Kruskal-Wallis H Critical Value Calculator
What Is the Kruskal-Wallis Critical Value Table?
The Kruskal-Wallis critical value table lists the minimum value the H statistic must reach for a result to be statistically significant at a given α level. It is organized by degrees of freedom (df = k − 1, where k is the number of groups) and by significance level.
For samples with at least 5 observations per group — the situation in most research — the H statistic approximates a chi-square distribution with df = k − 1. This means the chi-square critical value table serves as the Kruskal-Wallis critical value table for practical purposes.
Quick decision rule: H is what you compute from ranked data; Hcrit is what you look up. If H ≥ Hcrit, reject H₀ and conclude that at least one group distribution differs. The test does not identify which groups — that requires Dunn's post hoc test with Bonferroni correction.
Kruskal-Wallis H Test: What It Does and When to Use It
The Kruskal-Wallis H test, developed by William Kruskal and W. Allen Wallis in 1952, is a rank-based nonparametric alternative to one-way ANOVA. Rather than comparing group means, it converts all observations to ranks and compares rank distributions across groups.
The test answers one question: do all k groups come from populations with the same distribution? A statistically significant H means at least one group differs — but it does not say which one or by how much.
When to Use Kruskal-Wallis
- Three or more independent groups
- Ordinal or non-normal continuous data
- Small samples with non-normality
- Outliers that cannot be removed
- ANOVA normality assumption violated
Test Requirements
- k ≥ 3 independent groups
- Dependent variable at least ordinal
- Independent observations within groups
- Similar distributional shapes (for median comparison)
- No normality assumption required
Kruskal-Wallis Critical Value Table (Chi-Square Approximation)
All values below are Hcrit — the minimum H statistic for statistical significance using the chi-square approximation (valid when each group has ≥ 5 observations). Select the α level tab, then find the row matching your number of groups k. Click any cell to load it into the calculator above.
df = k − 1, where k = number of groups. Reject H₀ if H ≥ Hcrit. Values are chi-square critical values (χ²) with df = k − 1, valid when each group has ≥ 5 observations. For small samples (n < 5 per group), use exact Kruskal-Wallis tables from Kruskal & Wallis (1952, Journal of the American Statistical Association). See the full chi-square table.
The Kruskal-Wallis H Formula Explained
The H statistic measures how much the rank sums of each group deviate from what pure chance would produce if all groups came from the same population.
Standard H Formula
N = total number of observations across all groups
k = number of groups
nᵢ = sample size of group i
Rᵢ = sum of ranks assigned to observations in group i
Σ = summation across all k groups
Tie Correction Factor
When ties occur in the data, the standard H formula slightly underestimates the true test statistic. Dividing H by the correction factor C adjusts for this:
C = 1 − [Σ(tⱼ³ − tⱼ) / (N³ − N)]
Where tⱼ = the number of observations sharing the j-th tied rank. The correction is small unless there are many ties. Most software applies this correction automatically.
Degrees of Freedom
With k = 3 groups: df = 2. With k = 5 groups: df = 4. This df value is used to look up the critical value in the chi-square table above.
How to Read the Kruskal-Wallis Critical Value Table
Reading the table correctly takes two inputs: your number of groups k and your significance level α. The process below works whether you use the chi-square table above or an exact small-sample table.
Step 1 — Count Your Groups
Count k, the number of independent groups being compared. Three teaching methods: k = 3. Four medical treatments: k = 4. Five customer segments: k = 5. The number of groups determines df = k − 1.
Step 2 — Compute df = k − 1
Subtract 1 from the number of groups to get degrees of freedom. k = 3 → df = 2. k = 4 → df = 3. k = 6 → df = 5. This df value is the row index for your lookup in the chi-square table.
Step 3 — Choose α
Select your significance level before looking at the data. α = 0.05 is standard in most research. α = 0.01 is used when a stricter threshold is needed (medical research, for example). Your choice of α sets the column in the table.
Step 4 — Find H Critical
Locate the row for your df in the table above. Read the value in the column matching your α level. That number is Hcrit.
Step 5 — Apply the Decision Rule
If H < Hcrit → Fail to reject H₀ → Insufficient evidence of a group difference
A significant result tells you that groups differ — not which ones or how much. Follow up with Dunn's test and Bonferroni correction to identify specific pairwise differences.
Worked Example 1: Student Exam Scores Across Teaching Methods
Scenario: An education researcher tests whether three different teaching methods produce different exam scores. Students are randomly assigned: Method A (n=5), Method B (n=5), Method C (n=5). Scores are: A: 72, 68, 85, 74, 79 — B: 65, 58, 71, 63, 66 — C: 88, 92, 84, 95, 90. Data are not assumed to be normally distributed.
Solution — Step by Step
| Step | Action | Result |
|---|---|---|
| 1 | State H₀ and H₁ | H₀: All 3 methods produce same distribution; H₁: At least one differs; α = 0.05 |
| 2 | Rank all N=15 observations | 58(1), 63(2), 65(3), 66(4), 68(5), 71(6), 72(7), 74(8), 79(9), 84(10), 85(11), 88(12), 90(13), 92(14), 95(15) |
| 3 | Compute rank sums Rᵢ | R_A = 7+5+11+8+9 = 40; R_B = 3+1+6+2+4 = 16; R_C = 12+14+10+15+13 = 64 |
| 4 | Calculate H statistic | H = (12/[15×16]) × [(40²/5)+(16²/5)+(64²/5)] − 3×16 = (12/240)×[320+51.2+819.2] − 48 = H ≈ 10.42 |
| 5 | Find H critical (df = k−1 = 2) | df = 3−1 = 2; χ²crit(α=0.05, df=2) = 5.991 |
| 6 | Compare H vs Hcrit | 10.42 > 5.991 → Reject H₀ |
Interpretation
H(2) = 10.42, p < 0.05. Teaching methods produce statistically different score distributions. To identify which specific methods differ, run Dunn's test on all three pairwise comparisons (A vs B, A vs C, B vs C), applying Bonferroni correction with adjusted α = 0.05/3 = 0.017 per comparison. Based on rank sums alone, Method C (R = 64) appears highest and Method B (R = 16) lowest.
Worked Example 2: Patient Recovery Times Across Four Treatments
Scenario: A clinical researcher compares recovery times (days) for four treatment groups: Treatment A (n=6), B (n=6), C (n=6), D (n=6). Recovery data are right-skewed, ruling out ANOVA. The Kruskal-Wallis test is appropriate.
After ranking all N = 24 observations from 1 to 24 and summing ranks per group, the researcher obtains: R_A = 84, R_B = 105, R_C = 72, R_D = 39. Applying the H formula with N = 24, k = 4, all nᵢ = 6:
H = (12/600) × [1176+1837.5+864+253.5] − 75
H = 0.02 × 4131 − 75 = 82.62 − 75 = H ≈ 7.62
| k | df | H computed | H crit (α=0.05) | Decision |
|---|---|---|---|---|
| 4 | 3 | 7.62 | 7.815 | Fail to reject H₀ |
Interpretation
H(3) = 7.62, p > 0.05. The evidence is insufficient to conclude that recovery time distributions differ across the four treatments. Note how close H is to the critical value (7.815) — this is a borderline case. Increasing sample size would provide more power to detect differences if they exist in the population. This example demonstrates the importance of adequate sample size planning before data collection.
Quick Examples Across Research Fields
The examples below show how the same lookup process applies across different disciplines. In each case, k groups are compared using the Kruskal-Wallis test at α = 0.05.
Agriculture: Crop Yields Across Fertilizer Types
An agronomist tests five fertilizer formulas on wheat plots (k = 5, df = 4). Yield data are skewed due to weather variation. H = 11.23. Hcrit(α=0.05, df=4) = 9.488. Since 11.23 > 9.488, at least one fertilizer produces a different yield distribution. Dunn's test identifies which pairs differ.
Customer Satisfaction: Ratings Across Service Channels
A business analyst compares customer satisfaction scores (1–10 scale, ordinal data) across phone, email, chat, and in-person service channels (k = 4, df = 3). H = 8.91. Hcrit(α=0.05, df=3) = 7.815. The result is significant, suggesting channel type affects satisfaction ratings.
Psychology: Pain Ratings After Three Interventions
A clinical psychologist uses Likert-scale pain ratings (ordinal data) across three treatment groups (k = 3, df = 2). The normality assumption clearly fails. H = 6.84. Hcrit(α=0.05, df=2) = 5.991. Reject H₀ — at least one intervention produces different pain outcomes.
Manufacturing: Defect Rates Across Production Lines
A quality engineer compares defect counts from six production lines (k = 6, df = 5). Count data violate ANOVA normality. H = 13.14. Hcrit(α=0.01, df=5) = 15.086. Fail to reject H₀ at α = 0.01, but the result is significant at α = 0.05 (Hcrit = 11.070). Choice of α matters here — report which level was pre-specified.
Running the Kruskal-Wallis Test in R, Python, and SPSS
Statistical software computes H, the p-value, and the tie correction automatically. The critical value table above confirms the decision, but software p-values are typically more precise for reporting.
R (base and dunn.test)
kruskal.test(score ~ group, data = df)
# Output: Kruskal-Wallis chi-squared = 10.42, df = 2, p-value = 0.0054
# Dunn's post hoc test
install.packages("dunn.test")
library(dunn.test)
dunn.test(df$score, df$group, method = "bonferroni")
Python (SciPy)
from scikit_posthocs import posthoc_dunn
# Kruskal-Wallis test
stat, p = stats.kruskal(group_a, group_b, group_c)
print(f"H = {stat:.4f}, p = {p:.4f}")
# Dunn's post hoc test
# pip install scikit-posthocs
result = posthoc_dunn([group_a, group_b, group_c], p_adjust='bonferroni')
SPSS
In SPSS: Analyze → Nonparametric Tests → Legacy Dialogs → K Independent Samples → Select Kruskal-Wallis H → Move grouping variable to "Grouping Variable" box → Define groups → OK. Output includes the H statistic, df, and asymptotic significance (p-value). For post hoc tests, use Analyze → Nonparametric Tests → Independent Samples → Pairwise Comparisons.
The RANK Framework: A Memory Aid for Kruskal-Wallis
The five steps of a Kruskal-Wallis analysis map onto the letters R-A-N-K-S — a useful sequence when working through problems without software.
Kruskal-Wallis vs One-Way ANOVA: Which Test to Use
Both tests compare multiple independent groups, but they differ in assumptions, what they test, and when they apply. Use this table to decide which test fits your data.
| Feature | Kruskal-Wallis | One-Way ANOVA |
|---|---|---|
| Type | Nonparametric | Parametric |
| Normality required? | No | Yes |
| Data level | Ordinal or continuous | Continuous |
| What it compares | Rank distributions | Group means |
| Test statistic | H (≈ χ²) | F |
| Statistical power | ~95% of ANOVA (when applicable) | Higher when assumptions met |
| Post hoc test | Dunn's test | Tukey HSD, Bonferroni |
| Effect size | η² = H/(N−1) | η² or ω² |
Choosing Between Kruskal-Wallis and Related Nonparametric Tests
The right test depends on your study design, the number of groups, and whether observations are independent or related.
| Test | Groups | Independent? | Parametric equivalent |
|---|---|---|---|
| Kruskal-Wallis | k ≥ 3 | Yes | One-way ANOVA |
| Mann-Whitney U | k = 2 | Yes | Independent samples t-test |
| Friedman Test | k ≥ 3 | No (related) | Repeated measures ANOVA |
| Wilcoxon Signed-Rank | k = 2 | No (paired) | Paired samples t-test |
| Dunn's Test | Any pair | Yes | Tukey HSD (post hoc) |
Use the statistical test selector for guided test selection based on your data type, group count, and independence structure.
Common Mistakes When Using the Kruskal-Wallis Test
These are the errors that most frequently appear in student work and published research. Each one affects the validity of the conclusion.
Skipping Post Hoc Testing
A significant Kruskal-Wallis result only tells you that groups differ. Without Dunn's test or another pairwise comparison procedure, you cannot say which groups differ. Stopping at the omnibus test is incomplete reporting.
Using Chi-Square Table for Small Samples
The chi-square approximation is only accurate when each group has at least 5 observations. With smaller groups, the approximation is unreliable — use the exact critical value tables from Kruskal and Wallis (1952) instead.
Ignoring Tied Ranks
When observations share the same value, they receive averaged ranks. Ignoring this and assigning sequential ranks to ties inflates H slightly. When ties are frequent, apply the tie correction formula to H before comparing against the critical value.
Applying to Related Samples
Kruskal-Wallis requires independent groups. Repeated measures on the same subjects violate this independence assumption. Use the Friedman test for related samples across three or more conditions.
Using It for Only Two Groups
While mathematically possible, applying Kruskal-Wallis to exactly two groups is non-standard. The Mann-Whitney U test is the correct choice for two independent groups and gives an equivalent result.
Treating "Not Significant" as "No Difference"
Failing to reject H₀ does not prove groups are equal. It means the data collected did not provide sufficient evidence of a difference at the chosen α level. Low statistical power (small n) can produce non-significant results even when true group differences exist.
Symbol and Concept Glossary
Every symbol involved in the Kruskal-Wallis H test, with its definition and the most common misinterpretation.
| Symbol | Name | Definition | Common Error |
|---|---|---|---|
| H | Kruskal-Wallis statistic | Test statistic measuring rank-sum differences across groups | Interpreting H magnitude without comparing to Hcrit |
| k | Number of groups | Count of independent groups being compared | Using k instead of df for the table lookup |
| df | Degrees of freedom | df = k − 1; used to look up Hcrit | Using N − 1 or N − k instead of k − 1 |
| N | Total sample size | Sum of all nᵢ across all groups | Confusing N with nᵢ (individual group size) |
| Rᵢ | Rank sum for group i | Sum of all ranks assigned to observations in group i | Ranking within groups instead of across all observations |
| Hcrit | Critical value | χ² critical value at df = k − 1 and chosen α | Looking up by k instead of df = k − 1 |
| α | Significance level | Probability of Type I error; commonly 0.05 or 0.01 | Setting α after seeing the data (p-hacking) |
| p | p-value | Probability of observing H ≥ computed H if H₀ is true | Treating p as the probability that H₀ is true |
Frequently Asked Questions
Can the Kruskal-Wallis test be used with unequal group sizes?
Yes. The H formula handles unequal sample sizes (nᵢ differs across groups) without any adjustment. Unequal sizes reduce statistical power compared to balanced designs, but the test remains valid. Software implementations handle this automatically.
What effect size is used for Kruskal-Wallis?
The most common effect size for Kruskal-Wallis is eta-squared: η² = H / (N − 1). Values around 0.01 suggest a small effect, 0.06 a medium effect, and 0.14 or above a large effect, following Cohen's (1988) guidelines. Report this alongside the H statistic and p-value for a complete account of results.
How is the Kruskal-Wallis test related to the Mann-Whitney U test?
The Mann-Whitney U test is a special case of the Kruskal-Wallis test when k = 2. When applied to two groups, the Kruskal-Wallis H statistic equals the square of the Mann-Whitney standardized test statistic, and both tests produce identical p-values. For two groups, Mann-Whitney U is preferred; for k ≥ 3, use Kruskal-Wallis.
What happens after a significant Kruskal-Wallis result?
After rejecting H₀, run Dunn's test to determine which specific pairs of groups differ. Dunn's test performs pairwise comparisons using a pooled variance estimate from all groups and applies Bonferroni correction (or Holm correction for more power) to control family-wise Type I error. Report all pairwise p-values and state the correction method used.
Does the Kruskal-Wallis test compare medians?
Technically, Kruskal-Wallis tests whether groups come from populations with identical distributions — not specifically identical medians. The test can be interpreted as a median comparison only when all groups have similarly shaped distributions. If shapes differ substantially, a significant H means the distributions differ, not necessarily that medians do.
Is the Kruskal-Wallis test available in Excel?
Excel does not include a built-in Kruskal-Wallis function. The test can be performed manually in Excel by sorting observations, assigning ranks using RANK.AVG, computing group rank sums, and applying the H formula. Third-party Excel add-ins such as Real Statistics or XLSTAT provide a direct Kruskal-Wallis function. R, Python SciPy, and SPSS all include built-in implementations.
Decision Guide: Which Test Fits Your Research Design
Three or More Independent Groups — Non-Normal Data
The Kruskal-Wallis test is the right choice. It requires no normality assumption, works with ordinal data, and handles unequal group sizes. It is the nonparametric equivalent of one-way ANOVA.
Three or More Related Samples
When the same subjects or matched groups are measured under multiple conditions, use the Friedman test instead. It is the nonparametric equivalent of repeated-measures ANOVA and handles the within-subject correlation structure.
Normal Data, Equal Variance, Three or More Groups
When ANOVA assumptions are satisfied — normality and homogeneity of variance — one-way ANOVA has more statistical power than Kruskal-Wallis. Use ANOVA and report F, df, p-value, and η² or ω².
References and Further Reading
The following peer-reviewed sources and educational resources contain the statistical theory and software documentation underlying this page.
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. — The original paper introducing the H test and exact critical value tables for small samples.
Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric Statistical Methods (3rd ed.). Wiley. — Comprehensive textbook covering Kruskal-Wallis, Dunn's test, and related rank-based methods.
NIST Engineering Statistics Handbook: Kruskal-Wallis Test — Government reference covering the H formula, assumptions, and interpretation with worked examples.
SciPy Documentation: scipy.stats.kruskal — Official Python implementation reference for the Kruskal-Wallis H test, including parameters, return values, and usage examples.
R Documentation: kruskal.test — Base R function documentation with parameters and worked examples for the Kruskal-Wallis rank sum test.
Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3), 241–252. — Source paper for Dunn's test, the standard post hoc procedure following a significant Kruskal-Wallis result.