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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.

A histogram

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

Blockquote

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.

enter image description here

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  • $\begingroup$ "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. $\endgroup$
    – Dave
    Commented Jun 7, 2021 at 21:29
  • $\begingroup$ 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 $\endgroup$
    – Mathias
    Commented Jun 7, 2021 at 21:38
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    $\begingroup$ 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. $\endgroup$ Commented Jun 7, 2021 at 22:15
  • $\begingroup$ 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). $\endgroup$
    – Glen_b
    Commented Jun 8, 2021 at 4:08

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As a person who teaches something close in spirit to Mathematical Statistics, I would say that the first histogram from your question actually does resemble the normal distribution that was cut off at zero. In finite samples one should not expect a "perfect" bell curve from a histogram, so this is already quite a similar shape.

You actually need to think about what your professor expects from you. If this is a simple yes/no question, than I would answer "no" because the support for the distribution is clearly limited (we consider only non-negative values). But if you are supposed to give your thoughts on this topic, then this does resemble a normal distribution.

From my experience, I would expect something with either two bumps on a histogram or with divergence in the middle of the q-q plot (not in the tails) to be almost certain that the data does not come from a normal distribution. But still: almost.

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  • $\begingroup$ Hi. Thanks for the comment. I will take that in mind. $\endgroup$
    – Mathias
    Commented Jun 7, 2021 at 22:05
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    $\begingroup$ " to be almost certain that the data does not come from a normal distribution" We can be absolutely sure that no real data are normal without even looking at the data. (Unless by "normal" we actually mean "not quite normal but not too far away".) $\endgroup$ Commented Jun 7, 2021 at 22:26
  • $\begingroup$ @Lewian I see you comment above and I agree in the philosophical sense. However, this question (in my inderstanding) was more about a practical side of these considerations, specifically, about studying for the exams. And I believe that for the exams "normal" means "not quite normal but not too far away". $\endgroup$
    – Misius
    Commented Jun 7, 2021 at 22:32
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    $\begingroup$ @Misius: You may well be right, however I find it important to promote a correct use of terminology, even if it is used in a wrong or misleading way by many professors and examiners. The idea that we need to somehow confirm normality misrepresents a basic concept of statistics. $\endgroup$ Commented Jun 7, 2021 at 22:43

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