Effects of assumption violations for Pearson correlation I read that there are a few assumptions that need to be met before applying Pearson correlation to data. Yet, I fail to find explanation as to the consequences of those violations. In particular, how would any  violation affect the value of the Pearson Correlation?
The assumptions are as follow. And I look forward for a point-by-point explanation (perhaps from a more mathematical perspective)  of the impacts on the Pearson Correlation if they are violated.

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*the variables must be normally distributed

*no significant outliers in the variables

*variables should be continuous

*homoscedasticity

*continuous data, not ordinal

Also, please feel free to dismiss any of the assumptions above if in practice no one cares about the violation.

These "assumptions" are mostly not assumptions at all. They aren't even ideal conditions. What source or sources make these statements?
The main task of Pearson correlation is to answer a question: how well can data be summarized by a straight line?  Pearson answers that with a number between $0$ and $1$ and as a bonus indicates by sign, positive or negative, whether the straight line would be rising or falling (which you should tell from a scatter plot any way).
Thus if $x = 2, 3, 5, 7, 11, 13, 17$ and $y = 10x$, the correlation is exactly $1$ and it's immaterial that neither variable is continuous or normally distributed and that the variables differ in variance, so are not homoscedastic. That's a matter of principle. Naturally in practice the focus is on noisy relationships where the correlation will be less than $1$ in magnitude.
More generally, homoscedasticity is a complete red herring as

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*There is no reason why $y$ and $x$ need to have the same units of measurement and so even in principle their variances need not be comparable.


*As its formula shows, Pearson correlation adjusts for the variance of each variable and delivers a scale-free measure of strength of relationship.
Working backwards,

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*Whether variables are continuous is mostly irrelevant, as already shown. Pearson correlation can make sense when variables are both binary for example. Pearson can be applied to ordinal data too; Pearson correlation applied to ranks has another name, which is Spearman correlation.


*I wouldn't regard it as an assumption that there are no significant outliers when you calculate a Pearson correlation. In any case what is an outlier and how do you establish significance are serious questions. It's good advice to plot data for a correlation and check at least informally for outliers, because correlations can be very sensitive to outliers, but even when true that is a fact to be thought about and not a violation of assumptions.


*Normality of either distribution is not strongly prerequisite either. If there is an interest in a confidence interval or P-value for a correlation, bootstrapping or permutation testing is long since possible to get around any distributional assumptions that are being made implicitly or explicitly.
What can be true is that correlations may make more sense if one or both variables are transformed, but that is a matter of the associated science (social science, medicine, engineering, whatever) as much as a statistical question. Most commonly, one or both variables may be better worked with on logarithmic scale.
It's widespread (although far from universal) practice to treat correlations as possibly helpful descriptive statistics and to let regression modelling (in a wide sense of "regression") be the machinery for inferences. There are, however, occasions (e.g. in genetics) when correlations are of direct interest.
