the number of events in a given time interval. We showed that, in the case of the measurement error following a normal distribution, the performance of the EWMA-M chart is significantly affected. We suggest letting for keeping the chart with the same detection ability; otherwise, the chart’s detection ability would be much poor than the charts with no measurement errors.