Statistical tests for normality
There are statistical tests for normality such as the Shapiro-Wilk and Kolmogorov-Smirnoff tests but for small sample sizes (n < 20), the tests are unlikely to detect non-normality and for larger sample sizes (n > 50), the tests can be too sensitive. They are also sensitive to outliers so use histograms (large samples) or QQ plots (small samples).
Parametric test
What to check for normality
Independent t-test
Dependent variable by group
Paired t-test
Paired differences
One-way ANOVA
Residuals
Repeated measures ANOVA
Residuals at each time point
Pearson’s correlation coefficient
Both variables are normally distributed
Simple linear regression
Residuals