Although many control charts for variables data are formally based on the assumption of normality, you can still obtain good results with nonnormal data if you collect data in subgroups. The relationship between robustness to nonnormality and sample size is based on the central limit theorem. As long as your subgroups are independent, larger subgroup sizes will tend to result in subgroup means that are more normally distributed. Although the required subgroup size depends on how nonnormal the data are, in practice, any subgroup at all is often adequate.
While transformations are not usually necessary for control charts with subgroups, if the data are very skewed, you may want to consider a Box-Cox transformation.
If you are uncertain about whether the data from your process require transformation, compare control charts with transformed and untransformed data. Then, consider whether the charts give different out-of-control signals and which signals are more useful for describing the process.
Data should be moderately normal.
Moderate departures from normality do not significantly affect the results of the chart. However, severe departures from normality can increase the number of false out-of-control signals.
If the data are very skewed, you could try a Box-Cox transformation to see if that corrects the nonnormal condition. If your process naturally produces nonnormal data and the transformation is effective, you can use the chart of the transformed data to assess the stability of your process.