A Xbar control chart maps the variation of any process between identical samples. For example the number of defects per release cycles is a valid candidate for study of inherent process variations. The control chart uses two limits, UCL and LCL. The UCL for all practical purposes is Mean + 3 * Sigma and LCL is Mean – 3 * Sigma where Mean is the average of all processes and sigma is the standard deviation.
Here the underlying processes are assumed to follow a normal distribution. By virtue of the normal distribution there is a 0.27% probability that a point will fall outside the control limits. So when a process point falls outside the control limits there is no proof to say that it is caused due to a process failure. It may well be that the point falls outside due to the chance that it may be outside the control limits as the normal distribution is not a closed form distribution there is a chance for infinite and -infinite limits to occurrence.
Second if the underlying distribution is not normal or is heteroscedastic where the groups are unequal in breadth or width then a false negative or a false positive can occur. These errors are classified as Type I and Type II errors in Hypothesis testing. In such cases the normal distribution has an inherant skewness or kurtosis where a point outside the normal curve or within the area enclosed by the normal curve can be constrained to be outside or inside leading to Type I or Type II errors.
The extent of nonnormality is given by the second moment and the third moment of a probability distribution. These are termed as Skew and Kurtosis respectively. Any analysis using control charts should also be accompanied by testing the skew and the kurtosis. These may be determined by the functions skew and kurt in MS-EXCEL. If the skew and the kurt values are not zero that one has to undertake skew and kurtosis correction measures for identifying TYPE I and TYPE II errors.
Another anomaly is the shift of mean and the standard deviation that can occur between different time periods. Skewness or kurtosis can be introduced between samples taken at different time periods that it can signal a false positive or a false negative.
Although control charts are a good statistical analytical tool they must be used with caution. Hetrodascedasticity has to be handled using non parametric statistics and mostly the false inferments arise from tests of hypotheses that assess normality of the underlying distribution.