What to Do When Your Data Are Not Normally Distributed

You ran the Shapiro-Wilk test on your thesis data, and the output reads W = .91, p = .001. The normality assumption is violated. Your first instinct might be to panic, scrap the planned t-test, and start over. Don't. A significant Shapiro-Wilk result is a starting point, not a dead end. You have at least four solid paths forward, and which one you pick depends on your sample size, how badly the distribution is skewed, and what test you were planning to run.

First Response: Look at Your Data Before You React

A p-value alone does not tell you how non-normal your data are. The Shapiro-Wilk test is sensitive. With a sample of 400 or more, it will flag distributions that are only slightly off from a perfect bell curve. A skewness of 0.3 in a sample of 500 will often produce p < .05, even though that degree of asymmetry would not meaningfully affect a t-test or ANOVA. Before making any decision, open the histogram and the Q-Q plot.

In the histogram, look for obvious asymmetry, floor effects, or multiple peaks. Overlay a normal curve in SPSS (double-click the chart, then Elements > Show Distribution Curve) and see how far the bars diverge from it. In the Q-Q plot, check whether the points follow the diagonal reference line. A few points drifting at the tails is common and usually harmless. A systematic curve or S-shape across the entire plot signals a real problem. For a detailed walkthrough of reading these plots, see our guide to normality testing.

Also check the skewness and kurtosis statistics. Curran, West, and Finch (1996) consider |skewness| < 2 and |kurtosis| < 7 acceptable for parametric methods. If your values sit comfortably within those bounds and your Q-Q plot looks reasonable, the significant Shapiro-Wilk result may be a statistical artifact of your large sample rather than a genuine threat to your analysis.

Option 1: Transform the Variable

Data transformations apply a mathematical function to every score so that the resulting distribution is closer to normal. They work best when the non-normality has a clear, consistent pattern like right skew or left skew. Each type of skew calls for a different transformation.

For right-skewed data (a long tail stretching toward high values, common with reaction times, income, or biomarker concentrations), the log transformation is the standard first choice. In SPSS: COMPUTE log_variable = LG10(original_variable). If any values are zero, add a constant first: COMPUTE log_variable = LG10(original_variable + 1). The square root transformation handles moderate right skew where the log is too aggressive: COMPUTE sqrt_variable = SQRT(original_variable). For left-skewed data, you reflect the variable before transforming. Find the maximum value, add 1, and subtract each score: COMPUTE reflected = (MAX_VALUE + 1) - original_variable. Then apply the log or square root to the reflected variable. Remember that reflection reverses the direction of your scale, so higher transformed values now correspond to lower original scores. Account for this when interpreting results.

Here is the mistake we see in about one of every six projects that use transformations: the student applies a log transformation and moves on to the parametric test without checking whether the transformation actually worked. Always run Shapiro-Wilk again on the transformed variable and inspect the new Q-Q plot. If the distribution is still non-normal after transformation, you have not solved the problem. Switch to a nonparametric test instead.

Option 2: Use Nonparametric Tests

Nonparametric tests do not assume any particular distribution shape. They operate on ranks instead of raw values, which makes them immune to skew and outliers. The cost is a modest reduction in statistical power compared to parametric tests when the data actually are normal, typically 5% to 15% depending on the test and sample size.

The mapping is straightforward. If you planned an independent-samples t-test, run the Mann-Whitney U test instead. If you planned a paired-samples t-test, use the Wilcoxon signed-rank test. One-way ANOVA becomes Kruskal-Wallis H. Repeated-measures ANOVA becomes the Friedman test. Pearson correlation becomes Spearman rank correlation. In SPSS, all of these live under Analyze > Nonparametric Tests > Legacy Dialogs.

Report medians and interquartile ranges instead of means and standard deviations. The Mann-Whitney output, for example, should appear in APA format as: "A Mann-Whitney U test indicated that anxiety scores were significantly higher in the clinical group (Mdn = 38) than in the control group (Mdn = 29), U = 187, z = -2.61, p = .009, r = .31." Calculate the effect size yourself using r = z / √N. With z = -2.61 and N = 70, that gives r = 2.61 / 8.37 = .31, a medium effect by Cohen's benchmarks (.10 small, .30 medium, .50 large).

Option 3: Bootstrapping

Bootstrapping takes a different approach entirely. Instead of assuming a theoretical distribution, it builds an empirical one by resampling your actual data with replacement thousands of times. Each resample produces a test statistic, and the collection of those statistics forms a distribution that the procedure uses to calculate confidence intervals and p-values.

In SPSS (version 21 and later), bootstrapping is available as a checkbox within most analysis procedures. Go to Analyze > Compare Means > Independent-Samples T Test, click the Bootstrap button, check "Perform bootstrapping," and set the number of samples to at least 1,000. For a thesis, 5,000 resamples provides more stable estimates and is still fast on modern hardware, finishing in under 10 seconds for most datasets under 1,000 cases. The bootstrapped output gives you bias-corrected accelerated (BCa) confidence intervals that do not depend on normality.

Bootstrapping is especially useful for regression and mediation analyses where the alternatives (rank-based regression is uncommon, and nonparametric mediation is messy) are less well-established. If your supervisor or committee is familiar with bootstrapping, it can be the cleanest path. Academic Stats Agent can run bootstrap analyses on your data and produce publication-ready confidence intervals automatically.

Option 4: Proceed with the Parametric Test Anyway

This option surprises many students, but it has solid theoretical backing. ANOVA and the t-test are both based on the sampling distribution of the mean. The Central Limit Theorem guarantees that this sampling distribution approaches normality as the sample size grows, regardless of the shape of the raw data. Glass, Peckham, and Sanders (1972) showed that the F-test in ANOVA maintains accurate Type I error rates under substantial non-normality when group sizes are equal and each group contains at least 25 to 30 observations.

The conditions matter. Equal (or near-equal) group sizes are the key requirement. When groups are balanced, both the t-test and ANOVA tolerate skewed distributions well. When group sizes are very unequal and the data are also skewed, the actual Type I error rate can deviate from the nominal .05 level, sometimes reaching .08 or higher. Similarly, heavy-tailed distributions (kurtosis above 3 or 4) can inflate error rates even with moderate samples. The safest rule: if you have 30+ participants per group, equal group sizes, and the skewness falls below about 1.5 in absolute value, you can proceed with the parametric test and note the assumption violation in your results section.

Decision Guide: Matching the Fix to the Problem

The right approach depends on two things: how severe the violation is and how large your sample is. The table below maps these factors to a recommended strategy.

Violation Severity	Small Sample (n < 30 per group)	Medium Sample (30-100 per group)	Large Sample (n > 100 per group)
Mild (|skew| < 1, Q-Q near diagonal)	Nonparametric test	Proceed with parametric test	Proceed with parametric test
Moderate (1 ≤ |skew| < 2, some Q-Q deviation)	Nonparametric test	Try transformation; if it fails, nonparametric	Proceed with parametric test (equal groups) or bootstrap
Severe (|skew| ≥ 2, floor/ceiling effects, heavy tails)	Nonparametric test	Nonparametric test or bootstrap	Nonparametric test or bootstrap

A few notes on reading this table. "Proceed with parametric test" assumes roughly equal group sizes, following the Glass, Peckham, and Sanders (1972) findings. If your groups differ by more than a 2:1 ratio (say, n = 45 vs. n = 90), drop down one row in severity and follow that recommendation. Bootstrapping is listed alongside nonparametric tests for moderate-to-severe cases because it offers a flexible middle ground: you keep the original parametric framework but relax the distribution assumption.

Reporting the Assumption Violation

Whatever path you choose, document it in the results section of your thesis. A single sentence is enough when the violation is mild: "Shapiro-Wilk tests indicated non-normality in the treatment group (W = .93, p = .02); however, with n = 48 per group and |skewness| = 0.74, one-way ANOVA was retained given its known resistance to mild violations under balanced designs (Glass et al., 1972)." When you switch to a nonparametric test, explain why: "Given the significant deviation from normality in Group B (W = .85, p < .001, skewness = 1.82), a Mann-Whitney U test was used in place of the independent-samples t-test." This kind of transparent reporting shows your committee that you understood the assumptions and made a deliberate choice rather than ignoring the problem. For more guidance on structuring these justifications, see our article on writing the results section.