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Nonparametric repeated-measures rank test

Friedman Test: 7 Essential Steps, Formula and Worked Example

The Friedman test is a nonparametric repeated-measures procedure for comparing three or more related conditions, occasions, or treatments. This complete guide explains the Friedman test assumptions, within-block ranking formula, tie correction, Kendall’s W effect size, post hoc comparisons, and a worked analysis of G1, G2, and G3 grades in Python, R, SPSS, and Excel.

Three related measurements
Complete-block ranks
Tie-corrected chi-square
Kendall’s W effect size
Python + R + SPSS + Excel

Student blocks649

Friedman Q167.3283

Kendall’s W0.1289

DecisionSignificant change

Quick answer

Grades differed significantly across G1, G2, and G3.

In this worked Friedman test example, the same 649 students contributed grades at three occasions: G1, G2, and G3. Grades were ranked separately within every student block, with tied values receiving average ranks. The tie-corrected Friedman statistic was Q = 167.3283 with df = 2 and p = 4.63 × 10⁻³⁷. Kendall’s coefficient of concordance was W = 0.1289, indicating a statistically clear but modest repeated-measures effect.

Substantive finding: mean ranks rose from G1 = 1.7612 to G2 = 1.8983 and G3 = 2.3405. Follow-up paired Wilcoxon tests with Holm adjustment showed that all three grade pairs differed significantly.

1

What does the Friedman test measure?

A rank-based comparison of three or more related measurements.

The Friedman test, also called the Friedman rank test, Friedman statistical test, or nonparametric repeated-measures ANOVA, evaluates whether several related conditions have the same distributional location. It is designed for repeated observations on the same subjects or for randomized complete-block designs in which every block receives every treatment.

The core research question

The Friedman test asks whether the rank pattern across related conditions is too systematic to be explained by chance. Each participant or block forms its own comparison set. Values are ranked within that block, so stable between-person differences are removed from the test statistic. A student who usually earns higher grades than another student does not dominate the analysis merely because of a higher overall level; the method focuses on the ordering of G1, G2, and G3 inside each student’s record.

This blocking principle is closely related to the purpose of repeated-measures ANOVA, but the Friedman test uses ranks and does not require a normal residual model. It therefore belongs to the broader family described in parametric vs nonparametric tests.

What the Friedman test does not measure

The Friedman test is not an independent-groups test. When the groups contain different people, the appropriate omnibus rank procedure is usually the Kruskal–Wallis test rather than the Friedman test. It is also not a test of equal variances, a test of correlation, or a general model of growth over time.

The test statistic establishes that at least one related condition differs. It does not by itself identify the exact pairs responsible for the result. Pairwise follow-up analysis is therefore a separate step, just as pairwise comparisons after ANOVA are separate from an omnibus F test.

Interpretive scope: when the condition distributions have comparable shapes, the Friedman test is often explained as a test of median differences. More generally, it detects systematic changes in within-block rank position. Readers reviewing mean, median and mode should distinguish a raw median comparison from the pooled evidence produced by repeated within-block ranks.

2

When should the Friedman test be used?

Use it for one-factor repeated or blocked designs with at least three related conditions.

The search question when to use Friedman test is answered by the study design. The Friedman test is appropriate when the same subjects are measured repeatedly, matched sets are observed under several conditions, or blocks receive every treatment exactly once. The outcome should be ordinal or continuous and capable of meaningful ranking.

Related samples?

The same subjects, matched units, or blocks must contribute to every condition.

Three or more conditions?

The standard Friedman test compares k related samples, usually k ≥ 3.

Rankable outcome?

Ordinal scores and numeric measurements can be ranked within each block.

Complete blocks?

Every block should contain one usable value for every condition in the main analysis.

Omnibus question?

The initial goal is to determine whether any condition differs before testing pairs.

Typical Friedman test applications

Pain scores measured before treatment, after treatment, and at follow-up.

The same products rated by every member of a consumer panel.

Three teaching approaches evaluated within matched classrooms or student blocks.

G1, G2, and G3 grades recorded for the same students over successive occasions.

These designs differ from the independent-group settings covered by one-way ANOVA, Welch’s ANOVA, and the Kruskal–Wallis procedure.

Designs requiring another approach

Two related conditions are commonly analyzed with the sign test or a paired Wilcoxon procedure.

Independent groups require an independent-samples method rather than the Friedman test.

Binary repeated outcomes are better aligned with Cochran’s Q test.

Complex longitudinal data with missing occasions or covariates may require mixed-effects regression or generalized estimating equations.

3

Friedman test assumptions and data requirements

The method avoids normality but still depends on a valid block structure.

The assumptions of Friedman test are primarily about dependence, measurement scale, completeness, and the meaning of the repeated conditions. The procedure is nonparametric, but the design must still support the intended inference.

Related observations

Measurements inside a block are related because they come from the same subject or matched set. Blocks should be independent of one another. In the grade example, G1, G2, and G3 are linked within students, while one student’s record is treated as independent of another student’s record.

Ordinal or continuous outcome

The outcome must support ordering. Grades are numeric and therefore rankable. The method can also handle ordered ratings when the categories carry a clear sequence.

One value per block-condition cell

The classical Friedman design contains one response for every block and every condition. The workbook contains 649 complete rows, each with G1, G2, and G3.

Meaningful condition labels

G1, G2, and G3 represent three ordered assessment occasions. The labels must correspond to comparable measurements rather than unrelated variables placed together for convenience.

Within-block ranking

Ranks are assigned separately within every row. Ties receive average ranks, a principle also used in many procedures described under Spearman rank correlation.

Comparable distributional interpretation

A median-shift interpretation is strongest when condition distributions have broadly comparable shapes. Otherwise, the result is best described as a change in distributional rank position.

Normality and sphericity: the Friedman test does not require a normal outcome model and does not use the sphericity assumption tested by Mauchly’s test. That is a major difference from traditional repeated-measures ANOVA and its Greenhouse–Geisser or Huynh–Feldt corrections.

Ties are expected in discrete grade data. They do not invalidate the Friedman test. Average ranks and a tie correction preserve the correct scale of the test statistic. The grade workbook contains many blocks in which two or all three occasions are equal.

4

Friedman test hypotheses, variables, and block structure

The null concerns the related condition distributions within a complete-block design.

A precise Friedman test report names the outcome, repeated factor, blocks, and null hypothesis. In this analysis, the outcome is the grade score, the repeated factor is assessment occasion, and each student is a block.

Statistical hypotheses

H₀: G1, G2, and G3 have the same distributional location across the population of student blocks.

H₁: at least one assessment occasion differs in distributional location or systematic rank position.

The omnibus alternative does not identify a specific direction. Direction is interpreted from the mean ranks, descriptive summaries, and post hoc paired comparisons.

Variables used in the worked analysis

BlockStudent; n = 649 complete blocks.

Condition 1G1, first-period grade.

Condition 2G2, second-period grade.

Condition 3G3, final grade.

Outcome scaleNumeric grades observed from 0 to 19.

Alpha0.05 for the omnibus and adjusted follow-up decisions.

Connection with hypothesis testing: the decision compares the p-value with alpha, following the logic explained in hypothesis testing, null and alternative hypotheses, and p-value, significance level and test statistic.

5

Friedman test formula, ranks, and tie correction

The statistic is built from condition rank sums after ranking within each block.

The Friedman chi square test starts by ranking the k condition values separately inside each of the n blocks. The condition rank sums are then compared with the equal-rank pattern expected under the null hypothesis.

Step 1: rank the conditions within each block

For block i, assign ranks 1 through k to the k condition values. The smallest value receives rank 1 and the largest receives rank k. Tied values receive the average of the ranks they would occupy.

Illustrative student	G1	G2	G3	Within-student ranks
Student 1	0	11	11	1.0, 2.5, 2.5
Student 2	9	11	11	1.0, 2.5, 2.5
Student 3	12	13	12	1.5, 3.0, 1.5
Student 4	14	14	14	2.0, 2.0, 2.0

Step 2: add the ranks for each condition

R_j = Σ_i=1ⁿ r_ij

R_j is the sum of within-block ranks for condition j. In the worked dataset, the rank sums are 1,143 for G1, 1,232 for G2, and 1,519 for G3.

Step 3: calculate the uncorrected statistic

Q₀ = 12/[n k(k+1)] × Σ_j=1^k R_j² − 3n(k+1)

With n = 649 and k = 3, the uncorrected value is Q₀ = 118.9861.

Step 4: correct for within-block ties

C = 1 − [Σ_iΣ_g(t_ig³ − t_ig)]/[n(k³ − k)]

For each tied group g inside block i, t_ig is the number of tied observations. The total tie term is 4,500, giving C = 0.711094.

Q = Q₀/C = 167.3283

The corrected statistic is compared with a chi-square reference distribution having k − 1 = 2 degrees of freedom.

Why the tie correction is large here

Grades are discrete and repeated. Many students have equal values at two occasions, and some have G1 = G2 = G3. Without correction, those ties reduce the variability of the possible rank arrangements and would leave the statistic on the wrong scale. The correction raises Q from 118.9861 to 167.3283.

This issue connects with the broader role of frequency distributions and tied values in rank-based analysis.

Reference distribution

For a sufficiently large number of complete blocks, Q is approximated by a chi-square distribution with k − 1 degrees of freedom. The current n = 649 is large, and the resulting p-value is approximately 4.63 × 10⁻³⁷.

The reference distribution can be reviewed alongside chi-square test, chi-square assumptions, and the chi-square calculator.

6

Worked Friedman test example using G1, G2, and G3

Three related grade occasions are analyzed as a complete-block design.

This example of Friedman test uses 649 complete student records. The repeated conditions are G1, G2, and G3, so every student contributes a three-value block. The analysis examines whether grade position changes systematically across the three occasions.

G1Mean 11.399Median 11; SD 2.745; IQR 3; mean rank 1.761

G2Mean 11.570Median 11; SD 2.914; IQR 3; mean rank 1.898

G3Mean 11.906Median 12; SD 3.231; IQR 4; mean rank 2.341

Blocks649 studentsEvery block contains all three grade occasions.

Descriptive pattern

The raw means rise gradually from G1 to G2 to G3. The medians remain 11 for G1 and G2, then rise to 12 for G3. The within-block mean ranks show the same ordering, with G3 clearly above the first two occasions.

The Friedman test does not simply test these three raw means. It asks whether the within-student ordering is consistent enough across 649 blocks to reject equal condition distributions. That distinction separates the method from summaries such as descriptive statistics, standard deviation, and interquartile range.

Verified omnibus result

Q = 167.328

p < .001

The null hypothesis of equal grade distributions across G1, G2, and G3 is rejected. The repeated grade occasion is associated with a systematic change in within-student rank position.

Observed direction: the rank totals are 1,143 for G1, 1,232 for G2, and 1,519 for G3. G3 occupies the highest within-student position most consistently, while G1 occupies the lowest position most consistently.

7

Exact Friedman test results and calculation audit

The omnibus statistic, effect size, and supporting rank totals agree across the verified calculations.

A full Friedman test interpretation should report the statistic, degrees of freedom, p-value, condition ranks, and an effect size. The following table connects each number to its role in the final conclusion.

Result item	Value	Interpretation
Complete blocks	649	All students contributed G1, G2, and G3.
Number of occasions	3	The repeated factor has three levels.
G1 rank sum / mean rank	1,143 / 1.761171	Lowest average within-student position.
G2 rank sum / mean rank	1,232 / 1.898305	Intermediate average position.
G3 rank sum / mean rank	1,519 / 2.340524	Highest average position.
Uncorrected Q	118.986133	Statistic before accounting for ties.
Tie correction C	0.711094	Adjustment for repeated grades inside blocks.
Corrected Friedman Q	167.328277	Primary omnibus test statistic.
Degrees of freedom	2	k − 1 for three occasions.
p-value	4.63 × 10−37	Extremely strong evidence against equal condition distributions.
Kendall’s W	0.128912	Modest repeated-condition effect or concordance.

Statistical conclusion

Because p is far below .05, the null hypothesis is rejected. At least one of G1, G2, and G3 differs in distributional location. The p-value is not the probability that the null is true; it is the probability of obtaining a statistic at least this extreme under the reference model, a distinction developed in p-value.

Practical conclusion

The effect is statistically unmistakable because the sample contains 649 blocks. Kendall’s W shows that the magnitude is not correspondingly large. This illustrates why statistical significance and effect size should be interpreted together.

8

Kendall’s W effect size for the Friedman test

W converts the omnibus rank statistic into a standardized measure from 0 to 1.

The Friedman test p-value answers whether a systematic condition effect is present. Kendall’s coefficient of concordance, W, describes the strength of that repeated rank pattern.

W = Q/[n(k − 1)]

With Q = 167.328277, n = 649, and k = 3, W = 167.328277/[649(2)] = 0.128912.

Near 0

Condition ranks vary with little systematic agreement across blocks. The repeated factor explains little of the rank ordering.

Current value: 0.1289

The grade occasions show a consistent but modest ordering. G3 tends to outrank G1 and G2, yet many blocks contain ties or different individual trajectories.

Near 1

Blocks show nearly identical condition ordering, producing strong concordance and a large repeated-condition effect.

Magnitude language is contextual. Conventional small, medium, and large labels are rough summaries rather than universal laws. Subject-matter importance, measurement reliability, sample size, and the consequences of grade change remain central to interpretation. The distinction parallels discussions of statistical power and Type I and Type II error.

9

Friedman test in Python: results and chart interpretation

Python reproduces the tie-corrected omnibus result and provides a transparent path to follow-up analysis.

A Friedman test in Python can be calculated with SciPy after arranging the repeated conditions in aligned columns. The three arrays must represent the same subjects in the same row order. Related tutorials include categorical data analysis in Python, regression in Python, and correlation in Python.

Pythonfrom scipy.stats import friedmanchisquare

q, p = friedmanchisquare(data["G1"], data["G2"], data["G3"])
print(q) # 167.32827735644634
print(p) # 4.625154438692089e-37

kendalls_w = q / (len(data) * (3 - 1))
print(kendalls_w) # 0.12891238625304033

10

Friedman test in R: results and chart interpretation

R’s base implementation treats the data as an unreplicated complete-block design.

The keywords Friedman test in R, Friedman test R, and R Friedman test refer to the same workflow: arrange G1, G2, and G3 as related columns or reshape the data into response, group, and block variables. Related R resources include categorical data analysis in R, ANOVA in R, and correlation in R.

Rgrades <- as.matrix(data[c("G1", "G2", "G3")])
result <- friedman.test(grades)
print(result)

Q <- unname(result$statistic)
n <- nrow(grades)
k <- ncol(grades)
W <- Q / (n * (k - 1))
W

11

How to do a Friedman test in SPSS

SPSS treats G1, G2, and G3 as several related samples.

Common searches include how to do a Friedman test in SPSS, how to run Friedman test in SPSS, and how to interpret Friedman test results. In wide-format data, each repeated occasion appears in a separate variable column and each row represents one subject.

SPSS menu structure

The related-samples nonparametric procedure compares G1, G2, and G3 inside each row. The output normally provides the number of cases, mean rank for each variable, the Friedman chi-square statistic, degrees of freedom, and asymptotic significance.

The same principles used in ANOVA in SPSS and categorical data analysis in SPSS apply to data preparation: variable labels, missing-value definitions, and row alignment must match the design.

SPSS syntax

SPSSNPAR TESTS
/FRIEDMAN = G1 G2 G3
/MISSING ANALYSIS.

The expected output is approximately χ²(2) = 167.328, p < .001, with mean ranks of 1.76, 1.90, and 2.34.

SPSS interpretation: the omnibus table establishes a difference across the related grade variables. Mean ranks identify the direction, and separate paired follow-up tests determine which occasions differ while controlling multiplicity.

12

How to do the Friedman test in Excel

The workbook exposes every rank, tie term, intermediate quantity, and reporting value.

The query how to do Friedman test in Excel reflects the value of an auditable calculation. Excel does not need to hide the method behind a single function: each student row can be ranked, condition rank sums can be added, the tie correction can be calculated, and the chi-square p-value can be obtained from the corrected Q statistic.

Workbook sheet	Role in the Friedman test analysis	Key content
Guide	Defines design and variables	Three repeated grades with students as complete blocks.
Data_Input	Stores unchanged raw values	G1, G2, and G3 for 649 students.
Working	Shows row-level lineage	Within-student ranks, range ranks, weighted scores, and tie terms.
Calculations	Builds omnibus statistics	Rank sums, mean ranks, Friedman Q, Kendall’s W, Page and Quade sensitivity quantities.
Diagnostics	Documents design checks	Blocking, tie handling, and method identity.
Reporting	Cross-checks verified results	Q = 167.328277 and W = 0.128912 with zero calculation difference.

Row rank formulas

Each G1:G3 row is ranked in ascending order. Average ranks are required for ties. Column sums of the three rank columns produce R_G1, R_G2, and R_G3.

Excel workflows can be reviewed alongside regression in Excel, correlation in Excel, and t-test in Excel.

P-value formula

After computing the tie-corrected Q, the right-tail chi-square probability is obtained with 2 degrees of freedom. The workbook result rounds to zero at ordinary display precision, while the more precise probability is approximately 4.63 × 10⁻³⁷.

Displaying p < .001 is clearer than reporting p = 0, because a continuous reference probability is extremely small rather than literally zero.

13

Post hoc comparisons after a significant Friedman test

The omnibus result establishes a difference but does not identify every pair.

After rejecting the Friedman test null hypothesis, related-samples pairwise comparisons can be performed with Wilcoxon signed-rank tests and a multiplicity adjustment such as Holm. The same subjects must remain paired in every comparison.

Pair	Wilcoxon statistic	Raw p-value	Holm-adjusted p-value	Direction
G1 vs G2	43,328.0	0.0003473	0.0003473	G2 is generally higher than G1.
G1 vs G3	25,595.5	5.57 × 10−24	1.11 × 10−23	G3 is generally higher than G1.
G2 vs G3	13,701.0	7.26 × 10−25	2.18 × 10−24	G3 is generally higher than G2.

G1 to G2

Among 649 students, 273 had G2 above G1, 187 had G2 below G1, and 189 were tied. The pairwise shift is smaller than the changes involving G3 but remains significant after Holm adjustment.

G1 to G3

G3 exceeded G1 for 344 students, was lower for 120, and tied for 185. The median paired difference was 1 grade point, and the adjusted p-value was far below .001.

G2 to G3

G3 exceeded G2 for 290 students, was lower for 72, and tied for 287. Despite many ties, the positive direction was highly systematic.

Multiplicity control: Holm adjustment protects the familywise error rate while retaining more power than a simple Bonferroni division in many settings. The adjustment logic is explained further in Holm–Bonferroni method and Bonferroni correction.

14

Friedman test vs repeated-measures ANOVA and related methods

Method choice follows the dependence structure, outcome scale, and research question.

The difference between Kruskal Wallis and Friedman test is especially important: Kruskal–Wallis compares independent groups, whereas the Friedman test compares related conditions or treatments within blocks.

Method	Design	Outcome / model focus	Relationship to Friedman test
Friedman test	Three or more related conditions	Within-block ranks	Primary method in this article.
Repeated-measures ANOVA	Three or more related conditions	Parametric mean model	Uses normal-theory assumptions and may require sphericity corrections.
Kruskal–Wallis test	Three or more independent groups	Independent-group ranks	Not appropriate for repeated observations on the same students.
Wilcoxon signed-rank test	Two related conditions	Paired rank differences	Useful for post hoc pairs after Friedman.
Sign test	Two related conditions	Direction of paired differences	More minimal assumptions but often less powerful.
Cochran’s Q	Three or more related binary conditions	Repeated binary outcomes	Binary-data analogue of a repeated omnibus test.
Quade test	Complete blocks	Weighted within-block ranks	Can gain sensitivity when block ranges contain useful information.
Page test	Related ordered conditions	Prespecified monotonic trend	Targets an ordered alternative rather than any difference.

Quade sensitivity result

The workbook’s Quade calculation produced F = 91.1481 with p far below .001. It weights block rank differences by the range of scores within each student, giving more influence to blocks that show larger separation across occasions.

Page trend result

With the prespecified order G1 → G2 → G3, the workbook produced Page L = 8,164 and z = 10.4364, again indicating a strong upward ordered trend. The Page test addresses a narrower directional question than the omnibus Friedman test.

Researchers comparing model families may also review mixed ANOVA, multilevel regression, and ordinal logistic regression.

15

Friedman test diagnostics and design checks

Quality control focuses on block completeness, ties, ordering, and the meaning of the repeated occasions.

The Friedman test does not use residual normality diagnostics, but careful analysis still checks the data structure and the interpretability of the rank comparison.

Complete-block verification

All 649 rows contain G1, G2, and G3.

Every row represents one student block.

The same occasion order is used in every row.

No student contributes more than one row to the primary block analysis.

Missing repeated measurements can change the usable sample and may motivate a model-based longitudinal method. General missing-data structure is different from the outlier and influence issues covered under outlier detection and influence diagnostics.

Tie and shape assessment

Within-row ties receive average ranks.

The total tie term equals 4,500.

The correction factor equals 0.711094.

Raw distributions can be reviewed with histograms and box plots.

The descriptive distributions differ slightly in spread, especially for G3. The safest omnibus wording is therefore a difference in repeated grade distributions or rank positions, followed by the explicit descriptive direction.

Complete blocks, missing occasions, and analysis population

The classical complete-block calculation uses only subjects with one valid response at every repeated occasion. That requirement makes the analysis population explicit: the 649 students in this example are the students for whom G1, G2, and G3 can all be ranked within the same row. If an occasion is missing, there is no complete three-way ordering for that block, so the ordinary rank-sum formula cannot use the row unchanged.

Complete-case analysis is easy to audit, but its interpretation depends on why records are incomplete. When missingness is related to performance, attendance, treatment response, or another measured process, the complete blocks may differ systematically from the original cohort. Longitudinal modeling with mixed-effects regression, generalized estimating equations, or a suitable multilevel regression can retain partially observed subjects under stated modeling assumptions.

For the present grade analysis, the block count is stable across the workbook, Python, R, and SPSS outputs. That consistency is important because a changed case count would alter the rank sums, tie correction, degrees of freedom calculation, effect-size denominator, and every post hoc comparison.

Asymptotic inference, exact inference, and large samples

The chi-square reference distribution is an asymptotic approximation. It becomes accurate as the number of independent blocks grows, provided the design is valid and no small collection of unusual blocks dominates the ranking pattern. With 649 complete student blocks, the approximation is well supported, and the resulting probability is so small that minor reference-distribution differences would not change the decision.

Small studies can use exact or permutation-based reasoning when software supports it. Exact inference enumerates or otherwise evaluates the possible within-block rank arrangements under the null. Permutation methods repeatedly rearrange condition labels within blocks, preserving the dependence structure while measuring how unusual the observed rank sums are. Both approaches respect the essential blocking principle; labels are never shuffled freely between different students.

The distinction mirrors the broader difference between approximate and exact procedures discussed in Fisher’s exact test, Barnard’s exact test, and the binomial test. In every case, the reference method should match the sample structure and be stated in the report.

Interpreting an ordered pattern without overstating growth

The descriptive means and mean ranks increase from G1 through G3, and the Page trend calculation supports the prespecified order G1 < G2 < G3. Even so, the repeated grade pattern is not identical for all students. G2 is below G1 for 187 students, and G3 is below G1 for 120 students. The omnibus result therefore describes a population-level tendency rather than a universal individual trajectory.

This distinction is important in educational, medical, and behavioral repeated-measures research. A statistically significant average rank progression can coexist with stable cases, reversals, and heterogeneous response profiles. The effect-size value W = 0.1289 captures some of that heterogeneity: the occasions differ reliably, but the within-block ordering is far from perfectly concordant.

When individual trajectories are scientifically important, line plots, transition summaries, and subject-specific models add information beyond the omnibus rank result. The current student-block chart provides a visual starting point, while methods such as hierarchical linear models and random-effects regression can represent variation in starting level and change across students.

Robustness of the conclusion: Friedman Q, pairwise Wilcoxon comparisons, the directional Page result, and the weighted Quade result all point to systematic occasion differences. Their agreement supports the conclusion without implying that every student followed the same trajectory.

16

How to report Friedman test results in APA style

A complete report includes the design, descriptives, statistic, effect size, and adjusted follow-up findings.

Searches for how to report Friedman test results and how to report Friedman test results APA call for a concise paragraph that remains statistically complete.

APA-style example

A Friedman test showed that grades differed significantly across G1, G2, and G3, χ²_F(2, N = 649) = 167.33, p < .001, Kendall’s W = .129. Mean ranks increased from G1 (M_rank = 1.76) to G2 (M_rank = 1.90) and G3 (M_rank = 2.34). Holm-adjusted Wilcoxon signed-rank comparisons indicated significant differences between G1 and G2 (p = .00035), G1 and G3 (p < .001), and G2 and G3 (p < .001). Descriptively, mean grades were 11.40, 11.57, and 11.91, respectively.

Minimum reporting elements

• Name the Friedman test and repeated conditions.
• State the number of complete blocks.
• Report Q or χ²_F, df, and p.
• Report mean ranks for direction.
• Include Kendall’s W.
• Name the post hoc test and p-value adjustment.

Interpretation language

• Use “differed significantly across occasions” for the omnibus result.
• Use mean ranks and descriptives to explain direction.
• Use “p < .001” rather than p = 0.
• Describe W as a magnitude indicator, not another hypothesis test.
• Keep association language separate from causal claims unless the design supports causality.

Additional reporting foundations appear in confidence intervals, margin of error, and statistical power.

17

Friedman test downloads and related guides

Reports, workbook, and contextual statistical resources.

PDF

Python reportthe procedure metrics, ranks, charts, and verified interpretation.Open Python PDF

PDF

R reportRepeated-samples rank analysis and matching result summary.Open R PDF

PDF

SPSS outputSeveral-related-samples output for G1, G2, and G3.Open SPSS PDF

XLS

Excel workbookAuditable ranks, tie correction, effect size, and reporting sheets.Open Excel workbook

Related statistical guides

Repeated-measures ANOVA
Repeated-measures MANOVA
Mixed ANOVA
Mauchly’s test of sphericity
Greenhouse–Geisser correction
Huynh–Feldt correction
Sign test
Cochran’s Q test
Parametric vs nonparametric tests
Effect size
P-value
Hypothesis testing
Descriptive statistics
Box plot interpretation
Frequency distribution
Holm–Bonferroni method
Bonferroni correction
Mixed-effects regression

18

Frequently asked questions about the the procedure

Direct answers to the main the procedure search questions.

What is a the procedure?

The the procedure is a nonparametric omnibus procedure for comparing three or more related conditions. It ranks the condition values within each subject or block and tests whether the condition rank sums differ more than expected under the null hypothesis.

What is the the procedure used for?

It is used for repeated-measures or complete-block designs when the outcome is ordinal or continuous and a rank-based alternative to repeated-measures ANOVA is appropriate.

When should the the procedure be used?

Use the the procedure when the same subjects are observed under at least three conditions, every block contains each condition, and the research question concerns an omnibus difference across those related measurements.

Does the the procedure require normality?

No. The method uses within-block ranks and does not require normally distributed residuals. It still requires a valid related-samples structure and meaningful ranking.

What are the assumptions of the the procedure?

The observations must be related within blocks and independent between blocks; the outcome must be rankable; the same conditions must be measured for each complete block; and the condition labels must represent comparable repeated treatments or occasions.

What is the difference between Kruskal–Wallis and the procedure?

Kruskal–Wallis compares independent groups. The the procedure compares related conditions measured on the same subjects or matched blocks.

Is the the procedure a nonparametric ANOVA?

It is commonly described as the nonparametric alternative to one-way repeated-measures ANOVA. The description is useful, although the the procedure evaluates ranks rather than fitting the same mean-based model.

What is the null hypothesis of the the procedure?

The null states that the related conditions have the same distributional location or no systematic difference in rank position across blocks.

How is the Friedman statistic calculated?

Values are ranked within each block, ranks are summed by condition, and the squared condition rank sums are inserted into the Friedman formula. A correction is applied when ties occur.

What does a significant the procedure mean?

It means at least one related condition differs from the others. Post hoc paired comparisons are needed to determine the specific pairs.

How is the the procedure interpreted here?

G1, G2, and G3 differed significantly, Q(2) = 167.33, p < .001. G3 had the highest mean rank, and all three paired comparisons remained significant after Holm adjustment.

What is Kendall’s W in the the procedure?

Kendall’s W is a standardized effect-size or concordance measure. Here W = 0.1289, indicating a modest repeated-occasion effect.

How do ties affect the the procedure?

Tied values receive average ranks. A tie correction adjusts the statistic because ties reduce the variability of possible rank arrangements.

How do I run a the procedure in R?

Place the repeated conditions in aligned columns and use friedman.test() on the matrix, or supply response, group, and block variables in long format.

How do I run a the procedure in Python?

Use SciPy’s friedmanchisquare() with one aligned array per repeated condition.

How do I run a the procedure in SPSS?

Use the several-related-samples nonparametric procedure with G1, G2, and G3 as the related variables, or use the NPAR TESTS /FRIEDMAN syntax.

How can the the procedure be calculated in Excel?

Rank each row across the repeated conditions, sum the ranks by column, calculate Q, apply the within-row tie correction, and obtain the upper-tail chi-square probability with k − 1 degrees of freedom.

Which post hoc test follows the the procedure?

Pairwise Wilcoxon signed-rank tests with Holm adjustment are a common follow-up. Other specialized repeated-rank multiple-comparison procedures are also available.

The the procedure provides a clear rank-based analysis for repeated or blocked designs. In this grade example, 649 complete student blocks produced a highly significant omnibus result, a modest Kendall’s W, and significant G1–G2, G1–G3, and G2–G3 follow-up differences. The complete interpretation combines the block design, tie correction, condition ranks, effect size, and adjusted pairwise evidence.

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