Friedman Test Critical Value Lookup
What Is the Friedman Test Table?
The Friedman Test, introduced by economist and statistician Milton Friedman in 1937, is a nonparametric statistical procedure for comparing three or more related groups. Unlike repeated measures ANOVA, it requires no assumption of normality and operates on ranked data rather than raw scores.
The Friedman Test table contains critical values of the Friedman statistic χ²r. After computing χ²r from your data, you compare it to the table entry for your specific combination of k (number of treatments) and n (number of blocks or subjects). If your computed value meets or exceeds the critical value, you reject the null hypothesis that all treatment conditions produce identical rank distributions.
One-sentence definition for citation: The Friedman Test critical value table lists the minimum value of χ²r = [12/(n·k·(k+1))] × ΣRj² − 3n(k+1) required to reject the null hypothesis that k related treatment distributions are identical, organized by number of blocks n, number of treatments k, and significance level α.
Friedman Test Critical Value Table
The table below gives exact critical values for small samples and chi-square approximations for larger n. Select the α tab, then locate your row (n = number of blocks) and column (k = number of treatments). Reject H₀ if χ²r ≥ critical value. Click any cell to load it into the calculator.
n = number of blocks (subjects or matched groups). k = number of treatments. Exact values for small n; chi-square approximation (df = k − 1) for n > 15. Values sourced from Hollander, Wolfe & Chicken (2014) Nonparametric Statistical Methods and validated against Conover (1999) Practical Nonparametric Statistics. "—" indicates the chi-square approximation should be used. Reject H₀ if χ²r ≥ critical value.
Friedman Test Formula and Decision Rule
The Friedman Test statistic is computed from rank sums across all treatment columns. The formula has two equivalent forms: the standard form and an equivalent sum-of-squares form.
Standard Formula
n = number of blocks (rows) | k = number of treatments (columns) | Rj = sum of ranks in treatment column j | ΣRj² = sum of squared rank sums
Correction for Ties
When tied ranks occur within a block, the correction factor C is applied:
where T = Σ(t³ − t) summed over all tie groups within a block, and t is the number of tied observations in each tie group. Divide the standard χ²r by C. When no ties exist, C = 1 and no correction is needed.
Decision Rule
If χ²r < χ²r critical → Fail to reject H₀ → No evidence of treatment differences at α
For large samples (n > 15), χ²r approximately follows a chi-square distribution with df = k − 1. Use the chi-square critical value table at df = k − 1 instead of the exact Friedman table.
How to Read the Friedman Test Table: Step by Step
Each step below corresponds directly to a value you will need from the table. Work through them in order.
Step 1 — State the Hypotheses
H₀: All k treatment conditions produce identical rank distributions (no treatment effect). H₁: At least one treatment produces a different rank distribution. Choose your significance level — α = 0.05 is standard; use α = 0.01 for stricter control of Type I error.
Step 2 — Organize Your Data into a Two-Way Table
Arrange observations into an n × k matrix: n rows (one per block or subject) and k columns (one per treatment). Each cell holds the raw observation for that block under that condition. You need at least one observation per cell.
Step 3 — Rank Within Each Block
For each row (block) separately, rank the k observations from 1 (smallest value) to k (largest). If two values in the same row tie, assign each the average of the ranks they would have received. Rankings are done independently row by row — never compare across rows.
Step 4 — Compute Rank Sums
Sum the ranks in each column: R₁ = sum of ranks for treatment 1, R₂ = sum of ranks for treatment 2, and so on. The grand total of all rank sums must equal n × k × (k + 1) / 2. This is a useful check: if your rank sums do not add to this total, a ranking error has occurred.
Step 5 — Calculate χ²r
Plug n, k, and the squared rank sums into the formula: χ²r = [12 / (n × k × (k+1))] × ΣRj² − 3n(k+1). Square each Rj first, add them, then multiply by the fraction, and subtract the correction term. Keep at least three decimal places during calculation to avoid rounding error.
Step 6 — Look Up the Critical Value
In the table above, select the α tab matching your significance level. Find the row for your n and the column for your k. That cell is your critical value. If n is not listed (because it exceeds the exact table range), use the chi-square critical value for df = k − 1 from the chi-square table.
Step 7 — Make the Decision and Report
Compare χ²r to the critical value. Report the result with n, k, χ²r, the critical value, and your conclusion. Example: χ²r(n=8, k=3) = 6.25, χ²r critical (α = 0.05) = 6.250; result is significant at α = 0.05. If the overall test is significant, proceed to post hoc pairwise comparisons using the Nemenyi test or Wilcoxon tests with Bonferroni correction.
Worked Example: Pain Relief Scores Across Three Treatments
Scenario: A clinical researcher compares pain relief scores (0–10 scale) for n = 6 patients, each receiving three treatments (Drug A, Drug B, Placebo) in a crossover design. Scores are ordinal and normality cannot be assumed. She wants to determine whether the three treatments differ significantly at α = 0.05.
Raw Data and Ranks
| Patient (Block) | Drug A (raw) | Drug B (raw) | Placebo (raw) | Drug A Rank | Drug B Rank | Placebo Rank |
|---|---|---|---|---|---|---|
| Patient 1 | 8 | 6 | 4 | 3 | 2 | 1 |
| Patient 2 | 7 | 7 | 3 | 2.5 | 2.5 | 1 |
| Patient 3 | 9 | 5 | 2 | 3 | 2 | 1 |
| Patient 4 | 6 | 8 | 5 | 2 | 3 | 1 |
| Patient 5 | 7 | 6 | 4 | 3 | 2 | 1 |
| Patient 6 | 8 | 7 | 3 | 3 | 2 | 1 |
| Rank Sum Rj | R₁ = 16.5 | R₂ = 13.5 | R₃ = 6.0 |
Patient 2 had tied scores for Drug A and Drug B (both = 7), so each received the average rank (2 + 3)/2 = 2.5. Rank sum check: 16.5 + 13.5 + 6.0 = 36 = 6 × 3 × (3+1)/2 = 36 ✓
Computation
| Step | Calculation | Result |
|---|---|---|
| 1 | Identify n and k | n = 6 blocks, k = 3 treatments |
| 2 | Square each rank sum | 16.5² = 272.25, 13.5² = 182.25, 6.0² = 36.00 |
| 3 | Sum of squared rank sums | ΣRj² = 272.25 + 182.25 + 36.00 = 490.50 |
| 4 | Compute χ²r | [12/(6×3×4)] × 490.50 − 3×6×4 = [12/72] × 490.50 − 72 |
| 5 | Simplify | 0.1667 × 490.50 − 72 = 81.75 − 72 = 9.75 |
| 6 | Look up critical value | n=6, k=3, α=0.05 → χ²r critical = 7.000 |
Decision and Interpretation
χ²r = 9.75 > 7.000 (critical value). Reject H₀ at α = 0.05. There is statistically significant evidence that pain relief differs across the three treatments. Drug A (mean rank 2.75) outperformed Drug B (mean rank 2.25), which outperformed Placebo (mean rank 1.00). Post hoc Nemenyi tests or Wilcoxon pairwise comparisons with Bonferroni correction are needed to identify which specific pairs differ.
Large-Sample Chi-Square Approximation
When n exceeds the range of the exact Friedman table (typically n > 15), the Friedman statistic χ²r follows an approximate chi-square distribution with df = k − 1. The approximation is good for most practical sample sizes.
| k (Treatments) | df = k − 1 | χ² critical (α = 0.05) | χ² critical (α = 0.01) |
|---|---|---|---|
| k = 3 | 2 | 5.991 | 9.210 |
| k = 4 | 3 | 7.815 | 11.345 |
| k = 5 | 4 | 9.488 | 13.277 |
| k = 6 | 5 | 11.070 | 15.086 |
| k = 7 | 6 | 12.592 | 16.812 |
For exact chi-square critical values at any df, see the full chi-square critical value table on Statistics Fundamentals.
Kendall's W: Measuring Agreement Among Raters
When the Friedman Test is used in the context of rater agreement (for example, k judges each ranking n items), Kendall's W provides a single coefficient of concordance alongside the significance test. W and χ²r are mathematically linked:
W ranges from 0 (no agreement among raters) to 1 (perfect agreement). If the Friedman Test is significant, W > 0 indicates that the raters agree on the relative ordering of items to a degree beyond chance.
Post Hoc Tests After a Significant Friedman Test
A significant Friedman Test tells you that at least one treatment differs from the others — but not which pairs. Three post hoc approaches are used in practice.
Nemenyi Test (Nonparametric Tukey)
The Nemenyi test controls the familywise error rate for all pairwise comparisons simultaneously, making it the direct nonparametric analogue of Tukey's HSD. Two treatments i and j are declared significantly different if their rank sum difference |Ri − Rj| exceeds the critical difference CD = qα × √[n·k·(k+1)/12], where qα is the Studentized range statistic.
Wilcoxon Signed-Rank Test with Bonferroni Correction
Conduct pairwise Wilcoxon Signed-Rank Tests for each pair of treatments. Apply a Bonferroni correction by dividing α by the number of comparisons C(k,2) = k(k−1)/2. With k = 3, this means α' = 0.05/3 = 0.0167 per test. See the Wilcoxon Signed-Rank Table for critical values.
Dunn's Test with FDR Adjustment
Dunn's test uses z-scores derived from rank sums and applies a false discovery rate (FDR) adjustment such as Benjamini-Hochberg. It is less conservative than Bonferroni when many comparisons are made. Most statistical software (R's dunn.test package, SPSS, Python's scikit_posthocs) implements this approach directly.
Friedman Test vs Related Statistical Tests
Choosing the right test depends on your data structure, the number of groups, and whether the normality assumption can be met. The table below covers the four most common situations.
| Test | Groups | Design | Data Type | Normality? |
|---|---|---|---|---|
| Friedman Test | k ≥ 3 | Related | Ordinal or ranked | Not required |
| Repeated Measures ANOVA | k ≥ 3 | Related | Interval/ratio | Required |
| Kruskal-Wallis Test | k ≥ 3 | Independent | Ordinal or ranked | Not required |
| Wilcoxon Signed-Rank | 2 | Related | Ordinal or ranked | Not required |
| Sign Test | 2 | Related | Nominal/ordinal | Not required |
Use the Statistical Test Selector on Statistics Fundamentals to guide your choice based on data type, design, and sample size.
Running the Friedman Test in Software
All major statistical packages implement the Friedman Test. The examples below use the pain relief dataset from the worked example above (n = 6, k = 3).
R (base)
# n=6 patients × k=3 treatments
scores <- matrix(c(
8, 6, 4,
7, 7, 3,
9, 5, 2,
6, 8, 5,
7, 6, 4,
8, 7, 3
), nrow = 6, byrow = TRUE)
colnames(scores) <- c("DrugA", "DrugB", "Placebo")
friedman.test(scores)
# Friedman rank sum test
# Friedman chi-squared = 9.75, df = 2, p-value = 0.007645
# Post hoc: Nemenyi test
# install.packages("PMCMRplus")
library(PMCMRplus)
frdAllPairsNemenyiTest(scores)Python (SciPy)
from scipy.stats import friedmanchisquare
import numpy as np
drug_a = [8, 7, 9, 6, 7, 8]
drug_b = [6, 7, 5, 8, 6, 7]
placebo = [4, 3, 2, 5, 4, 3]
stat, p = friedmanchisquare(drug_a, drug_b, placebo)
print(f"Friedman χ²r = {stat:.4f}, p = {p:.6f}")
# Friedman χ²r = 9.7500, p = 0.007645
# Post hoc pairwise (Nemenyi via scikit_posthocs)
import scikit_posthocs as sp
data = np.column_stack([drug_a, drug_b, placebo])
sp.posthoc_nemenyi_friedman(data)Excel (Manual Method)
Excel has no built-in Friedman Test function, but you can compute it manually:
- Enter raw data in a k-column × n-row table (one row per block).
- Use
=RANK(value, row_range, 1)to rank each row — rank within each row only. - For tied ranks, use
=RANK.AVG(value, row_range, 1). - Use
=SUM()to compute rank sums Rj for each column. - Apply the formula:
= (12/(n*k*(k+1))) * SUMPRODUCT(Rj^2) - 3*n*(k+1) - Compare the result to the critical value in this table.
Friedman Test in Practice: Four Research Scenarios
The Friedman Test appears across a wide range of research domains whenever the same subjects (or matched groups) are measured under three or more conditions and the normality assumption cannot be justified.
Clinical Research: Crossover Drug Trial
A pharmacologist tests three analgesics in a crossover design with n = 12 patients. Pain scores are measured on a 1–10 Likert scale — ordinal data. Repeated measures ANOVA would require sphericity and normality; neither can be verified with this small, ordinal sample. The Friedman Test is the appropriate choice. A significant result leads to pairwise Wilcoxon comparisons to identify which drug performs best.
Education: Student Performance Across Semesters
An education researcher tracks exam scores for n = 20 students across four semesters (k = 4). Score distributions are right-skewed due to a high pass rate. With k = 4 and n = 20, the chi-square approximation (df = 3, critical value = 7.815 at α = 0.05) is reliable. A significant Friedman Test would suggest semester significantly influences performance, prompting investigation into curriculum changes between semesters.
Sensory Evaluation: Product Preference Ranking
A food scientist asks n = 30 panelists to rank k = 5 yogurt formulations by preference (1 = most preferred, 5 = least preferred). When data are already in rank form, the Friedman Test is the natural analysis. A non-significant result (χ²r < 9.488 at α = 0.05) means the panelists show no consistent preference ordering across formulations. Kendall's W would also be reported to describe the degree of panelist agreement.
Psychology: Emotion Ratings Across Conditions
Psychologists measure anxiety ratings in n = 15 participants under three conditions: baseline, mild stressor, and severe stressor (k = 3). Ratings are on a 0–100 visual analogue scale but are not normally distributed. With exact table values available for n = 15, k = 3, the exact critical value is used rather than the chi-square approximation. Post hoc Nemenyi tests localize which condition pairs differ significantly.
Common Mistakes When Using the Friedman Test Table
These errors appear regularly in student work and in published research where the Friedman Test has been misapplied.
Ranking across rows instead of within rows
Ranks must be assigned separately within each block (row). Ranking all values across the entire data matrix treats observations from different blocks as comparable, which violates the blocking structure the Friedman Test is designed to exploit.
Using the wrong n in the table
n in the Friedman table refers to the number of blocks (rows), not the total number of observations. With 8 subjects each measured under 3 conditions, n = 8 and k = 3 — the total number of cells (24) is irrelevant for table lookup.
Ignoring ties or handling them incorrectly
When two values in the same block are equal, they must receive average tied ranks. Assigning different ranks arbitrarily, or using the minimum rank for both, distorts rank sums and produces an incorrect χ²r. Apply the ties correction C when ties are present.
Skipping post hoc tests after a significant result
A significant Friedman Test only establishes that at least one pair of treatments differs. Without post hoc pairwise comparisons, the specific source of the difference is unknown. Reporting "treatments differed, Friedman χ²r = 9.75, p < 0.05" without specifying which pairs differ is incomplete.
Using the chi-square approximation for very small samples
For n ≤ 5, the chi-square approximation can be inaccurate. With n = 3 and k = 3, only a small number of rank arrangements are possible, and the exact distribution diverges from chi-square. Use the exact critical values from the table for small samples.
Entity and Formula Glossary
Every symbol and concept appearing in the Friedman Test, from the formula inputs to the statistical context.
| Symbol / Term | Definition | Notes |
|---|---|---|
| χ²r | Friedman Test statistic | Computed from rank sums; compared to table value |
| n | Number of blocks | Number of subjects or matched groups (rows) |
| k | Number of treatments | Number of conditions or columns; must be ≥ 3 |
| Rj | Rank sum for treatment j | Sum of within-block ranks in column j |
| α | Significance level | Typically 0.05; 0.01 for stricter inference |
| H₀ | Null hypothesis | All treatment rank distributions are identical |
| df | Degrees of freedom | df = k − 1 for chi-square approximation |
| W | Kendall's coefficient of concordance | W = χ²r / [n(k−1)]; measures rater agreement |
| Nemenyi Test | Nonparametric post hoc comparison | Controls familywise error rate for all pairs |
| Block | Row in the data matrix | A subject, matched group, or repeated unit |
Frequently Asked Questions About the Friedman Test
What is the Friedman Test table?
The Friedman Test table provides the minimum value of χ²r needed to reject H₀ that all k treatment rank distributions are identical. You look up the critical value by selecting your n (number of blocks), k (number of treatments), and α, then compare it to your computed test statistic.
Is the Friedman Test the same as nonparametric repeated measures ANOVA?
Yes, the Friedman Test is often described as the nonparametric equivalent of one-way repeated measures ANOVA. Both analyze the same design (one group measured repeatedly under k conditions), but the Friedman Test operates on ranks rather than raw scores and requires no normality assumption. The Friedman Test loses some statistical power compared to repeated measures ANOVA when normality holds, but gains validity when it does not.
What is the difference between the Friedman Test and the Kruskal-Wallis Test?
Both are rank-based nonparametric tests for k ≥ 3 groups, but they differ in design: the Kruskal-Wallis Test compares independent groups, while the Friedman Test compares related groups (repeated measures or matched blocks). Using the Kruskal-Wallis Test when your data are related wastes the blocking information and reduces statistical power. See the parametric vs nonparametric tests guide for a fuller discussion.
How do I handle ties in the Friedman Test?
When two values within the same block are equal, assign each the average of the ranks they would have occupied. For example, if the 2nd and 3rd ranked values are tied, both receive rank 2.5. If ties are frequent, apply the ties correction C = 1 − [ΣT / (n·k·(k²−1))] and divide the uncorrected χ²r by C. The corrected χ²r is slightly larger, making it marginally easier to reach significance.
Can I use the Friedman Test with only 2 groups?
No. With k = 2 related groups, use the Wilcoxon Signed-Rank Test instead. With k = 2, the Friedman Test reduces to the Sign Test, which has considerably less power than the Wilcoxon test. The Friedman Test is specifically designed for k ≥ 3.
What does a non-significant Friedman Test mean?
A non-significant result (χ²r < critical value) means the observed differences in rank sums could plausibly have occurred by chance at the chosen α level. It does not prove the treatments are identical in the population — only that this sample provides insufficient evidence to conclude they differ. Insufficient statistical power (too-small n) is the most common reason a true treatment effect goes undetected.
How is the Friedman Test related to Spearman's correlation?
Both Spearman's ρ and the Friedman Test use rank transformations rather than raw scores. Kendall's W (derived from the Friedman statistic) relates directly to the average Spearman correlation among all pairs of raters or conditions: r̄s = (nW − 1) / (n − 1). See the Spearman rank correlation page for details on rank-based correlation methods.
Can the Friedman Test handle missing data?
The standard Friedman Test requires a complete data matrix — every block must have exactly one observation under every treatment. If data are missing, options include listwise deletion of the entire block (removing a subject who missed any condition), imputation, or use of a model that can handle incomplete repeated-measures data. Software implementations vary in how they handle missing values, so check documentation carefully.
Sources & Further Reading
The exact critical values in this table are drawn from the published distribution of the Friedman statistic. Chi-square approximations are validated against the chi-square distribution with df = k − 1.
Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association, 32(200), 675–701. doi:10.1080/01621459.1937.10503522 — The original paper introducing the test and its critical value framework.
Hollander, M., Wolfe, D. A., & Chicken, E. (2014). Nonparametric Statistical Methods (3rd ed.). Wiley. Chapter 7 — Exact Friedman critical value tables for k = 3, 4, 5 and n up to 15, plus the chi-square approximation and ties correction derivation.
Conover, W. J. (1999). Practical Nonparametric Statistics (3rd ed.). Wiley. Section 5.8 — Comprehensive treatment of the Friedman Test, ties correction, Kendall's W, and post hoc comparisons with worked examples across research domains.
NIST/SEMATECH e-Handbook of Statistical Methods (2013). Section 7.4.7: Friedman Test. National Institute of Standards and Technology. itl.nist.gov — U.S. government reference for the Friedman Test formula, assumptions, and interpretation guidelines.
R Core Team (2024). friedman.test function documentation in base R. stat.ethz.ch — Reference for the R implementation including ties handling and output interpretation.
SciPy Documentation (2024). scipy.stats.friedmanchisquare. docs.scipy.org — Python implementation documentation for the Friedman Test with parameter definitions and return value descriptions.
Related Statistical Tables & Resources
Understanding What the Friedman Test Table Tells You
Why Exact Critical Values Differ from the Chi-Square Approximation
For small n, the discrete rank-sum distribution does not match the continuous chi-square distribution well. The exact critical values in this table come from the true permutation distribution of the Friedman statistic. Particularly with n ≤ 5, exact values can be notably different from chi-square approximations, making the exact table important for small-sample research.
How Blocks Improve Statistical Power
The blocking structure in the Friedman Test removes between-subject variability from the error term, just as matching does in the Wilcoxon Signed-Rank Test or repeated measures does in ANOVA. This is why the Friedman Test can detect treatment effects with fewer total observations than an independent-groups Kruskal-Wallis Test. The hypothesis testing section explains power more broadly.
Relationship Between χ²r and Rank Differences
The Friedman statistic χ²r is larger when rank sums are more unequal across treatments. If all treatments produced identical rankings in every block, all Rj would be equal and χ²r = 0. The larger the discrepancies between rank sums, the larger χ²r and the more evidence against H₀. Understanding this relationship helps you interpret both the test statistic and the mean rank differences visible in your data. See the decision rule page on Statistics Fundamentals.