The joint monitoring of the process mean and variance can be achieved by running what is termed a joint scheme. The process is deemed out-of-control whenever a signal is observed on either individual chart of a joint scheme. Thus, the two following events are likely to happen:
- a signal is triggered by the chart for the mean although it is on-target and the standard deviation is off-target;
- the mean is out-of-control and the variance is in-control, however, a signal is given by the chart for the standard deviation.
Signals such as these are called misleading signals (MS) and can possibly send the user of the joint scheme to try to diagnose and correct a nonexistent assignable cause. Thus, the need to evaluate performance measures such as:
- the probability of a misleading signal (PMS); and
- the number of sampling periods before a misleading signal is given by the joint scheme, the run length to a misleading signal (RLMS).
We present some striking and instructive examples that show that the occurrence of misleading signals should be a cause of concern in practice.
We also establish stochastic monotonicity properties for RLMS and monotone behaviors for PMS, which have important implications in practice in the assessment of the performance of joint schemes for the mean and variance of process output.

CEMAT - Center for Computational and Stochastic Mathematics