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.enter image description here 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 enter image description here. 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.

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    $\begingroup$ Please tell us what you are assuming has a Normal distribution and what role that assumption might play in your analysis. (It is unusual for any statistical procedure to require the data to be normal.) $\endgroup$ – whuber Feb 28 '18 at 21:28
  • $\begingroup$ I would expect that the responses to have a normal distribution. I have three groups and would like to know if I should be an ANOVA or non-parametric test $\endgroup$ – AWrath Feb 28 '18 at 21:31
  • $\begingroup$ ANOVA does not require the responses to have a Normal distribution, nor does it require the responses conditional on the explanatory variables to be Normal. It only supposes that the conditional responses will be close enough to Normal to support the usual assertions about the distributions of test statistics. That will be an issue in this case for an ANOVA F statistic, for instance, but not for the t-statistic for any individual parameter estimate. $\endgroup$ – whuber Feb 28 '18 at 21:53

I will assume you ask about the normality of your change scores. About how to analyze such data. You could do much better than just plotting a histogram, which will depend on the binning. See here: Is there a rule of thumb to choose a normality test?

Other graphical methods is qqplot Interpreting QQplot - Is there any rule of thumb to decide for non-normality? which you could try. And, if there really is an appreciable number of exact zeros in your differences, you should investigate why. But maybe it is just about binning.

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