Dear José Sánchez Santamaría

In large samples, parametric tests can often still be used even if normality tests fail, as they are generally robust to violations of normality due to the Central Limit Theorem, which states that the distribution of sample means approximates a normal distribution as the sample size increases, regardless of the population's distribution. However, it is important to consider the extent of non-normality and the specific assumptions of the test being used; in cases of extreme skewness or outliers, non-parametric alternatives might be more appropriate despite the large sample size. Additionally, visual inspections of data (e.g., histograms, Q-Q plots) and other diagnostic tools should complement formal normality tests to make an informed decision.

I hope this helps.

You can use non-parametric tests on any data in theory :)

Dear José Sánchez Santamaría

That's a very interesting question!

There are several academic papers that might be of interest to you. For the specific question you raised, the work of Lumley et al. (2002) would be a good reference. Additionally, the publication by Field and Wilcox (2017) offers a very interesting discussion on the topic. Personally, I have found Andy Field's works, such as Discovering Statistics Using IBM SPSS Statistics (2024), to be incredibly helpful as a starting point for delving into these issues. I always recommend them.

I hope this helps!

Best regards,

Alvaro

References:

Field, A. (2024). Discovering statistics using IBM SPSS statistics. Sage publications limited.

Field, A. P., & Wilcox, R. R. (2017). Robust statistical methods: A primer for clinical psychology and experimental psychopathology researchers. Behaviour research and therapy, 98, 19-38. https://doi.org/10.1016/j.brat.2017.05.013

Lumley, T., Diehr, P., Emerson, S., & Chen, L. (2002). The importance of the normality assumption in large public health data sets. Annual review of public health, 23(1), 151-169. https://doi.org/10.1146/annurev.publhealth.23.100901.140546

You're right to raise this — it's a common debate. In large samples, the Central Limit Theorem suggests that the sampling distribution of the mean tends to be normal regardless of the data’s distribution, making parametric tests generally robust. Many statisticians argue that parametric tests (like the t-test or ANOVA) are fairly resistant to violations of normality in large samples (n > 30 or 50), especially when there are no extreme outliers or skewness.

Studies like Lumley et al. (2002) and Rasch et al. (2011) support this robustness. That said, it's wise to check homogeneity of variance and consider non-parametric alternatives if assumptions are severely violated. Practical judgment matters too!