I am having difficulty in exactly understanding several statistical tests, such as the t-test and ANOVA test. These tests require that the data we use be normally distributed.
However, whilst sharing my experience in analyzing data a bit, I have analyzed several data from numerous sources online (web scraping, open-accessed data sources online, etc.), with considerably high number of samples (hundreds, thousands). An example of the data in question is the amount of donation given to certain campaigns in a fixed periods of time (day 1 at 1pm, day 2 at 1pm, etc.).
And when I tested whether the normality distribution of the data, using visual aids (histograms, Q-Q plots) and Shapiro-Wilks test, they all showed me that the data is not normal. For example, Shapiro-Wilks test gave a p-value of so small (less than 0.00000000000000022), of course the null hypothesis has to be rejected, i.e. the data is nor normally distributed.
Because I read in articles like in this link, it says
However, even if the distribution of the individual observations is not normal, the distribution of the sample means will be normally distributed if your sample size is about 30 or larger
So naturally, I am confused, is my data normally distributed or not? How often do you encounter normal and not-normal distribution, in real-life data?
Many posts and forums also agree that normality in the data is quite rare. But if that is the case, then are parametric tests such as Chi-Square, ANOVA, t-tests, etc., by nature rarely applicable, and therefore useless? An example of this discussion that supports this is here.