What is the Kruskal-Wallis test?

The Kruskal-Wallis test is a non-parametric statistical test used to compare three or more independent groups. It tests whether samples come from the same distribution without requiring normality. It ranks all data together and compares rank sums between groups.

When should I use the Kruskal-Wallis test instead of one-way ANOVA?

Use the Kruskal-Wallis test when your data violates the normality assumption required by ANOVA, when you have ordinal data, when you have small sample sizes, or when your data contains significant outliers. ANOVA is preferred when normality holds, as it has more statistical power.

What does the H statistic mean in Kruskal-Wallis?

The H statistic (also called the Kruskal-Wallis statistic) measures how much the rank distributions of the groups differ from what would be expected by chance. A larger H value indicates greater differences between groups. H follows a chi-square distribution with k−1 degrees of freedom.

How do you interpret the p-value in Kruskal-Wallis?

If p < 0.05 (at α = 0.05), you reject the null hypothesis and conclude at least one group differs significantly from the others. If p ≥ 0.05, you fail to reject the null hypothesis. A significant result requires Dunn's post-hoc test to identify which specific pairs differ.

What is Dunn's post-hoc test?

Dunn's test is a pairwise comparison procedure used after a significant Kruskal-Wallis result. It compares each pair of groups using rank-based statistics and applies Bonferroni or Holm correction for multiple comparisons to control the family-wise error rate.

What is the effect size for the Kruskal-Wallis test?

The most common effect size for Kruskal-Wallis is eta-squared (η²), calculated as (H − k + 1) / (N − k). Values near 0.01 = small, 0.06 = medium, 0.14 = large (Cohen, 1988). Epsilon-squared (ε²) is another option that is less biased for small samples.

What are the assumptions of the Kruskal-Wallis test?

The Kruskal-Wallis test assumes: (1) observations are independent within and between groups, (2) the dependent variable is at least ordinal, (3) groups have similar distribution shapes (for median comparison), and (4) groups consist of at least 5 observations for the chi-square approximation to be reliable.

How do I report Kruskal-Wallis results in APA format?

Report as: H(df) = [value], p = [value], η² = [value]. Full example: A Kruskal-Wallis test indicated a significant difference in scores across groups, H(2) = 8.34, p = .015, η² = .18 (large effect). Dunn's post-hoc tests revealed that Group A differed significantly from Group C (p = .012).

Can I use Kruskal-Wallis with only two groups?

Technically yes, but the Mann-Whitney U test is preferred for two-group comparisons. The Kruskal-Wallis test with k=2 is mathematically equivalent to Mann-Whitney U, producing identical p-values. Use Mann-Whitney U directly for two groups as it is the standard choice.

What post-hoc test should I use after Kruskal-Wallis?

Dunn's test is the most widely recommended and used post-hoc test after Kruskal-Wallis. Other options include pairwise Mann-Whitney U with Bonferroni correction, the Conover-Iman test (slightly more powerful), and Nemenyi's test. Dunn's test with Holm-Bonferroni correction balances power and Type I error control.

Is the Kruskal-Wallis test suitable for small samples?

Yes, the Kruskal-Wallis test works with small samples. However, when each group has fewer than 5 observations, the chi-square approximation may not be accurate. In such cases, exact p-values should be computed using permutation methods rather than relying on the asymptotic distribution.

What is the difference between Kruskal-Wallis and Friedman test?

The Kruskal-Wallis test is for independent groups (between-subjects design), while the Friedman test is for related groups (within-subjects or repeated measures design). Use Friedman when the same subjects are measured under multiple conditions, and Kruskal-Wallis when each group consists of different participants.

Kruskal Wallis Test Calculator – Free Online Non-Parametric ANOVA Tool | StatsUnlock

Kruskal-Wallis Test Calculator – Free Online H-Test Tool | StatsUnlock

📥 Enter Your Data

Sample Dataset

Upload file (.csv, .txt, .xlsx, .xls):

Headers auto-detected. Select which columns map to each group.

Enter values directly in the table. Leave cells blank for unequal group sizes.

Significance Level (α)

Post-Hoc Correction

Tie Correction

📊 Full Results Table

📋 Group Summary

🔍 Dunn's Post-Hoc Pairwise Comparisons

📈 Visualizations

Box Plot — Median & IQR by Group

Mean Rank Comparison

Distribution Overview (Violin-style KDE)

Ranked Data Scatter per Group

📎 Copy Summary to Clipboard

✅ Assumption Checks

💡 Detailed Interpretation of Results

✍️ How to Write Your Results in Research

Choose the writing style that matches your output — dissertation, journal article, plain-language summary, conference poster, or pre-registration. All templates are auto-filled with your computed statistics.

①

Kruskal-Wallis H Statistic

H = [12 / N(N+1)] × Σ(Rᵢ² / nᵢ) − 3(N+1)

HTest statistic approximating a chi-square distribution with k−1 df

NTotal number of observations across all groups

kNumber of groups being compared

RᵢSum of ranks assigned to group i (all data ranked together 1 to N)

nᵢSample size of group i

②

Tie Correction Factor

C = 1 − [Σ(tˇ³ − tˇ) / (N³ − N)]

H_cTie-corrected H = H / C (always ≥ uncorrected H)

tˇNumber of tied observations in each tie group

CCorrection factor ≤ 1; closer to 1 when ties are rare

RuleIf no ties exist, C = 1 and corrected H = uncorrected H

③

Degrees of Freedom

df = k − 1

dfDegrees of freedom for the chi-square distribution approximation

kNumber of groups; df = 2 for three groups, df = 3 for four groups, etc.

NoteChi-square approximation is valid when each group n ≥ 5

④

Effect Size — Eta-Squared (η²)

η² = (H − k + 1) / (N − k)

η²Proportion of variance in ranks explained by group membership

HKruskal-Wallis test statistic (tie-corrected if applicable)

Benchmarks0.01 = small · 0.06 = medium · ≥0.14 = large (Cohen, 1988)

⑤

Effect Size — Epsilon-Squared (ε²) — Less Biased

ε² = H / [(N² − 1) / (N + 1)]

ε²Alternative effect size less biased for small N; ranges 0–1

NTotal sample size across all groups

BenchmarksSame cutoffs as η²: 0.01 small · 0.06 medium · 0.14+ large

⑥

Dunn's Post-Hoc Z Statistic

zᵢˇ = (R̄ᵢ − R̄ˇ) / SEᵢˇ

R̄ᵢMean rank of group i based on combined ranks across all N observations

SEᵢˇSE = √[N(N+1)/12 × (1/nᵢ + 1/nˇ)] adjusted for ties

p-valueTwo-tailed p from standard normal; multiply by number of comparisons (Bonferroni)

⑦

Mean Rank per Group

R̄ᵢ = Rᵢ / nᵢ

R̄ᵢAverage rank of observations in group i; higher = higher data values

RᵢSum of all ranks assigned to observations in group i

nᵢNumber of observations in group i

RuleExpected mean rank under H₀ = (N+1)/2 for all groups

Enter your data

Type or paste comma-separated values for each group, upload a CSV/Excel file, or use the manual entry table. Each group represents one category of your independent variable.

Name your groups

Click the editable group name box above each textarea (e.g., "Control", "Low Dose", "High Dose") to label your groups meaningfully. Names appear in all outputs.

Add or remove groups

Use the "+ Add Group" and "− Remove Last Group" buttons. You can compare 3 to 6 groups. The test requires at least 3 groups; for 2 groups, use the Mann-Whitney U test instead.

Try a sample dataset

Select one of the 5 built-in sample datasets from the dropdown to explore how the tool works before entering your own data. Each sample represents a real-world research scenario.

Set significance level (α)

Choose 0.01, 0.05, or 0.10. The most common choice in academic research is α = 0.05, meaning a 5% probability threshold for declaring a result significant.

Choose post-hoc correction

Holm-Bonferroni is recommended as it controls family-wise error while being more powerful than standard Bonferroni. Select "None" to see raw p-values without adjustment.

Run the analysis

Click "Run Kruskal-Wallis Test". Results appear immediately — H-statistic, degrees of freedom, p-value, effect size (η²), and group summary statistics including median and mean rank.

Read the four charts

The box plot shows medians and IQR. The rank chart compares mean ranks per group. The KDE violin plot shows distribution shape. The scatter plot shows individual ranked observations.

Interpret your results

Read the detailed interpretation section. It tells you whether differences are statistically significant, explains the effect size in practical terms, and highlights which pairs differ post-hoc.

Copy a write-up template

Click "📋 Copy" on any write-up card (APA, Thesis, Plain Language, Abstract, Pre-Registration) to copy a fully auto-filled results paragraph for your paper or report.

✅ Use Kruskal-Wallis When…

You have 3 or more independent groups
Your data violates normality (skewed, ordinal)
Group sizes are small (n < 20 per group)
Your dependent variable is ordinal
Significant outliers are present in the data
You are comparing ranks rather than means

❌ Avoid Kruskal-Wallis When…

Data is normally distributed (use one-way ANOVA)
Groups are related / paired (use Friedman test)
You have only 2 groups (use Mann-Whitney U)
Your dependent variable is binary (use chi-square)
You need to include covariates (use ANCOVA)

📌 Real-world examples:

• Ecology: Comparing species richness across three habitat types (forest, grassland, wetland)
• Clinical: Comparing pain scores (0–10 ordinal scale) across four treatment arms
• Education: Comparing test performance across five teaching methods with non-normal score distribution
• Biology: Comparing plant growth across three fertilizer types when data is right-skewed

❓ Frequently Asked Questions

📚 References

The Kruskal-Wallis test calculator on this page is based on the original rank-sum methodology for non-parametric one-way ANOVA, Dunn's pairwise post-hoc comparisons, and eta-squared effect size estimation. Key peer-reviewed sources are listed below.

Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260), 583–621. https://doi.org/10.1080/01621459.1952.10483441

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3), 241–252. https://doi.org/10.1080/00401706.1964.10490181

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates. https://doi.org/10.4324/9780203771587

Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2), 65–70. https://www.jstor.org/stable/4615733

Conover, W. J., & Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. The American Statistician, 35(3), 124–129. https://doi.org/10.1080/00031305.1981.10479327

Vargha, A., & Delaney, H. D. (1998). The Kruskal-Wallis test and stochastic homogeneity. Journal of Educational and Behavioral Statistics, 23(2), 170–192. https://doi.org/10.3102/10769986023002170

Dinno, A. (2015). Nonparametric pairwise multiple comparisons in independent groups using Dunn's test. The Stata Journal, 15(1), 292–300. https://doi.org/10.1177/1536867X1501500117

Field, A. (2013). Discovering statistics using IBM SPSS statistics (4th ed.). SAGE Publications. ISBN: 978-1446249185

Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods (3rd ed.). John Wiley & Sons. https://doi.org/10.1002/9781119196037

Tomczak, M., & Tomczak, E. (2014). The need to report effect size estimates revisited. An overview of some recommended measures of effect size. Trends in Sport Sciences, 1(21), 19–25. Link

Corder, G. W., & Foreman, D. I. (2014). Nonparametric statistics: A step-by-step approach (2nd ed.). John Wiley & Sons. https://doi.org/10.1002/9781118592908

McDonald, J. H. (2014). Kruskal–Wallis test. In Handbook of biological statistics (3rd ed., pp. 157–164). Sparky House Publishing. http://www.biostathandbook.com/kruskalwallis.html

Kassambara, A. (2017). Practical statistics in R for comparing groups: Numerical variables. STHDA. http://www.sthda.com/english/wiki/kruskal-wallis-test-in-r

Chambers, J. M., Cleveland, W. S., Kleiner, B., & Tukey, P. A. (1983). Graphical methods for data analysis. Wadsworth International Group. https://doi.org/10.1201/9781351072304

Zar, J. H. (2010). Biostatistical analysis (5th ed.). Prentice Hall. ISBN: 978-0131008465