Due to central limit theory, the assumption of normality implied in many statistical tests and estimators is not a problem.
The normal distribution is the basis of much statistical theory. Hypothesis tests and interval estimators based on the normal distribution are often more powerful than their non-parametric equivalents. When the distribution assumption can be met they are preferred as the increased power means a smaller sample size can be used to detect the same difference.
However, violation of the assumption is often not a problem, due to the central limit theorem. The central limit theorem states that the sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For many samples, the test statistic often approaches a normal distribution for non-skewed data when the sample size is as small as 30, and for moderately skewed data when the sample size is larger than 100. The
downside in such situations is a reduction in statistical power, and there may be more powerful non-parametric tests.
Sometimes a transformation such as a logarithm can remove the skewness and allow you to use powerful tests based on the normality assumption.
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