All parametric statistical tests assume that data meet certain conditions for their results to be valid. These assumptions are not mere theoretical formalities: when they are severely violated, the conclusions you draw may be erroneous. The problem is that many researchers in psychology either ignore the assumptions entirely (applying the test without verifying anything), verify them incorrectly (typically by applying tests that are not the most appropriate), or react inadequately when a violation is detected (automatically switching to a nonparametric test without considering alternatives).
The most important assumptions vary depending on the analysis, but three recur consistently: normality, homogeneity of variances (homoscedasticity), and independence of observations. Of these three, independence is by far the most important and the one whose violation carries the most severe consequences. If your observations are not independent (for example, because you assessed multiple students within the same classrooms without accounting for the nested structure), the standard errors will be too small, the confidence intervals too narrow, and the p-values too optimistic. Unlike the other assumptions, the violation of independence cannot be resolved with transformations or robust tests, but rather requires a change of model (for example, using multilevel models).
Normality: when it matters and when it does not
Normality is the assumption that worries researchers the most and, paradoxically, the one that should worry them the least in most situations. The t-test and ANOVA assume that the model residuals (not the variables themselves) are normally distributed. With reasonably large samples (say, more than 30 participants per group), the central limit theorem guarantees that the sampling distribution of the mean will be approximately normal even if the individual data points are not. This makes these tests quite robust to moderate violations of normality.
The most common error when assessing normality is relying on formal tests such as Shapiro-Wilk or Kolmogorov-Smirnov as the sole criterion (try our normality interpreter for a clear reading of your results). These tests have a fundamental problem: their power depends on sample size. With large samples (more than 100 participants), they reject normality for trivial deviations that have no practical effect on the validity of the test you intend to apply. With small samples, they may fail to detect important deviations. The best strategy is to combine visual assessment (histograms, Q-Q plots) with formal tests, and use judgment to decide whether the observed deviation is severe enough to justify a change in approach.
If you are unsure which test to use after checking assumptions, the statistical test selector helps you choose. When normality is severely violated (highly skewed distributions, extreme outliers, or bimodal distributions), the options include variable transformations (logarithmic, square root), robust tests that do not assume normality (such as those based on trimmed means available in the WRS2 package for R), or nonparametric tests (Mann-Whitney, Kruskal-Wallis). The choice among these options depends on the nature of the violation and the researcher's priorities. Transformations change the scale of the variable, which can complicate interpretation. Nonparametric tests answer slightly different questions (about medians or ranks, not means). Robust tests are probably the most flexible option, but they are less well known and may generate reluctance among reviewers.
Homogeneity of variances and independence
Homogeneity of variances (homoscedasticity) refers to the assumption that the variability of the dependent variable is similar across all groups being compared. Levene's test is the most commonly used method to assess it. When variances are unequal (heteroscedasticity), the consequences depend on the design: with equal group sizes, ANOVA is quite robust to heteroscedasticity; with unequal group sizes, it can produce inflated Type I error rates. The simplest solution is to use the Welch correction, which adjusts the degrees of freedom to compensate for the inequality of variances and is available for both the t-test (Welch's t) and ANOVA (Welch's ANOVA). In fact, some statisticians recommend always using the Welch version as the default option, since it produces results virtually identical to the classical version when variances are equal and is more accurate when they are not.
In regression, homoscedasticity refers to the residual variance being constant across the range of predicted values. It is assessed visually through the plot of standardized residuals against predicted values: if a funnel-shaped or fan-shaped pattern is observed, heteroscedasticity is present. The solution may involve using robust standard errors (White's sandwich estimator), which are valid regardless of whether homoscedasticity holds, or transforming the dependent variable.
A pragmatic approach
My general recommendation is to always verify the assumptions, but not to obsess over meeting all of them perfectly. The relevant question is not whether your data meet the assumptions (they almost never meet them exactly), but whether the violations are severe enough to affect your conclusions. To answer this question, it is useful to conduct sensitivity analyses: apply both the original parametric test and a robust or nonparametric alternative, and check whether the conclusions change. If both approaches produce the same results, you can be confident that your conclusions do not depend on strict adherence to the assumptions. If the results differ, that tells you something important about the robustness of your findings that deserves an honest discussion in your paper.
Before you submit: writing "assumptions were not met, so a nonparametric test was used" with no further justification is one of the lines a methodological reviewer underlines on the first read, along with a significant Shapiro-Wilk at n = 400 used as the excuse. If your manuscript is already written, run it through the AI Paper Reviewer: a free Reviewer 2 style pre-review that tells you what they will challenge about your assumption checks and your analytic decision while you can still change it.
When assumptions break in your dissertation data and you are unsure whether to keep the parametric test or report the alternative, my consulting for doctoral theses walks through the diagnostics with you and picks the approach you can defend later before the committee.