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

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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.
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