Kruskal-Wallis Test Calculator

The Kruskal-Wallis test compares k independent groups using ranks of the pooled data. Let N be the total sample size, n_i the size of group i, and R_i the sum of ranks in group i.

H = (12 / (N(N+1))) * Σ (Rᵢ² / nᵢ)  −  3(N+1)

Tie factor T = 1 − Σ(t³ − t) / (N³ − N)
H_corrected = H / T

p-value ≈ 1 − CDF_χ²(H_corrected, df = k − 1)

1. Enter values for each group (commas, spaces, or new lines work).
2. Choose alpha, tie correction, and post-hoc options.
3. Press Submit to compute H, df, p-value, and effect sizes.
4. Review per-group mean ranks and (optional) Dunn comparisons.
5. Download CSV or PDF for reporting or archiving.

Rank-based comparison at scale

The calculator supports 2 to 20 independent groups and aggregates all observations into a pooled rank list. For N up to 5,000 values, ranking remains responsive in typical hosting. When N grows, export CSV for auditing and replicate results in your workflow. Record the unit of measurement and the sampling window so the ranks remain comparable across teams and reporting periods.

Interpreting H and p-value

The H statistic summarizes between-group rank separation. Under the chi-square approximation with df = k−1, smaller p-values indicate stronger evidence that at least one group distribution differs. Common reporting uses alpha 0.05, but regulated work often uses 0.01. If p is near the threshold, rerun with sensitivity checks, such as removing obvious data entry errors or validating collection order.

Tie correction impact

Ties compress rank variance. The tie factor T = 1 − Σ(t³−t)/(N³−N) adjusts H upward via H/T when repeated values exist. If your data are rounded (for example, integer scores), enable correction to reduce false confidence. A low T indicates heavy duplication; consider collecting at higher precision or using a measurement instrument with finer resolution.

Effect sizes for practical meaning

Alongside significance, the page reports η² and ε² approximations from H. As rough benchmarks, values near 0.01 suggest small differences, 0.06 moderate, and 0.14 large. Use them to justify whether a detected shift matters operationally. For stakeholder communication, combine effect size with a plain-language statement about direction using mean ranks, and attach the ranked table for traceability.

Post-hoc comparisons

With three or more groups, optional Dunn tests estimate pairwise z-scores using mean-rank differences and a variance term based on N(N+1)/12 scaled by T. Bonferroni is conservative; Holm keeps familywise control while improving power. Report both raw and adjusted p-values, and focus follow-up investigation on comparisons with consistent direction and stable rank gaps across repeated samples.

Data hygiene and assumptions

Use independent samples, similar measurement scales, and consistent sampling windows. Extreme imbalance (for example, one group with n=3 and another with n=300) can distort interpretation even if p is small. Inspect ranked tables and graphs before concluding. For ongoing monitoring, track H, p, and ε² monthly and flag changes beyond pre-set practical thresholds, not only statistical thresholds in production dashboards.

What does the test measure?

It tests whether two or more independent groups come from the same distribution by comparing pooled ranks rather than raw values. It is robust to non-normality and unequal variances, but still assumes independent observations and comparable measurement scales.

When should I enable tie correction?

Enable it whenever duplicated values occur, especially with rounded or discrete data. The correction adjusts the variance of ranks so the chi-square approximation is more reliable and avoids underestimating p-values when ties are frequent.

Do I need equal sample sizes?

No. The method allows unequal n, but extremely imbalanced groups can make results harder to interpret. Consider collecting more observations for very small groups or supplementing with descriptive summaries such as medians and mean ranks.

Why run Dunn post-hoc tests?

Kruskal-Wallis indicates at least one difference, but not which groups differ. Dunn comparisons quantify pairwise mean-rank gaps with a z statistic. Use adjusted p-values to control familywise error when testing multiple pairs.

How should I report results?

Report H (tie-corrected if used), df, p-value, alpha, and an effect size such as epsilon-squared. Add a per-group mean-rank table and note the adjustment method for post-hoc comparisons, if performed.

What data formats are accepted?

Paste numbers separated by commas, spaces, tabs, or new lines. Scientific notation is supported. Non-numeric tokens are ignored. Use the example loader to confirm formatting, then download CSV or PDF to keep a record of inputs and outputs.