I have a sample of n=77 change over time scores (T2-T1). In a lot of cases a score of 0 (no change over time) is modal, at times 33% of the sample. I am wondering if this impacts the assumption of normality at all, due to the high frequency, compared to other scores. I should note that when I make the bins smaller, there is still one response that remains the mode by quite a bit.
Here is another screenshot where there may or may/not be left skew . It is difficult to tell with that mode and a single extreme value. I have avoided doing statistical tests of normality as I hear they can be very sensitive, and I'd prefer to use parametric tests if possible.