Shewhart, Cumulative sum and Exponentially weighted moving average control charts are commonly used to monitor the process mean and variance in quality control. Traditionally, the mean and variance charts are plotted and evaluated separately and this can cause false inferences. Sometimes special causes in the industry can result in changes in both mean and variance. Therefore, joint monitoring schemes for mean and variance have been developed for simultaneously monitoring the mean and variance of a process. Currently there are four Shewhart joint monitoring schemes available for monitoring mean and variance. In all these schemes, a state which is having very smaller variance when the mean is in-control, is taken as an out-of-control situation and issues false alarm. Actually this state is very desirable to the process. This study computes the false alarms for the existing Shewhart joint monitoring schemes under different shifts of mean and variance for the selected in-control ARLs setup. A new joint monitoring scheme is introduced with its design procedure to avoid the false alarming. The performance of the new scheme has been compared with existing Shewhart joint monitoring schemes. This new scheme performs better or equally well in most of the scenarios. The significance of this new scheme is, it has the higher ability to eliminate the false alarm.