# Arguments to explain whether/or not data can be described by a normal distribution

I am studying for my exam on Friday in Mathematical Statistics. One thing, I have trouble with is what arguments I have to use to say whether or not the data I am given can be described by a normal distrubition. For an example, consider the following histogram of some data. The question is: Is the data normally distributed? First of all, I would say that the histogram does not show a "perfect" bell curve, i.e. is symmetric because it is a little right skewed. Is this enough to conclude that the data can't be described by a normal distribution or do I need more?

From the QQplot we can also see deviations at the tails, thus showing that there are systematic deviations. Therefore I would end by concluding that data can't be described by a normal distribution.

Is the argumentation adequate? Or what would I need to say otherwise? Can you give explanations of cases where data can be described by a normal distribution with adequate arguments and a case where it is not with adequate arguments? It would help me a lot.

Furthermore, if we have a qqplot + a plot of the residuals, what would I need to consider when arguing for/against a normal distribution? Consider the down below, for an example. • "Mathematical statistics" in what sense? // Why not a formal distribution test? That can't show your data to come from a normal distribution, but you can cast doubt on the data coming from a normal distribution.
– Dave
Jun 7, 2021 at 21:29
• The course is actually called “Mathematical Stastistics” (translated from Danish). However, most of the times I am not asked to make formal distribution test but rather to comment on a histogram, QQ plot or residuals from some data I am given Jun 7, 2021 at 21:38
• This is of no help for your exam, but my experience is that students who have trouble with this question are usually among the better ones, as this is a very hard and even philosophical question and every straight answer is probably wrong (even if your examiner wants one). In fact, no real data (actually not even simulated data) are truly normal, and whether it makes sense to model them by a normal distribution regardless depends heavily on what the model will be used for. Jun 7, 2021 at 22:15
• Whether it's reasonable to use a normal approximation depends on what you're using it for, as well as your tolerance for approximation in whatever you're trying to attain. This very much relates to Box's "how wrong does it have to be to not be useful". Such things are not answered by simply looking at a histogram (... and much less so still by performing a goodness of fit test). Jun 8, 2021 at 4:08