Statistical tests were designed to test for relationships and differences on normally distributed variables

True, when analyzing data measured by a continuous quantitative variable, the statistical tests of estimation and contrast frequently used are based on supposing that a random sample has been obtained from a probability distribution of normal type, but in many cases this assumption is not valid, in these cases we have two possible mechanisms: The data can be transformed in such a way that they follow a normal distribution or you can resort to parametric or non-parametric tests.

You don’t list any options but the one that frets overspending and the consequences of that.