Many statistical tests (like t-tests and anova) require normality of data. However, there is a scope for subjective judgement also. Shapiro-Wilk test or Kolmogorov-Smirnov test (most commonly used normality tests) oftentimes may lead to the conclusion of non-normality. It is recommended to take care of data cleaning tasks first (like outliers and missing values). If there is still the normality violation, then look at the (i) skewness, (ii) kurtosis and (iii) sample size. If Skewness and Kurtosis are within acceptable limits (-2 to +2), and the sample size > 30, then you can still use "normality" assumption for further tests. For PLS-SEM, normality assumption is not a pre-requisite as it is a non-parametric method. (edit: some literature also puts the Kurtosis limit from -7 to +7, to assume normality).

Reference: https://www.smartpls.com/documentation/functionalities/excess-kurtosis-and-skewness

Thank you, Dr Khandoker Mahmudur Rahman, for this insightful clarification. Your balanced perspective on combining statistical tests with practical judgment, especially considering skewness, kurtosis, and sample size, ishighly valuable. Also, the note on PLS-SEM not requiring normality is a helpful reminder. Much appreciated!

Best,

Amine,

First, note that the paired-samples t-test assumes the difference scores are normally distributed, not raw scores from each individual group.

Second, it really makes no sense to use a hypothesis test (Shapiro-Wilk, KS) to make a decision about normality. I recommend constructing a Q-Q Plot of the residuals and check for any major deviations.

Third, I'd advise that you DON'T remove outliers simply to make the data more normally distributed (especially post hoc)! Not only could this be construed as p-hacking but may be a mistake. For instance, perhaps the variable you're measuring isn't actually normally distributed in the population and it would be more appropriate to analyze the data with a Wilcoxon signed-rank test or a test that assumes a different distribution (e.g., gamma) that describes the data better.

Both Shapiro-Wilk test and skewness/kurtosis statistics are commonly used to assess normality in SPSS, but neither should be treated as the definitive test in isolation. Each has limitations, and their usefulness depends on sample size and the broader context of the analysis.

Shapiro-Wilk test is statistically powerful, particularly for small samples, but it has a known drawback:

• · In large samples (e.g., N > 50–100), even trivial departures from normality can become statistically significant (Razali & Wah, 2011; Ghasemi & Zahediasl, 2012).
• · This can lead to false positives, prompting researchers to reject normality unnecessarily—even when the deviation has negligible practical implications.

Skewness and kurtosis values, along with their standardized z-scores or confidence intervals, offer a complementary perspective:

• · For large samples, small skewness/kurtosis is often tolerated, especially if |skew| < 2 and |kurtosis| < 7 (West, Finch, & Curran, 1995).
• · The absolute values of these indices can provide more practical information about the shape of the distribution than a binary hypothesis test.

Moreover, visual methods such as histograms, Q-Q plots, and boxplots should not be overlooked. They allow researchers to evaluate practical (rather than purely statistical) normality and detect issues like bimodality or extreme outliers.

Conclusion: No single method is “more important” universally. Instead:

Suggested references: