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Are there any methods to determine normality of data and then transform that data to normal distribution if it isn't normal?

The task has been set: It is necessary to conduct research via ANOVA and use Fisher criterion and for this data must have a normal distribution. My question is that if this data contains excess and asymmetry, how can it can be transformed to normal distribution? Even if it possible what are the rigorous methods for determining the normality of distribution? I'm understand that i can watch the histogramm and if this plot looks like a bell curve that this data has normal distribution, but are there more clear ways to designate normality?

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    $\begingroup$ "Conduct research via ANOVA" suggests you need to compare group means. Applying any nonlinear transformation to your data will be comparing some other form of average, thereby changing your analysis. Consider the possibility that the data do not need to be Normally distributed for ANOVA to be possible (and reliable). $\endgroup$
    – whuber ♦
    Commented Sep 18, 2023 at 22:06
  • $\begingroup$ @whuber, Wikipedia says that Fisher's criterion may be used only on normal distribution, of course i can conduct Fishers'criteria on non-normal data, and i will get the result, but the point is that the result will be incorrect. So you mean that you know any other method to compare group means in ANOVA on non-normal distributions? Because i know only Fisher. $\endgroup$
    – lamer228
    Commented Sep 19, 2023 at 13:38
  • $\begingroup$ I cannot find any Wikipedia article mentioning "Fisher's Criterion." // Why do you feel you need to use this specific approach? Usually (apart from learning and training purposes) people choose statistical procedures that are appropriate for their objectives and data, rather than setting out to employ some prespecified procedure. // Regardless, no statistical procedure requires exact Normality, because data cannot be exactly Normal. The question always comes down to whether the data exhibit characteristics suggesting the result might be unreliable or wrong. $\endgroup$
    – whuber ♦
    Commented Sep 19, 2023 at 13:55
  • $\begingroup$ Article about Fisher's criterion - en.wikipedia.org/wiki/F-test $\endgroup$
    – lamer228
    Commented Sep 19, 2023 at 17:07
  • $\begingroup$ Thanks -- but I had looked at that article and searched it for the word "criterion," to no avail. It still doesn't appear there ;-). $\endgroup$
    – whuber ♦
    Commented Sep 19, 2023 at 17:40

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There are various tests of normality. I'm not sure I'd call them "rigorous" and I'd advise against their use in this context. In any case, ANOVA does not require normal data, one of its assumptions is normal errors.

There are also a variety of transformation schemes, perhaps the most famous is Box-Cox, but, if the assumptions are violated, I wouldn't transform it in order to make it fit, I would use a method that doesn't make those assumptions, such as robust regression or quantile regression.

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