Quick answer: The Kruskal-Wallis critical value is found by computing df = k − 1 (where k = number of groups), then reading the table at your chosen α. For 3 groups (df = 2) at α = 0.05: critical value = 5.991. For 4 groups (df = 3) at α = 0.05: critical value = 7.815. Reject H₀ when H exceeds the table value.
Kruskal-Wallis Critical Value Lookup
Kruskal-Wallis Critical Value Table
Upper-tail critical values of H at α = 0.10, 0.05, and 0.01, based on the chi-square approximation with df = k − 1. Reject H₀ when the computed H exceeds the table value. Click any row to load it into the lookup tool above.
df = k − 1, where k = number of independent groups. All values are upper-tail chi-square critical values — mathematically identical to the chi-square distribution with the same df. Reject H₀ when H > critical value. Sourced and cross-verified against the chi-square distribution: Conover, W. J. (1999), Practical Nonparametric Statistics, 3rd ed., and NIST/SEMATECH e-Handbook of Statistical Methods.
What Is the Kruskal-Wallis Test?
The Kruskal-Wallis H test is a nonparametric procedure for comparing three or more independent groups on an ordinal or continuous outcome. It extends the logic of the Mann-Whitney U test to more than two groups.
Rather than working with raw data values, the test converts all observations to ranks — from 1 to N across all groups pooled — and then examines whether the rank sums differ more than chance would produce under H₀. Because no distributional assumptions about the data are required beyond a continuous or ordinal measurement scale, this test works in situations where one-way ANOVA cannot be applied.
A key distinction: The Kruskal-Wallis test formally tests whether the group distributions are identical. When distributions share a similar shape but may differ in location, this is often interpreted as a test of central tendency. Saying it simply tests medians understates what the test actually does.
What Does the Kruskal-Wallis Table Show?
The table contains critical values — specific thresholds that H must exceed for the result to be statistically significant. Each cell in the table represents the minimum H value that places the result in the rejection region at a given combination of df and α.
Degrees of Freedom (df)
df = k − 1, where k is the number of independent groups. With 3 groups, df = 2; with 5 groups, df = 4.
Significance Level (α)
The probability of rejecting H₀ when it is true (Type I error rate). Common choices are 0.10, 0.05, and 0.01.
Critical Value
The threshold. If H exceeds this value, the result is statistically significant at the chosen α. The test is upper-tailed only.
How to Use the Kruskal-Wallis Critical Value Table
The procedure below applies whether you're working by hand or verifying software output against the table.
Step 1 — State Hypotheses
H₀: All k populations have identical distributions. H₁: At least one population distribution differs from the others. Set α before data collection. The standard choice in most research is α = 0.05.
Step 2 — Rank All Observations
Pool all N observations across all k groups and rank them from 1 (smallest) to N (largest). When ties occur, assign the average of the tied ranks to each tied value.
Step 3 — Calculate H
N = total observations across all groups. nᵢ = number of observations in group i. Rᵢ = sum of ranks in group i. The sum Σ runs over all k groups.
Step 4 — Find df and Look Up Critical Value
Compute df = k − 1. Find the row in the table above for your df and read the critical value at your chosen α column.
Step 5 — Decision
If H ≤ critical value → Fail to reject H₀ → Not statistically significant
Step 6 — Report and Follow Up
Report H, df, and the p-value. Example: H(3) = 9.42, p = 0.024. If H₀ is rejected, conduct post-hoc pairwise comparisons to identify which specific groups differ.
Kruskal-Wallis Degrees of Freedom
The degree of freedom formula for the Kruskal-Wallis test is straightforward:
The most common error is using k instead of k − 1. This shifts df upward by 1 and gives the wrong critical value.
| Groups (k) | df = k − 1 | H critical (α = 0.05) | H critical (α = 0.01) |
|---|---|---|---|
| 3 | 2 | 5.991 | 9.210 |
| 4 | 3 | 7.815 | 11.345 |
| 5 | 4 | 9.488 | 13.277 |
| 6 | 5 | 11.070 | 15.086 |
| 7 | 6 | 12.592 | 16.812 |
| 8 | 7 | 14.067 | 18.475 |
| 9 | 8 | 15.507 | 20.090 |
| 10 | 9 | 16.919 | 21.666 |
| 11 | 10 | 18.307 | 23.209 |
The Kruskal-Wallis H Statistic
The H statistic quantifies how much the observed rank sums differ from what equal distributions would predict. The formula is:
| Symbol | Meaning |
|---|---|
| N | Total number of observations across all k groups |
| k | Number of independent groups |
| nᵢ | Number of observations in group i |
| Rᵢ | Sum of all ranks assigned to group i |
| Σ | Summation over all k groups |
Tied Ranks
When two or more observations share the same value, each receives the average of the ranks they would have occupied. For example, if values at positions 5 and 6 are tied, both get rank 5.5. A correction factor for ties exists but has little practical effect when ties are rare. Statistical software applies this correction automatically.
Worked Example: 3 Groups (df = 2)
Hypothetical scenario: A researcher tests whether pain-relief scores differ across three treatment conditions (A, B, C). Each group has n = 6 participants. After ranking all 18 scores and computing rank sums, the H statistic equals 7.42.
Solution
| Step | Action | Result |
|---|---|---|
| 1 | State hypotheses and α | H₀: identical distributions; α = 0.05 |
| 2 | Number of groups | k = 3 |
| 3 | Degrees of freedom | df = k − 1 = 3 − 1 = 2 |
| 4 | Computed H | H = 7.42 |
| 5 | Critical value (df=2, α=0.05) | 5.991 |
| 6 | Decision | 7.42 > 5.991 → Reject H₀ |
Interpretation
H(2) = 7.42, p < 0.05. The Kruskal-Wallis test indicates a statistically significant difference among the three treatment groups. At least one treatment produces a systematically different distribution of pain-relief scores. Post-hoc pairwise comparisons (e.g., Dunn's test) would identify which specific pairs differ.
Worked Example: 4 Groups (df = 3)
Hypothetical scenario: A psychology researcher compares satisfaction scores across four teaching methods. After ranking all observations and computing H, the result is H = 6.84. The researcher uses α = 0.05.
| Parameter | Value |
|---|---|
| Groups (k) | 4 |
| df = k − 1 | 3 |
| Computed H | 6.84 |
| Critical value (df=3, α=0.05) | 7.815 |
| Decision | 6.84 ≤ 7.815 → Fail to reject H₀ |
Interpretation
H(3) = 6.84, p > 0.05. The test does not provide sufficient evidence to conclude that the four teaching methods produce different satisfaction distributions. Note that failing to reject H₀ is not proof that groups are identical — it means the data do not provide enough evidence to detect a difference at this α level.
Worked Example: 5 Groups (df = 4)
Hypothetical scenario: A nutritionist compares hunger-rating scores across five diet types. H = 10.85, α = 0.05.
Decision and Interpretation
10.85 > 9.488, therefore reject H₀. H(4) = 10.85, p < 0.05. At least one diet type produces a systematically different hunger-rating distribution. Conducting Dunn's test with Bonferroni correction would identify which diets differ pairwise.
How α Changes the Decision
Consider a fixed result: H = 8.10, 4 groups (df = 3). The decision changes depending on the chosen α.
| α Level | Critical Value | H = 8.10 vs Critical | Decision |
|---|---|---|---|
| α = 0.10 | 6.251 | 8.10 > 6.251 | Reject H₀ |
| α = 0.05 | 7.815 | 8.10 > 7.815 | Reject H₀ |
| α = 0.01 | 11.345 | 8.10 < 11.345 | Fail to Reject H₀ |
The same H statistic is significant at α = 0.10 and α = 0.05 but not at α = 0.01. A stricter α demands a larger H to reach significance. The α level must be chosen before data collection, not selected post-hoc to achieve a desired outcome.
Kruskal-Wallis Critical Value vs P-Value
Two equivalent approaches reach the same statistical decision. The critical-value method uses a table; the p-value method uses software output. Both are mathematically equivalent.
Critical-Value Approach
Compute H, find the critical value from the table, compare directly.
H ≤ critical value → Fail to reject H₀
P-Value Approach
Software computes the exact p-value from the chi-square distribution with df = k − 1.
p > α → Fail to reject H₀
Statistical software (R, SPSS, Python, SAS) reports the p-value directly, making the table lookup unnecessary in practice. The table remains useful for understanding the test, for teaching, and for verifying software output.
Kruskal-Wallis Chi-Square Approximation
Under H₀ and with reasonably large samples, the H statistic follows an approximate chi-square distribution with df = k − 1. This is why the critical values in the Kruskal-Wallis table match the chi-square table exactly — they are the same distribution.
The approximation works well when each group has at least 5 observations. With smaller groups, the discrete rank structure of H causes the chi-square distribution to fit less precisely.
When to Use Exact Methods
If any group has fewer than 5 observations, the chi-square approximation may not be accurate. In those cases, exact permutation-based methods compute the true finite-sample distribution of H. Software packages such as R (with the coin package) and SPSS provide exact Kruskal-Wallis p-values. For very small samples, prefer exact p-values over the chi-square table.
Kruskal-Wallis Test Assumptions
Independence
Observations within each group are independent of one another, and groups do not share subjects.
Measurement Scale
The dependent variable must be ordinal or continuous so that a meaningful rank ordering is possible.
Random Sampling
Observations should be drawn from representative samples for inference to be valid.
Distribution Shapes
For the result to be interpreted as a test of location (central tendency), distributions should have roughly similar shapes. If shapes differ substantially, a significant H may reflect that difference rather than a location shift.
Kruskal-Wallis vs One-Way ANOVA
Both tests compare three or more independent groups, but they differ in approach, assumptions, and the type of question they answer.
| Feature | Kruskal-Wallis | One-Way ANOVA |
|---|---|---|
| Data type | Ordinal or continuous | Continuous |
| Normality required? | No | Yes (within groups) |
| Equal variances assumed? | Not required | Assumed (homoscedasticity) |
| Test statistic | H | F |
| Reference distribution | Chi-square (approx.) | F distribution |
| Works on ranks? | Yes | No (uses means) |
| Statistical power | Slightly lower when normality holds | Higher when assumptions met |
| Preferred when | Data are skewed, ordinal, or contain outliers | Data are continuous and approximately normal |
Post-Hoc Testing After a Significant Result
A significant Kruskal-Wallis result establishes that the groups are not all equivalent — but it does not say which pairs differ. Post-hoc pairwise comparisons are needed for that determination.
Dunn's Test
The most widely used post-hoc procedure for Kruskal-Wallis. Performs z-tests on pairwise rank-sum differences. Commonly applied with Bonferroni or Holm correction to control the familywise error rate.
Bonferroni Correction
Divides α by the number of pairwise comparisons. Conservative but straightforward. With 4 groups, there are C(4,2) = 6 pairs, so the adjusted α = 0.05/6 ≈ 0.0083.
Holm Correction
A step-down procedure that is less conservative than Bonferroni while still controlling the familywise error rate. Ranks p-values and adjusts sequentially.
Common Mistakes When Using the Kruskal-Wallis Table
Using df = k instead of df = k − 1
The single most common table error. With 4 groups, df = 3, not 4. Using the wrong df row gives the wrong critical value.
Treating Kruskal-Wallis as strictly a median test
The test formally compares distributions. Interpreting it as a median comparison is valid only when distributions have similar shapes.
Applying the chi-square approximation to very small samples
With fewer than 5 observations per group, the approximation breaks down. Use exact methods in software instead.
Stopping after a significant result without post-hoc testing
Knowing that groups differ is only step one. Post-hoc pairwise comparisons identify which groups drive the difference.
Reporting only "significant/not significant" without H, df, and p
Full reporting requires H(df) = value, p = value. Omitting any of these makes the result difficult to evaluate or reproduce.
Ignoring tied ranks
Many ties can slightly distort H. The correction factor for ties is built into most software but should be verified, especially with frequent ties.
Reporting Kruskal-Wallis Results
Statistical reporting should include the H statistic, degrees of freedom, and p-value. Effect size is increasingly expected in many fields.
Standard Format
Example (Hypothetical)
"Pain relief scores differed significantly across the three treatment groups, H(2) = 7.42, p = 0.024."
Effect Size: Epsilon-Squared (ε²)
One effect size measure for Kruskal-Wallis results is epsilon-squared:
Values near 0 indicate little effect; values near 1 indicate a large effect. This formula is one commonly cited version — definitions vary slightly across references. Report it alongside the p-value for a more complete picture.
Kruskal-Wallis Test in Statistical Software
All major statistical programs compute the Kruskal-Wallis H statistic, degrees of freedom, and p-value automatically. The critical-value table is used to understand or verify the output.
R
data = mydata)
Returns H, df, and p-value. Use the dunn.test or FSA package for Dunn's post-hoc test.
Python
kruskal(g1, g2, g3)
Returns the H statistic and p-value. The scikit_posthocs library provides Dunn's test.
SPSS
Analyze → Nonparametric Tests → Independent Samples → Kruskal-Wallis. Output includes H, df, exact and asymptotic p-values.
SAS
CLASS group;
VAR outcome;
The WILCOXON option provides the Kruskal-Wallis test when there are 3+ groups.
Sources and Verification
Conover, W. J. (1999). Practical Nonparametric Statistics (3rd ed.). John Wiley & Sons. The authoritative reference for Kruskal-Wallis critical values, assumptions, and tied-rank corrections. Critical values in this page are consistent with the chi-square distribution tables in this text.
NIST/SEMATECH e-Handbook of Statistical Methods. Section 7.2.3: Kruskal-Wallis Test. National Institute of Standards and Technology. itl.nist.gov — U.S. government statistical reference covering the H statistic formula, chi-square approximation, and decision rules.
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. doi:10.2307/2280779
Penn State STAT 415: Introduction to Mathematical Statistics. Kruskal-Wallis Test. Pennsylvania State University. online.stat.psu.edu/stat415 — Open-access course notes covering nonparametric rank-based tests.
Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric Statistical Methods (3rd ed.). John Wiley & Sons. Comprehensive treatment of the Kruskal-Wallis H statistic and exact distribution theory for small samples.
Related Statistical Tables & Resources
Frequently Asked Questions
What is a Kruskal-Wallis table?
A Kruskal-Wallis table lists critical values of the H statistic at different combinations of degrees of freedom and significance level α. Find the row for your df = k − 1 and the column for your α to get the threshold. Computed H above that threshold is statistically significant.
What is the critical value for 3 groups at α = 0.05?
With 3 groups, df = 2. The critical value at α = 0.05 is 5.991. Reject H₀ if H > 5.991.
What is the critical value for 4 groups at α = 0.05?
With 4 groups, df = 3. The critical value at α = 0.05 is 7.815.
What is the critical value for 5 groups at α = 0.05?
With 5 groups, df = 4. The critical value at α = 0.05 is 9.488.
How is degrees of freedom calculated for Kruskal-Wallis?
df = k − 1, where k is the number of independent groups. For 3 groups, df = 2. For 10 groups, df = 9. The most common mistake is using k instead of k − 1.
Is the Kruskal-Wallis test upper-tailed or two-tailed?
The Kruskal-Wallis test is always upper-tailed. Large values of H indicate that group rank sums differ more than chance predicts. The table gives upper-tail critical values only; reject H₀ when H exceeds the critical value.
Can I use a chi-square table for the Kruskal-Wallis test?
Yes. The Kruskal-Wallis critical values are identical to chi-square critical values with df = k − 1. A standard chi-square table gives exactly the same numbers. The chi-square approximation is accurate when each group contains at least 5 observations.
When should I use exact Kruskal-Wallis methods instead of the table?
When any group has fewer than 5 observations, the chi-square approximation may be imprecise. Statistical software can compute exact p-values through permutation methods (available in R's coin package and in SPSS). Use exact methods rather than the table in very small samples.
Does Kruskal-Wallis require normality?
No. The Kruskal-Wallis test makes no normality assumption. It requires independent observations, independent groups, and an ordinal or continuous dependent variable. This makes it appropriate when ANOVA assumptions cannot be met.
What does a significant Kruskal-Wallis result tell me?
It tells you that at least one group distribution differs from the others. It does not identify which specific groups differ — post-hoc pairwise comparisons (Dunn's test) are needed for that. It also does not establish what caused the difference.
Does the Kruskal-Wallis test compare medians?
Not exactly. The test formally compares distributions. When groups share a similar shape, a significant result can reasonably be interpreted as a difference in location (such as the median). If shapes differ substantially, the result reflects the distributional difference more broadly.
How is Kruskal-Wallis different from ANOVA?
ANOVA compares group means and assumes normally distributed residuals with equal variances. Kruskal-Wallis uses ranks instead of raw values, requires no normality assumption, and compares distributions rather than means. Use ANOVA when its assumptions are met; use Kruskal-Wallis otherwise.
What is Dunn's test?
Dunn's test is the standard post-hoc procedure used after a significant Kruskal-Wallis result. It performs pairwise comparisons between groups using z-statistics derived from rank sums. Multiple-comparison corrections such as Bonferroni or Holm are applied to control the familywise error rate.
How do I handle tied ranks?
When two or more observations share the same value, assign each the average of the ranks they would occupy. For example, if three values are tied at ranks 4, 5, and 6, each gets rank 5. A correction factor for ties can be applied to H; most statistical software handles this automatically.
Key Takeaways
df = k − 1, Always
With 3 groups, df = 2. With 4 groups, df = 3. Using k instead of k − 1 is the most common table-lookup error and produces a wrong critical value.
Upper-Tail Test Only
Reject H₀ when H exceeds the critical value. There is no lower-tail version. All Kruskal-Wallis critical values in the table are upper-tail thresholds.
Table = Chi-Square Table
The Kruskal-Wallis critical values are identical to chi-square critical values at the same df. A chi-square table works equally well when each group has at least 5 observations.
Significance ≠ Which Groups Differ
A significant result means at least one group distribution differs. Post-hoc pairwise testing (Dunn's test) is required to identify which specific groups drive that difference.
Not Strictly a Median Test
The Kruskal-Wallis test compares distributions. Interpreting it as a median comparison is reasonable only when groups share a similar distributional shape.
Small Samples Need Exact Methods
The chi-square approximation in the table requires at least 5 observations per group. For smaller groups, exact permutation-based p-values computed by software give more reliable results. Visit Statistics Fundamentals for more nonparametric guidance.