How to Test Data for Normality in Theses and Journal Articles

The single most prevalent conceptual error in empirical research is attempting to perform normality tests on individual ordinal Likert survey items (e.g., 1 = Strongly Disagree to 5 = Strongly Agree). Individual Likert responses are discrete, bounded integer categories that inherently cannot form a continuous Gaussian bell curve. Normality testing must be restricted to continuous variables measured on an interval or ratio scale, or to summated composite and mean scores derived from validated multi-item scales.

The second non-negotiable rule is the requirement for group-wise testing in comparative designs: When planning an independent samples t-test (e.g., Treatment vs. Control) or a One-Way ANOVA across multiple conditions, the distributional normality assumption applies to the dependent variable WITHIN EACH SUBGROUP SEPARATELY, not to the pooled aggregate sample. Pooling groups can artificially induce bimodal distributions or mask genuine within-group skewness, rendering inferential statistics invalid.

Two fundamental coefficients mathematically define the shape of your sample distribution: Skewness and Kurtosis. Skewness quantifies the asymmetry of the distribution relative to the mean. A coefficient of zero indicates perfect bilateral symmetry. Positive skewness reflects data concentrated on the lower end with a tail extending toward positive infinity; negative skewness reflects clustering on the high end with an elongated lower tail.

Kurtosis measures the tail weight and peakedness relative to a normal distribution. In modern statistical packages like SPSS and R, 'Excess Kurtosis' is computed where the normal curve equals zero (mesokurtic). Positive kurtosis indicates heavy tails and outlier proneness (leptokurtic), whereas negative kurtosis indicates light tails and platykurtic flatness. For graduate theses and dissertations, the standard citation is George & Mallery (2010, 2020), which deems skewness and kurtosis values between [-2.0, +2.0] as an acceptable approximation of univariate normality. For submissions to high-impact Q1/Q2 journals, Tabachnick & Fidell's (2019) stricter [-1.5, +1.5] benchmark is expected.

Statistical software generates two primary goodness-of-fit tests under normality screening: the Shapiro-Wilk test and the Kolmogorov-Smirnov test (with Lilliefors significance correction). In these procedures, the null hypothesis (H0) states that the empirical sample is drawn from a normally distributed population; the alternative hypothesis (H1) posits a significant departure from normality. Consequently, researchers seek a non-significant result (p > .05) to retain H0.

Rigorous Monte Carlo simulation studies by Razali & Wah (2011) demonstrate that the Shapiro-Wilk test possesses substantially greater statistical power than the Kolmogorov-Smirnov test across virtually all sample sizes and non-normal distribution geometries. For small to moderate samples (n < 50 and n ≤ 300), Shapiro-Wilk should always be preferred; Kolmogorov-Smirnov is prone to Type II errors, frequently failing to flag critical departures from normality.

Graphical diagnostics represent the most resilient safeguard against the dual risks of test underpowering in small samples and overpowering in large samples (Field, 2018). The Normal Q-Q (Quantile-Quantile) Plot plots observed sample quantiles against theoretical normal quantiles along a 45-degree diagonal reference line. Close adherence of points to this diagonal confirms univariate normality.

A bell-shaped curve overlay on the sample histogram verifies unimodality; distinct bimodal peaks signal that two unmodeled subpopulations (e.g., novices vs. experts) have been conflated. Furthermore, boxplots provide an objective screen for distribution symmetry and flag mild outliers (exceeding 1.5 × IQR, displayed as open circles) and extreme outliers (exceeding 3.0 × IQR, displayed as asterisks).

If numerical tests yield p < .05, skewness and kurtosis breach the [-2.0, +2.0] threshold, and Q-Q plots demonstrate pronounced departure from linearity, researchers must execute a structured fallback protocol. Three scientifically sound avenues exist: The first and most established path is transitioning to non-parametric equivalents: Mann-Whitney U for independent samples t-test, Wilcoxon Signed-Rank for paired samples, Kruskal-Wallis H for One-Way ANOVA, and Spearman's rho for Pearson correlation.

The second approach is mathematical data transformation (e.g., logarithmic log10, square root, or inverse transformation for positively skewed data; reflect-and-log for negative skewness). However, transformations carry a heavy interpretive cost: transformed variables lose their original physical units and become challenging to interpret substantively. The third and modern gold standard is Bootstrapping (1,000 to 5,000 resamples). By computing 95% Bias-Corrected and Accelerated (BCa) confidence intervals, researchers can retain their original parametric model without relying on the Gaussian distributional assumption.

The Publication Manual of the American Psychological Association (7th ed., Section 6.44 and Chapter 12 JARS-Quant) specifies exact formatting requirements for reporting distributional assumptions in academic manuscripts. A critical stylistic rule is the 'Leading Zero Rule': statistical values that cannot exceed 1.00 by definition (such as p-values, Shapiro-Wilk W, and correlation coefficients) must be reported WITHOUT a leading zero before the decimal point (e.g., W = .98, p = .184, NOT W = 0.98, p = 0.184).

All statistical symbols (N, n, M, SD, W, D, p, Z) must be italicized in both in-text narrative and tabular headers. Your method section must explicitly justify the normality threshold employed, citing foundational literature (e.g., George & Mallery, 2010 or Tabachnick & Fidell, 2019) to ensure resilience during thesis defenses and peer review.