Many companies now are focusing on a Six Sigma program to achieve improvements. Control charts are a natural for the Six Sigma project's control phase but it seems they make good tools in the Analyze stage as well. Is this a proper use of this tool?
You are absolutely right. As you suspect, the control charts ability to distinguish between common and special causes of variation also provides important direction during the Analyze phase of DMAIC. Our response to special causes of variation must be different from the response to common cause variation: For special cause variation, we can determine and remove the cause of instability; for common cause variation, we need to redesign the process. See also Control Chart Interpretation
Actually, control charts are necessary even before the Analyze stage. They are needed in the Measure stage of DMAIC to establish a process baseline. The process baseline is an essential part of DMAIC, because this baseline is later compared to the improved process to demonstrate the project results. If you are a project sponsor, you should not accept the baseline unless it includes a control chart analysis. Without a sound baseline analysis, there is credible argument that a project may not have achieved its objectives.
For example, consider a baseline analysis of a process using confidence intervals: If the process were out of control during the baseline sample period, the project team will proceed through DMAIC, and try to verify their improvement in the latter part of the Improve stage. If the special cause is no longer present, totally unrelated to any improvement effort of the team, the team would think they had achieved their objective, when in fact that special cause had simply not been present during the Improve stage. The variation they see between the Improve stage and the Measure stage is due to nothing more than random common cause variation. The special cause, unfortunately, will likely reappear at some point in the future, and the efforts were essentially wasted!
Unfortunately, there are some Six Sigma training programs that focus on classical statistics in the Measure stage, and that is simply wrong. Confidence intervals and hypothesis tests are inappropriate tools for analyzing process data, except in the context of controlled (designed) experiments. The differences between analytical and enumerative statistics were described by Shewhart nearly a hundred years ago and Deming in the 1980s.
Learn more about the Lean Six Sigma principles and tools for process excellence in Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller, in his online Lean Six Sigma DMAIC short course ($249), or his online Green Belt certification course ($499).