Choose. The test results indicate whether you should reject or fail to reject the null hypothesis that the data come from a normally distributed population. You can do a normality test and produce a normal probability plot in the same analysis. The normality test and probability plot are usually the best tools for judging normality, especially for smaller samples.
The following are types of normality tests that you can use to assess normality.
Anderson-Darling and Kolmogorov-Smirnov tests are based on the empirical distribution function. Ryan-Joiner (similar to Shapiro-Wilk) is based on regression and correlation.
All three tests tend to work well in identifying a distribution as not normal when the distribution is skewed. All three tests are less distinguishing when the underlying distribution is a t-distribution and nonnormality is due to kurtosis. Usually, between the tests based on the empirical distribution function, Anderson-Darling tends to be more effective in detecting departures in the tails of the distribution. Usually, if departure from normality at the tails is the major problem, many statisticians would use Anderson-Darling as the first choice.
If you are checking normality to prepare for a normal capability analysis, the tails are the most critical part of the distribution.