Consider, we want to study the stability of a process with respect to a current metric score of a process. It's a new process. We define control limits based on current metric score average. Based on the identified control limits and with the help of control chart we say the process is stable since there isn't any special cause variation. What if the defect rate against the metric is still high since the control limits are defined by process and client is not in picture here ? Provided the majority of people understanding in that process about a particular procedure is wrong which directly impacts the metric. In that case, how can we rely on control chart to determine process stability of a process ?

Excellent question, Senthilnathan. 

If the process is consistently generating high percentage of defects, it may pass the stability test and is considered to be in statistical control. However, it might fail badly on the Capability Assessment if customer is unhappy with the defect level. Now, I have used the word "might" because the unhappiness of customer depends on the competitive scenario or the novelty of the solution. 

To provide you an example, there is an app called "Skyscanner" for which you can compare prices for the same flights over different platforms. It is an aggregator of aggregators. In US, customers of Skyscanner are delighted because they are saving in most of their flights or stay although, the tool is not perfect and does show erroneous results sometimes. 

If Skyscanner is having a defect rate which is unusually high (compared to other aggregator apps) but still makes customer happy due to lack of competition as an aggregator of aggregators, its process may be considered stable as well as capable. 

By the way, Skyscanner works in India too and is worth a try ")

Process Stability refers to the consistent output of a process. Stable process implies that the process is in control and giving a consistent output. If the process performance is not meeting the customer requirements, then we will have to improve the process. Improving the process becomes easier if it is in control.