A junior researcher conducts an RCT comparing wound-healing time (in days) between two surgical techniques, with n=14 per group. She runs an independent samples t-test, reports t(26)=2.18, p=0.038, and submits. Her examiner asks: “Did you test the normality assumption before using a parametric test?” She hadn’t. When the Shapiro-Wilk test is subsequently run, both groups show significant departures from normality (p=0.029 and p=0.041). The correct test, Mann-Whitney U, gives U=58, p=0.092 — not significant. The entire conclusion of her results chapter must be rewritten. This scenario, repeated in countless theses and manuscripts, illustrates the single most consequential decision in applied biostatistics: selecting between parametric and non-parametric tests. The wrong choice doesn’t just change a p-value — it changes the scientific conclusion.
What Are Parametric Tests?
Parametric tests are statistical procedures that make explicit assumptions about the parameters (means, variances) of the population distribution from which data are sampled. The most fundamental assumption is normality — that the data, or the residuals from a model, follow a Gaussian (bell-shaped) distribution. Parametric tests analyse the raw data values directly, computing means, standard deviations, and variances to perform inference.
The most commonly used parametric tests in medical research are: the independent samples t-test (comparing means between two unrelated groups), the paired samples t-test (comparing two related measurements), one-way ANOVA (comparing means across three or more groups), Pearson correlation (measuring the linear relationship between two continuous variables), and linear regression (modelling the relationship between an outcome and one or more predictors).
When their assumptions are met, parametric tests are the more powerful choice — they use more information from the data (the actual numerical values) and detect true effects with fewer participants.
Parametric Tests
Distribution-Based
- Assume normality of data or residuals
- Analyse actual data values (means, variances)
- Require interval or ratio scale data
- More powerful when assumptions are met
- Sensitive to outliers and skewness
- Examples: t-test, ANOVA, Pearson r, linear regression
Non-Parametric Tests
Distribution-Free
- No assumptions about population distribution shape
- Convert data to ranks; analyse the ranks
- Suitable for ordinal, interval, or ratio data
- Robust to outliers and skewed distributions
- Slightly less power when normality holds
- Examples: Mann-Whitney U, Kruskal-Wallis, Spearman, Friedman
Assumptions of Parametric Tests
Parametric tests carry five key assumptions. Violating any one of them invalidates the test and can produce misleading p-values and conclusions. Critically, these assumptions must be tested for every dataset — they cannot be assumed to hold simply because the outcome variable is measured on a continuous scale.
Normality Shapiro-Wilk
The outcome variable (or residuals from a model) must follow an approximately normal distribution within each group. This is the primary and most tested assumption in medical research.
Homogeneity of Variance Levene’s
For tests comparing two or more groups (t-test, ANOVA), variance should be approximately equal across all groups. Violated when one group has substantially more spread than another.
Independence of Observations
Each data point must be independent of all others. Violated in repeated-measures designs, matched pairs, clustered samples, and family-based studies. Independence determines which test design applies.
Interval/Ratio Scale Data
Parametric tests require that the difference between data values is meaningful and equal in magnitude. Ordinal data (Likert scales, ranked scores) violate this assumption even when they look like numbers.
No Extreme Outliers
Parametric tests are based on means and variances, which are disproportionately influenced by extreme values. A single outlier can shift the mean enough to produce or mask a significant result.
Adequate Sample Size
With very small samples (n<15 per group), the central limit theorem provides insufficient protection. Even when the population is normal, small samples cannot establish normality reliably, and parametric tests are poorly calibrated.
Testing the Normality Assumption
The Shapiro-Wilk test is the recommended normality test for samples up to approximately n=50. It is the most powerful formal test for detecting non-normality in small to medium samples — more powerful than the Kolmogorov-Smirnov (K-S) test. A significant result (p<0.05) indicates significant deviation from normality, supporting the use of a non-parametric alternative. An non-significant result (p≥0.05) is evidence consistent with normality, supporting a parametric test — but it does not prove normality.
n < 50
Small samples
Use Shapiro-Wilk. Most powerful for this range. Significant p<0.05 → use non-parametric. In SPSS: Analyze → Descriptives → Explore → Plots → Normality.
n 50–200
Medium samples
Shapiro-Wilk still preferred. Support with Q-Q plot visual inspection. Formal tests become sensitive to minor deviations — combine p-value with visual assessment.
n > 200
Large samples
Formal tests are overly sensitive — any trivial deviation will be significant. Rely on Q-Q plots, histograms, and skewness/kurtosis ratios. CLT provides protection for parametric tests.
Visual Assessment Methods
Q-Q plot (Quantile-Quantile plot): Plots the quantiles of your data against the theoretical quantiles of a normal distribution. If data are normally distributed, points fall approximately along the diagonal reference line. S-shaped curves indicate skewness; banana curves indicate kurtosis. Q-Q plots are essential for large samples where formal tests are uninformative.
Histograms: A bell-shaped histogram supports normality. Right-skew (long tail to the right) is common in clinical variables like hospital length of stay, CRP values, enzyme levels, and survival times. Left-skew occurs in variables like scores on an easy test where most participants score near the maximum.
Skewness and kurtosis statistics: Commonly cited thresholds: skewness between −1.0 and +1.0 and kurtosis between −2.0 and +2.0 suggest approximate normality. More conservative thresholds of ±0.5 for skewness are sometimes applied. Always report these alongside the formal test result.
The Kolmogorov-Smirnov warning: SPSS reports the Kolmogorov-Smirnov test (with Lilliefors significance correction) alongside Shapiro-Wilk. For small samples, K-S is substantially less powerful than Shapiro-Wilk — it will often fail to detect non-normality that Shapiro-Wilk identifies. Do not rely on K-S alone. Always use Shapiro-Wilk as the primary normality test for n<50, and report both in your methods section.
The Central Limit Theorem: What It Does and Doesn’t Protect
The central limit theorem (CLT) states that the sampling distribution of the mean approaches normality as sample size increases, regardless of the distribution of the underlying raw data. This is why, with sufficiently large samples, parametric tests remain valid even when the raw data are moderately non-normal: the means, not the individual values, are what t-tests and ANOVA actually compare.
The CLT offers genuine protection from approximately n≥30 per group for mildly skewed data. However, several scenarios render the CLT insufficient:
- Severely skewed distributions (hospital LOS, serum CRP in sepsis, surgical blood loss) — may require n>100 per group for the CLT to adequately normalise the sampling distribution
- Heterogeneity of variance — unequal variances between groups are not fixed by large sample size; Welch’s t-test (which does not assume equal variances) should be used instead of the standard t-test in these cases
- Ordinal data — a 5-point Likert scale does not become interval-scale data with a larger sample; ordinal data cannot fulfill the interval-scale requirement regardless of n
- Bimodal distributions — where the CLT convergence is much slower than for unimodal distributions
Rule of thumb: The CLT provides reasonable protection for parametric tests with n ≥ 30 per group and mild skewness. For moderate-to-severe skewness or distributions with many outliers, use non-parametric tests until n ≥ 80–100 per group, or transform the data (log transformation is appropriate for right-skewed positive data such as CRP, LOS, or enzyme levels) and verify approximate normality of the transformed values before applying parametric tests.
Parametric vs Non-Parametric Test Equivalents
| Parametric Test | ⟶ | Non-Parametric Equivalent | Design | Post-Hoc |
|---|
| Independent t-test | → |
Mann-Whitney U (Wilcoxon rank-sum) |
2 independent groups |
— |
| Paired t-test | → |
Wilcoxon signed-rank |
2 related/matched measurements |
— |
| One-way ANOVA | → |
Kruskal-Wallis H |
≥3 independent groups |
Dunn test (Bonferroni/BH) |
| Repeated-measures ANOVA | → |
Friedman test |
≥3 related measurements |
Wilcoxon + Bonferroni |
| Pearson correlation | → |
Spearman rank correlation |
Association between 2 variables |
— |
| Linear regression | → |
Quantile regression / rank-based regression |
Outcome predicted by ≥1 variable |
— |
| One-sample t-test | → |
Wilcoxon signed-rank (vs hypothetical median) |
1 sample vs known value |
— |
Statistical Power: The Real Cost of Choosing Non-Parametric
A persistent concern among researchers is that non-parametric tests “waste data” by converting values to ranks and discarding the magnitude information. This is true — but the power loss is smaller than most researchers assume, and under non-normal conditions, non-parametric tests can actually be more powerful than their parametric equivalents.
The Asymptotic Relative Efficiency (ARE) quantifies the power of a non-parametric test relative to its parametric equivalent as sample size grows:
Asymptotic Relative Efficiency (ARE) — Non-Parametric vs Parametric
Mann-Whitney U vs t-test (normal data)
0.955
Wilcoxon SR vs paired t-test (normal data)
0.955
Kruskal-Wallis vs one-way ANOVA (normal)
0.955
Mann-Whitney U vs t-test (exponential data)
>1.0
Mann-Whitney U vs t-test (uniform data)
1.00
ARE = power of non-parametric / power of parametric equivalent. ARE >1.0 means non-parametric is MORE powerful.
The practical implication: using the Mann-Whitney U test instead of a t-test when data are actually normal costs you approximately 4.5% power — equivalent to losing about 4.5% of your sample size. In an underpowered study of 30 participants, this is non-trivial. But for studies with adequate sample sizes, the power penalty is small and the gain in robustness under non-normality more than compensates.
Decision Framework: Choosing the Right Test
→ Step-by-Step Test Selection Guide
1
What is the measurement scale of your outcome variable?
Nominal/categorical → Use chi-square, Fisher’s exact, McNemar (not parametric or non-parametric ordinal tests)
Ordinal or continuous → Continue to Step 2
2
Is the data truly continuous (interval/ratio) or ordinal (Likert, ranked)?
Ordinal (Likert, pain scale, rank) → Non-parametric test. Skip normality testing — ordinal data cannot fulfill the interval-scale assumption regardless of distribution shape.
Continuous (interval/ratio) → Continue to Step 3
3
How many groups/conditions are you comparing?
2 groups → t-test family or Mann-Whitney U / Wilcoxon SR
≥3 groups → ANOVA family or Kruskal-Wallis / Friedman
Association → Pearson r or Spearman r
4
Are observations independent or related (paired/repeated)?
Independent → t-test / Mann-Whitney U / one-way ANOVA / Kruskal-Wallis
Related/paired → paired t-test / Wilcoxon SR / repeated-measures ANOVA / Friedman
5
Test normality using Shapiro-Wilk (n<50) or visual methods (n≥50)
p≥0.05 (normal) + no extreme outliers + n≥15/group → Parametric test appropriate. Also check homogeneity of variance (Levene’s test) for group comparisons.
p<0.05 (non-normal) OR small n<15 OR extreme outliers present → Non-parametric equivalent
6
For parametric tests: check homogeneity of variance (Levene’s test)
Levene’s p≥0.05 (equal variances) → Standard independent t-test / ANOVA
Levene’s p<0.05 (unequal variances) → Welch’s t-test or Welch’s ANOVA (not the standard versions)
Clinical Examples
1
Hospital Length of Stay: Mann-Whitney U Outperforms the t-test
A retrospective study compares length of hospital stay (LOS, in days) between patients managed with an enhanced recovery protocol (ERP, n=45) vs standard care (SC, n=45). LOS is a classic right-skewed variable: most patients are discharged within a few days; a minority stay for weeks.
n=90
Total patients (45/arm)
W=0.821
Shapiro-Wilk SC (p=0.002)
W=0.847
Shapiro-Wilk ERP (p=0.004)
Skew=2.8
SC arm (severely right-skewed)
Descriptive Statistics
Standard care: mean=8.3 days (SD=6.1), median=6 days (IQR=4–11), range 2–38
Enhanced recovery: mean=5.9 days (SD=4.2), median=5 days (IQR=3–8), range 1–24
Normality Assessment (supports non-parametric)
Shapiro-Wilk, SC: W=0.821, p=0.002 → SIGNIFICANT non-normality
Shapiro-Wilk, ERP: W=0.847, p=0.004 → SIGNIFICANT non-normality
Skewness SC: +2.8 (severely right-skewed; threshold: |1.0|)
Q-Q plot: both groups show marked deviation from diagonal reference line
Independent t-test (incorrect — normality violated)
t(88) = 2.17, p = 0.033, Cohen's d = 0.46
Mean difference: 2.4 days (95% CI: 0.2–4.6 days)
→ Statistically significant — but the means are distorted by outliers
Mann-Whitney U (correct — distribution-free)
U = 736, z = 2.48, p = 0.013, r = 0.26 (small-to-medium effect)
Median difference: 1 day (bootstrapped 95% CI: 0–3 days)
→ Also statistically significant — and appropriately robust
Critical demonstration: effect of a single outlier
When 3 patients with LOS=34, 36, 38 days are in SC arm (not unusual):
t-test: t(88)=1.89, p=0.062 → NOT significant (outliers inflate SD)
Mann-Whitney U: U=770, z=2.71, p=0.007 → SIGNIFICANT (ranks unaffected)
The t-test is controlled by 3 extreme values out of 90 patients.
✓ Mann-Whitney U (Correct)
U=736, z=2.48
p=0.013, r=0.26
Median: 6 vs 5 days
✗ Independent t-test (Incorrect)
t(88)=2.17
p=0.033, d=0.46
Mean: 8.3 vs 5.9 days
The key lesson: Both tests reach significance here, but the t-test is methodologically invalid because its assumptions are violated. With the 3-outlier scenario, the t-test switches to non-significant while Mann-Whitney U remains clearly significant — revealing that the t-test conclusion depends on which extreme LOS values happen to be in the dataset. LOS, CRP, enzyme levels, surgical blood loss, and waiting times are almost always right-skewed in clinical datasets. Always test normality before applying a t-test, regardless of sample size.
2
Pain VAS Scores Pre/Post Surgery: Wilcoxon Signed-Rank vs Paired t-test
A pre-post study assesses pain using a Visual Analogue Scale (VAS, 0–100mm) in n=22 patients before and after a physiotherapy intervention for chronic low-back pain. The researcher plans a paired t-test on the change scores. VAS is measured on a 100mm line — technically continuous, but with n=22 the sample is too small to rely on the CLT.
62→38
Median VAS (pre→post)
W=0.887
Shapiro-Wilk (p=0.017)
−22mm
Median change (IQR: −31 to −14)
Normality of Change Scores
Shapiro-Wilk on change scores: W = 0.887, p = 0.017
→ SIGNIFICANT non-normality at p < 0.05
Skewness = +0.72 (mild positive skew in change scores)
Q-Q plot: 3 points deviate from the reference line at the lower tail
Paired t-test (methodologically questionable)
Mean change: −22.4mm (SD = 22.5mm)
t(21) = 4.68, p < 0.001, Cohen's d = 0.996
95% CI for mean change: [−32.4mm, −12.4mm]
Wilcoxon Signed-Rank (correct choice for n=22, non-normal)
T+ = 191 (sum of positive ranks), T− = 42 (sum of negative ranks)
z = 3.41, p = 0.001, r = 0.73 (large effect)
[17 of 22 participants showed improvement; 4 worsened; 1 no change]
Same substantive conclusion — but methodology is defensible
Both: significant improvement in pain VAS following intervention
Difference: Wilcoxon is robust to the non-normal change distribution
The paired t-test gives d=0.996; Wilcoxon gives r=0.73 (large by both scales)
At thesis defence: "Why did you choose Wilcoxon?" has a clean, evidence-based answer.
Model Reporting — Wilcoxon Signed-Rank
“Pain VAS scores (0–100mm) were compared before and after intervention using the Wilcoxon signed-rank test, as change scores showed significant non-normality on the Shapiro-Wilk test (W=0.887, p=0.017). Median pain score decreased significantly from 62mm (IQR: 48–74) at baseline to 38mm (IQR: 25–52) following intervention (z=3.41, p=0.001, r=0.73), indicating a large treatment effect.”
The examiner’s question: “You got the same conclusion either way — does the choice of test really matter?” Yes. First, the tests don’t always agree — as demonstrated in Example 1. Second, using an invalid test undermines the methodological credibility of the entire chapter. A thesis examiner who notices a violated assumption — and most experienced examiners will — questions not just this result but the researcher’s statistical competence throughout the thesis. The correct test is always the defensible test, regardless of whether the conclusion changes.
3
Likert Composite Wellbeing Score Across 4 Treatment Groups: Kruskal-Wallis
An RCT compares four treatment groups on a composite wellbeing score: the sum of 10 Likert items, each rated 1–5, giving a total score ranging from 10 to 50. n=20 per group. The researcher considers one-way ANOVA, noting that the composite score “looks like a continuous variable.”
10–50
Score range (ordinal composite)
2/4
Groups failing Shapiro-Wilk
p=0.040
Levene’s test (unequal variances)
Descriptive Statistics (Median, IQR)
Group A (placebo): median=28 (IQR=22–33)
Group B (low dose): median=31 (IQR=26–38)
Group C (high dose): median=36 (IQR=30–42)
Group D (combination): median=39 (IQR=33–45)
Assumption Checking
Shapiro-Wilk normality:
Group A: W=0.872, p=0.041 → NON-NORMAL
Group B: W=0.921, p=0.168 → Normal (just)
Group C: W=0.893, p=0.078 → Borderline (p<0.10)
Group D: W=0.851, p=0.022 → NON-NORMAL
Levene's test for homogeneity of variances:
F(3,76) = 2.89, p = 0.040 → SIGNIFICANT heterogeneity
Two violations identified: (1) non-normality in 2 of 4 groups
(2) heterogeneity of variance
Critical issue: The score is a sum of ordinal items = ordinal composite.
The numbers 10–50 are scores, not measurements on a true interval scale.
Parametric tests are inappropriate regardless of distribution shape.
One-Way ANOVA (inappropriate)
F(3,76) = 7.23, p < 0.001, η² = 0.22
Post-hoc Tukey: D vs A p<0.001, D vs B p=0.012, C vs A p=0.031
Kruskal-Wallis (correct)
H(3) = 19.47, p < 0.001, η²H = 0.21
Post-hoc Dunn test (Bonferroni-corrected):
D vs A: z=3.82, p(adj)<0.001
D vs B: z=2.64, p(adj)=0.033
C vs A: z=2.41, p(adj)=0.064 [n.s. after correction]
C vs B: z=1.47, p(adj)=0.562 [n.s.]
Same omnibus conclusion (p<0.001); different post-hoc pairwise findings.
Kruskal-Wallis: H(3) = 19.47, p < 0.001, η²H = 0.21. Dunn post-hoc (Bonferroni): combination therapy (Group D) was significantly higher than placebo (p<0.001) and low-dose (p=0.033); high-dose (Group C) vs placebo reached p=0.064 after correction — not significant after adjusting for multiple comparisons.
The composite score trap: Composite Likert scores are ordinal, not continuous — even when they span a wide numerical range. The individual items are rated 1–5 on a discrete scale where the “distance” between ratings is assumed but not proven to be equal. Adding 10 ordinal items produces an ordinal composite. ANOVA assumes interval-scale data with equal measurement intervals — an assumption that no Likert-based composite can guarantee. The Kruskal-Wallis H test is the correct analysis, and the post-hoc comparison using Dunn’s test with Bonferroni correction reveals that the ANOVA would have over-counted significant pairwise differences (C vs A was significant by Tukey but not after Bonferroni on the rank-based test). This difference in post-hoc conclusions can directly affect the interpretation of which treatment groups differ.
Thesis Writing Recommendations
Postgraduate researchers are frequently uncertain about whether to present their parametric/non-parametric test selection rationale in the Methods chapter or the Results chapter. The answer: both, in different ways.
Model Methods Paragraph — Statistical Analysis
“Continuous variables were assessed for normality using the Shapiro-Wilk test (for n<50) and by inspection of Q-Q plots and histograms. Normally distributed data are presented as mean ± SD; non-normally distributed data are presented as median and interquartile range (IQR). Between-group comparisons were made using the independent samples t-test (normally distributed continuous data with equal variances confirmed by Levene’s test), Welch’s t-test (unequal variances), or the Mann-Whitney U test (non-normally distributed or ordinal data). Pre-post comparisons were made using the paired t-test (normal distribution of change scores) or the Wilcoxon signed-rank test. For comparisons across three or more groups, one-way ANOVA with Tukey post-hoc testing was applied where assumptions were met; the Kruskal-Wallis H test with Dunn post-hoc correction (Bonferroni) was applied otherwise. Effect sizes were reported as Cohen’s d (parametric) or r = z/√N (non-parametric). All tests were two-tailed; α = 0.05.”
Model Results Disclosure — Justifying the Non-Parametric Choice
“Hospital length of stay data were significantly non-normally distributed in both the standard care group (Shapiro-Wilk W=0.821, p=0.002) and the enhanced recovery group (W=0.847, p=0.004), with positive skewness of +2.8 and +1.9 respectively. Accordingly, between-group comparison was performed using the Mann-Whitney U test. Median LOS was significantly shorter in the enhanced recovery group (5 days, IQR=3–8) compared with standard care (6 days, IQR=4–11; U=736, z=2.48, p=0.013, r=0.26).”
Always Report the Normality Test Result
In your Results chapter, briefly report the Shapiro-Wilk statistic and p-value that justified your choice of test. Examiners cannot verify your decision without it. One sentence per variable is sufficient — you don't need a separate table unless testing many variables simultaneously.
Match Descriptive Statistics to the Test Used
If you use Mann-Whitney U or Wilcoxon SR, report median and IQR — not mean and SD. Using mean±SD alongside a non-parametric test tells an examiner that you don't understand why you chose the non-parametric test. The non-parametric test is appropriate because the mean is a distorted measure for your distribution.
Always Report a Non-Parametric Effect Size
For Mann-Whitney U and Wilcoxon SR, report r = z / √N. For Kruskal-Wallis, report η²H. These are required for a complete result: a p-value alone does not tell readers how large the effect was. Many published papers and theses omit non-parametric effect sizes — a gap your examiner will notice and reward you for filling.
Pre-Specify Your Statistical Approach for Multiple Outcomes
If your study has multiple outcome variables, test normality for each variable separately (not just for the primary outcome). Different variables in the same study may require different tests. State in your methods that each variable will be tested individually and report the outcome for each. This prevents post-hoc test selection.
Common Mistakes Researchers Make
Mistake 1: Skipping Normality Testing Entirely
The most common error by a wide margin. Many researchers simply apply a t-test or ANOVA without testing whether the normality assumption is met, citing precedent (“previous studies in this field used a t-test”) or data type (“the outcome is continuous so I used a t-test”). Neither justification is valid. Previous studies may also have been wrong, and continuous data is necessary but not sufficient for a parametric test.
Fix: Run Shapiro-Wilk and inspect Q-Q plots as a mandatory step before any parametric test. Document both in your Statistical Analysis section in Methods, and report the Shapiro-Wilk W and p-value in Results wherever it justifies your test selection.
Mistake 2: Using Non-Parametric Tests “to be safe” Without Justification
The reverse error: using Mann-Whitney U or Kruskal-Wallis for all analyses regardless of whether normality holds, on the grounds that non-parametric tests are “safer.” This is statistically incorrect and wasteful. Non-parametric tests are less powerful than their parametric equivalents when normality holds. In underpowered studies, the 4.5% power reduction from using Mann-Whitney U instead of a t-test can push a marginal result from significant to non-significant.
Fix: Test normality for each variable and choose the test that the data justify. Do not default to non-parametric tests. If an examiner or reviewer asks why you chose a non-parametric test, your answer should always be: “Because normality was violated [evidence].”
Mistake 3: Treating Likert Scale Data as Continuous
Individual Likert items (e.g., “Rate your satisfaction from 1 to 5”) are ordinal. So are composite Likert scores — because the equal-interval property of a true interval scale cannot be guaranteed for Likert responses. Running a t-test or ANOVA on Likert data is a violation of the interval-scale assumption that parametric tests require, regardless of the apparent “spread” of the composite score across a wide range.
Fix: Use Mann-Whitney U (two groups), Kruskal-Wallis (three or more groups), or Wilcoxon SR (pre-post within subjects) for Likert and ordinal data. Present medians and IQRs, not means and SDs. Note that some statisticians accept ANOVA for Likert composites with 7+ items and approximately symmetrical distributions — if you take this position, explicitly justify it and show the distribution is symmetric.
Mistake 4: Applying the Standard t-test When Variances Are Unequal
The standard independent samples t-test assumes equal variances in both groups (homoscedasticity). When Levene’s test shows significantly unequal variances (p<0.05), the standard t-test produces inflated Type I error rates, especially when group sizes are unequal. Many SPSS users report the standard t-test result and ignore the Welch’s t-test row, which is valid when variances differ.
Fix: Always check Levene’s test output when using the independent samples t-test. If Levene’s p<0.05, report the Welch’s t-test (SPSS: “Equal variances not assumed” row). State this choice explicitly: “Welch’s t-test was applied due to significant heterogeneity of variance (Levene’s F=5.12, p=0.027).”
Mistake 5: Using Mann-Whitney U as a “Test for Medians”
A persistent misconception is that the Mann-Whitney U test compares medians. It does not — technically, it tests stochastic superiority: the probability that a randomly selected value from Group A exceeds a randomly selected value from Group B. The test compares the full distributions of the two groups, not specifically their medians. When two groups have similar medians but different distributions, Mann-Whitney U can be significant even though medians are nearly equal.
Fix: Report that the Mann-Whitney U test was used to compare the distribution of ranks between groups. Present medians and IQRs as the descriptive summary for non-parametric data, but clarify in your methods that the Mann-Whitney U “tests whether one group tends to have higher values than the other” rather than specifically testing for a difference in medians. This nuance is frequently tested in PhD vivas.
Mistake 6: Forgetting Post-Hoc Correction After Kruskal-Wallis
Kruskal-Wallis H tells you that a significant difference exists somewhere among three or more groups — it does not tell you which specific pairs differ. Conducting Kruskal-Wallis and then reporting which individual groups differ without a post-hoc multiple-comparison correction is equivalent to conducting multiple unprotected pairwise tests, inflating the Type I error rate. With four groups, there are six possible pairwise comparisons.
Fix: Always follow a significant Kruskal-Wallis result with Dunn’s test using Bonferroni or Benjamini-Hochberg (false discovery rate) correction for multiple comparisons. Bonferroni is more conservative (reduces Type I error); BH is more powerful (less Type II error). Report the adjusted p-values, not the uncorrected pairwise values. Specify which correction was applied in the Methods.
Scientific Reporting Standards
- For parametric tests (independent t-test): “The treatment group (mean±SD: 62.4 ± 8.3) showed significantly higher scores than the control group (54.1 ± 9.7); t(88)=4.73, p<0.001, Cohen’s d=0.91 (large effect).”
- For parametric tests (one-way ANOVA): “A significant between-group difference was found, F(3,76)=7.23, p<0.001, η²=0.22 (large effect). Tukey post-hoc tests showed Group D was significantly higher than Groups A (p<0.001) and B (p=0.012).”
- For non-parametric tests (Mann-Whitney U): “Median LOS was significantly shorter in the ERP group (5 days, IQR=3–8) vs standard care (6 days, IQR=4–11); U=736, z=2.48, p=0.013, r=0.26 (small-to-medium effect).”
- For non-parametric tests (Kruskal-Wallis): “A significant difference in wellbeing scores was found across groups, H(3)=19.47, p<0.001, η²H=0.21. Dunn’s post-hoc tests (Bonferroni-corrected) revealed Group D was significantly higher than Groups A (p adj<0.001) and B (p adj=0.033).”
- Effect size for Wilcoxon SR: “Wilcoxon signed-rank test: z=3.41, p=0.001, r=0.73 (large effect; r=z/√N=3.41/√44=0.73).”
- Reporting the normality rationale: Always state which normality test was used, its result, and how it influenced test selection. Reviewers in high-impact journals will ask; examiners in thesis defences will ask. Having this documented pre-empts the question.
Practical Guidance for Postgraduate Researchers
Run Your Full Normality Check Before Any Analysis
Before testing any hypothesis, generate a normality report for every continuous variable: Shapiro-Wilk W and p-value, skewness, kurtosis, and a Q-Q plot. In SPSS: Analyze → Descriptive Statistics → Explore → select your variables, then Plots → Normality plots with tests. This gives you everything you need to justify your test selection for each variable.
When in Doubt, Report Both Analyses
For borderline cases (Shapiro-Wilk p between 0.04 and 0.08, or n between 25 and 35 per group), some supervisors recommend running both parametric and non-parametric tests and reporting both with a clear statement of which was primary and why. If both agree, the conclusion is robust; if they disagree, investigate why and report the discrepancy transparently.
Consider Log-Transformation Before Non-Parametric Tests
For right-skewed positive data (CRP, enzyme levels, LOS, income), log-transformation (natural log or log₁₀) often achieves approximate normality — which restores the power advantages of parametric tests on the transformed scale. If the transformed data pass Shapiro-Wilk, you can use a parametric test on the transformed values. Back-transform the result (geometric mean, ratio) for clinical interpretation.
Know When to Escalate to a Biostatistician
The parametric vs non-parametric decision is your first-level choice, but complex designs (clustered data, mixed models, survival analysis, multiple correlated outcomes) require methodology beyond this framework. If your study has repeated measurements on the same patients over multiple time points, correlated outcomes, or a hierarchical data structure (patients within hospitals), seek biostatistical advice — these scenarios require multilevel models or GEE, not standard non-parametric tests.
Frequently Asked Questions
What is the difference between parametric and non-parametric tests? +
Parametric tests make specific assumptions about the population distribution — most critically that data or residuals are normally distributed. They analyse actual data values and compare means, variances, or regression coefficients. Examples: t-test, ANOVA, Pearson correlation. Non-parametric tests are distribution-free: they make no assumptions about the population distribution shape. They convert data to ranks and analyse those ranks. Examples: Mann-Whitney U, Kruskal-Wallis, Wilcoxon signed-rank, Spearman correlation. When parametric assumptions are met, parametric tests are more powerful. When assumptions are violated — especially with small samples, skewed data, or ordinal measurements — non-parametric tests are more appropriate and can match or exceed parametric power.
When should I use a non-parametric test? +
Use a non-parametric test when: (1) Data are ordinal (Likert scales, ranked outcomes, pain scores on a discrete scale) — ordinal data cannot fulfill the interval-scale requirement of parametric tests. (2) Normality is violated — Shapiro-Wilk p<0.05 in small samples, or marked deviation on Q-Q plot in any sample size. (3) Small sample (n<15 per group) — the CLT provides insufficient protection for very small samples. (4) Extreme outliers that cannot be removed with justification — non-parametric tests are substantially more outlier-robust. (5) Floor or ceiling effects — when many participants score at the minimum or maximum, the distribution is non-normal by design. (6) Distribution is known to be right-skewed (LOS, CRP, enzyme levels, survival time) — assume non-normality unless proven otherwise in small clinical samples.
What are the assumptions of parametric tests? +
The major assumptions are: (1) Normality — data or residuals follow a normal distribution within each group (Shapiro-Wilk, Q-Q plots). (2) Homogeneity of variance — variance is approximately equal across groups (Levene’s test; if violated, use Welch’s t-test or Welch’s ANOVA). (3) Independence of observations — each data point is independent; violated in repeated-measures, matched-pair, or clustered designs. (4) Interval or ratio scale — the difference between values is meaningful and equal; violated by ordinal data. (5) No extreme outliers — outliers distort means and variances disproportionately. All assumptions must be tested; they cannot be assumed.
Which normality test should I use? +
Shapiro-Wilk is the recommended test for n up to approximately 50 — it is the most sensitive test for normality in small to medium samples. For large samples (n>200), formal normality tests become overly sensitive and flag trivially small deviations as significant. For large samples, rely primarily on Q-Q plot inspection combined with skewness (acceptable range: −1.0 to +1.0) and kurtosis (acceptable range: −2.0 to +2.0). Never rely solely on the Kolmogorov-Smirnov test — it is substantially less powerful than Shapiro-Wilk for small samples. SPSS reports both automatically in the Explore procedure; always use the Shapiro-Wilk result as primary for n<50.
Can I always use a non-parametric test to be safe? +
No. This is a common misconception. Using non-parametric tests when parametric assumptions are met discards information: the test only uses the ranks of the data, not the actual values. The power penalty is approximately 4.5% (Asymptotic Relative Efficiency of Mann-Whitney U vs t-test = 0.955 under normality). In an underpowered study with 30 participants where you needed 32 to reach 80% power, this margin matters. The correct decision rule: test assumptions first, then choose the test the data justify. Both over-use of parametric tests (ignoring violated assumptions) and over-use of non-parametric tests (ignoring met assumptions) are methodological errors.
What is the non-parametric equivalent of a t-test? +
Independent samples t-test → Mann-Whitney U test (also called Wilcoxon rank-sum). Compares the distribution of ranks between two independent groups; tests whether values in one group tend to exceed values in the other. Effect size: r = z/√N. Paired samples t-test → Wilcoxon signed-rank test. Ranks the absolute values of the within-pair differences and tests whether positive differences outweigh negative ones. Appropriate for pre-post designs and matched-pair studies with non-normal change scores. One-sample t-test → Wilcoxon signed-rank test (testing whether the sample median differs from a hypothetical value). Note that Mann-Whitney U tests for stochastic superiority, not strictly for a difference in medians.
What is the non-parametric equivalent of ANOVA? +
One-way ANOVA → Kruskal-Wallis H test. Compares ranked outcomes across three or more independent groups. If H is significant (p<0.05), conduct Dunn post-hoc tests with Bonferroni or Benjamini-Hochberg correction. Effect size: η²H = (H − k + 1) / (N − k). Repeated-measures ANOVA → Friedman test. Compares ranked outcomes across three or more related measurements (same participants at multiple time points). Post-hoc: Wilcoxon signed-rank with Bonferroni correction, or Conover post-hoc. Effect size: Kendall’s W = χ² / [N(k−1)]. A significant Kruskal-Wallis result without post-hoc testing is an incomplete analysis.
Does the central limit theorem mean I can always use a t-test with large samples? +
The CLT provides genuine protection for parametric tests with large samples and mild skewness — the sampling distribution of the mean normalises regardless of the raw data distribution. However, the CLT does not override the interval-scale requirement (ordinal data remain ordinal at any n), does not fix heterogeneity of variance (Welch’s t-test is needed), and is insufficient for severe skewness until n is very large (100+ per group for exponential distributions). For clinical variables known to be severely right-skewed (LOS, CRP, troponin), confirm approximate normality visually before applying parametric tests even with n=60 per group. The “n≥30 rule” is a reasonable heuristic for mild skewness, not a universal exemption from assumption checking.
How do I report parametric and non-parametric test results? +
Parametric: “Mean±SD; t(df)=X, p=X, Cohen’s d=X.” Non-parametric: “Median (IQR); U=X [or H=X, T+=X], z=X, p=X, r=X.” Always report: the test name, test statistic, degrees of freedom (for F tests) or exact statistic, p-value (two-tailed unless pre-specified one-tailed), and effect size. Report the Shapiro-Wilk result that justified the test choice in your Results (“As normality was violated [W=0.821, p=0.002], the Mann-Whitney U test was applied”). Match descriptive statistics to the test: mean±SD for parametric; median (IQR) for non-parametric. Never report mean±SD alongside a non-parametric result.
What effect sizes should I report for non-parametric tests? +
Mann-Whitney U / Wilcoxon signed-rank: r = z / √N (where N = total observations). Interpretation: |r| = 0.1 small, 0.3 medium, 0.5 large. Formula: r = z / √N; SPSS gives the z-statistic in the output. Kruskal-Wallis: η²H = (H − k + 1) / (N − k), where H = Kruskal-Wallis statistic, k = number of groups, N = total n. Interpretation similar to η²: 0.01 small, 0.06 medium, 0.14 large. Friedman: Kendall’s W = χ² / [N × (k − 1)]. Range 0–1; W=0.1 slight, W=0.3 moderate, W=0.7 strong agreement/concordance. Do not report Cohen’s d or η² for non-parametric tests — these are based on means and standard deviations, which are not the quantities non-parametric tests analyse.
Run Your Statistical Tests
Normality testing, parametric and non-parametric comparisons, effect size calculation, and CONSORT-aligned reporting — all in one tool.
Open StatClinic →