What is the Wilcoxon signed-rank test critical value table?

The Wilcoxon signed-rank critical value table provides threshold values for the W (or T) statistic based on sample size (n) and significance level (α). If the calculated test statistic W is less than or equal to the tabled critical value, you reject the null hypothesis. A dash (—) in the table means statistical significance is mathematically impossible at that combination of n and α.

How do you find the critical value for a Wilcoxon signed-rank test?

To find the critical value: (1) Calculate the difference for each pair (Di = Xi − Yi). (2) Remove any pairs where Di = 0 — your remaining count is effective n. (3) Choose your significance level α (e.g., 0.05 two-tailed). (4) Locate the row for your n in the table. (5) Read the value under your chosen α column. That integer is W_critical. Reject H₀ if your computed W ≤ W_critical.

What is the decision rule for the Wilcoxon signed-rank test?

The decision rule is: Reject H₀ if W_calculated ≤ W_critical. Here, W_calculated = min(W⁺, W⁻), the smaller of the sum of positive ranks and the sum of negative ranks. If W is less than or equal to the tabled critical value, the difference between the paired groups is statistically significant.

What do you do with zero differences in the Wilcoxon signed-rank test?

When a paired difference equals zero (Di = 0), that pair is excluded from the analysis entirely. The effective sample size n is reduced by the count of zero pairs before ranking begins. Only non-zero differences are ranked and used to compute W.

When should you use the normal approximation for the Wilcoxon test?

Use the normal Z-score approximation when the effective sample size exceeds 20 (n > 20), or n > 30 for increased precision. As n grows, the sampling distribution of W converges to normal. The approximation formula is: Z = (W − μW) / σW, where μW = n(n+1)/4 and σW = √[n(n+1)(2n+1)/24].

Is the Wilcoxon signed-rank test for independent or dependent samples?

The Wilcoxon signed-rank test is strictly for dependent (paired) samples — before-and-after measurements on the same subjects, matched pairs, or repeated measures on a single group. For two independent groups, use the Mann-Whitney U test instead.

What is the minimum sample size for a Wilcoxon signed-rank test?

The theoretical minimum effective sample size is n = 5. For n < 5, it is mathematically impossible to achieve statistical significance at α = 0.05 (two-tailed), because the maximum possible W statistic is too small to exceed any meaningful threshold.

What is the difference between W and T in the Wilcoxon signed-rank test?

They refer to the same test statistic. W typically represents the overall signed-rank statistic, while T often denotes the smaller of the sum of positive ranks (W⁺) or negative ranks (W⁻). Both use the same critical value table. The decision rule is identical: reject H₀ if the statistic ≤ the tabled critical value.

How do you handle tied ranks in the Wilcoxon signed-rank test?

If two or more absolute differences are equal, assign them the average (mean) of the ranks they would have occupied. For example, if the 3rd and 4th ranked observations are tied, each receives rank 3.5. Ties also require a variance correction to the standard error when using the normal approximation formula.

Can the Wilcoxon signed-rank test be used for ordinal data?

Yes. The Wilcoxon signed-rank test is specifically designed for ordinal (ranked) data or continuous data that violates the normality assumption required by the paired t-test. It requires only that the data can be ranked and that the differences are symmetric around the median under H₀.

Wilcoxon Signed-Rank Test Table | Critical Values of W (n = 5 to 50) + PDF

Wilcoxon Critical Value Lookup

Sample Size (n)

Test Direction

Significance Level (α)

W_critical =

—

Decision rule: Reject H₀ if W_calculated ≤ tabled critical value. A dash (—) means statistical significance is impossible at that n and α. Click any cell to highlight it.

How to Read the Wilcoxon Critical Value Table: Step-by-Step

Follow these steps in sequence every time you use this table. The most common error researchers make is using the raw sample size instead of the effective n after excluding zero-difference pairs.

1 Calculate all paired differences (D_i = X_i − Y_i). List Before minus After (or Post minus Pre, depending on your hypothesis direction). If any difference equals zero exactly, discard that pair entirely — it contributes no rank information.

2 Count your effective n. After dropping zeros, n is the number of remaining pairs. This is the n you use for table lookup — not your original sample count.

3 Rank the absolute values |D_i| from smallest (rank 1) to largest (rank n). If two or more absolute differences are equal, assign each the mean of the ranks they would have received (average rank for ties).

4 Compute W. Re-attach the original sign of D_i to each rank. Sum all ranks with a positive sign → W⁺. Sum all ranks with a negative sign → W⁻. Your test statistic is W = min(W⁺, W⁻) — the smaller of the two sums.

5 Choose your α and test direction. Select α = 0.05 two-tailed for standard academic research. Use one-tailed only when your hypothesis predicts a specific direction of change before data collection.

6 Find the critical value W_critical. Locate the row for your effective n. Move across to your α column. That integer is your threshold.

7 Apply the decision rule. If W ≤ W_critical → Reject H₀ (statistically significant difference). If W > W_critical → Fail to reject H₀ (no significant difference detected). Note: the critical value is an inclusive bound — if W equals the tabled value exactly, you still reject H₀.

Key Formulas

Paired Difference

D_i = X_i − Y_i

Raw arithmetic difference for each pair. Drop if D_i = 0.

Test Statistic W

W = min(W⁺, W⁻)

The smaller of the sum of positive ranks and the sum of negative ranks.

Verification Check

W⁺ + W⁻ = n(n+1)/2

The two rank sums must always total n(n+1)/2. Use this to catch arithmetic errors.

Worked Example: Wilcoxon Signed-Rank Test Step by Step

Scenario: A clinical psychologist measures anxiety scores (0–20 scale; higher = more anxious) for 8 patients before and after a 6-week cognitive-behavioural therapy (CBT) programme. She cannot assume normality because of the small sample, so she uses the Wilcoxon signed-rank test in place of a paired t-test.

Step 1 — Raw Data & Paired Differences

Patient	Before (X)	After (Y)	D = X − Y	\|D\|	Rank of \|D\|	Signed Rank
1	14	13	+1	1	1	+1
2	12	10	+2	2	2	+2
3	16	13	+3	3	3	+3
4	11	15	−4	4	4	−4
5	15	10	+5	5	5	+5
6	17	11	+6	6	6	+6
7	13	6	+7	7	7	+7
8	18	10	+8	8	8	+8

No zero differences — effective n = 8. Patient 4 had higher anxiety after therapy (D = −4), giving a negative signed rank.

Step 2 — Compute W

Sum of positive ranks: W⁺ = 1 + 2 + 3 + 5 + 6 + 7 + 8 = 32

Sum of negative ranks: W⁻ = 4 = 4

Verification: W⁺ + W⁻ = 32 + 4 = 36 = 8 × 9 / 2 = 36 ✓

Test statistic: W = min(W⁺, W⁻) = min(32, 4) = W = 4

Step 3 — Table Lookup & Decision

At n = 8, two-tailed α = 0.05, the table gives W_critical = 3.

Decision: W = 4 > W_critical = 3 → Fail to reject H₀ at α = 0.05 (two-tailed).

One patient worsened enough to pull the positive rank sum down. The majority did improve, but the evidence is not quite strong enough to claim significance at the 5% level with only 8 observations.

Note: At one-tailed α = 0.05, W_critical = 5, and W = 4 ≤ 5 → reject H₀. One-tailed testing would be warranted only if the directional hypothesis (CBT reduces anxiety) was formally specified before data collection.

Step-by-step diagram of Wilcoxon signed-rank test calculation: paired differences, ranking, and W statistic computation for anxiety therapy example

Large Samples: The Wilcoxon Normal Approximation (n > 20)

When n exceeds 20, the distribution of W converges toward a normal distribution. Rather than extending the critical value table indefinitely, researchers compute a standard Z-score and compare it to the standard normal distribution — a method endorsed by the NIST/SEMATECH e-Handbook of Statistical Methods.

Mean of W under H₀

μ_W = n(n + 1) / 4

Expected value of the W statistic when the null hypothesis is true (no systematic difference).

Standard Error of W

σ_W = √[n(n+1)(2n+1) / 24]

Standard deviation of W under H₀. Assumes no ties; apply the variance correction formula when ties are present.

Z-Score Approximation

Z = (W − μ_W) / σ_W

Compare |Z| to 1.960 (two-tailed α = 0.05) or 2.576 (two-tailed α = 0.01) from the standard normal table.

Tie Variance Correction

σ_W,corrected = √[n(n+1)(2n+1)/24 − Σ(t_j³ − t_j)/48]

t_j = number of observations in tie group j. Apply when absolute differences share ranks (ties in |D|).

Quick Example (n = 25)

Suppose n = 25 and W = 150. Then μ_W = 25 × 26 / 4 = 162.5 and σ_W = √(25 × 26 × 51 / 24) = √(1381.25) ≈ 37.17. Z = (150 − 162.5) / 37.17 ≈ −0.34. Since |−0.34| < 1.96, fail to reject H₀ at α = 0.05 (two-tailed). The deviation from the expected rank sum is not statistically significant.

Wilcoxon Signed-Rank Test vs Paired Student's t-Test

Both tests analyse paired data, but they make fundamentally different assumptions. Choosing the wrong test inflates the Type I error rate or squanders statistical power. The guide below — grounded in the framework described by Conover (1999) in Practical Nonparametric Statistics — helps researchers select correctly.

Feature / Assumption	Wilcoxon Signed-Rank	Paired Student's t-Test
Data Type Required	Ordinal, Interval, or Ratio	Continuous Interval or Ratio
Distribution Shape	Distribution-free (non-parametric)	Differences must be ~Normal
Sensitivity to Outliers	Highly robust (ranks reduce impact)	Sensitive — outliers distort the mean
Basis of Comparison	Median of paired differences	Mean of paired differences
Best Sample Size	Small or non-normal (n < 30)	Normally distributed (n ≥ 30)
Relative Power (normal data)	~95% of t-test (asymptotic relative efficiency)	Full power on normal data
Sample Type	Dependent / paired samples only	Dependent / paired samples only

The asymptotic relative efficiency (ARE) of the Wilcoxon signed-rank test relative to the paired t-test is 3/π ≈ 0.955 under normal distributions — meaning the Wilcoxon test requires only about 5% more observations to achieve the same power as the t-test on perfectly normal data. For heavy-tailed or skewed distributions, the Wilcoxon test is often more powerful. This result is derived in Lehmann & D'Abrera (2006), Nonparametrics: Statistical Methods Based on Ranks.

Decision flowchart: when to use the Wilcoxon signed-rank test versus the paired Student's t-test based on normality, sample size, and data type

Wilcoxon Signed-Rank Test: Symbol & Formula Glossary

Every symbol used in the Wilcoxon signed-rank test is defined below. Bookmark this as a quick reference for exams, research write-ups, or software output interpretation.

Symbol	Full Name	Formula / Rule	Context
D_i	Paired difference	X_i − Y_i	Computed per pair; drop if D_i = 0
n	Effective sample size	Original n − count(D_i = 0)	Row lookup in this table
W⁺	Sum of positive ranks	Σ Rank(\|D_i\|) for D_i > 0	Reflects magnitude of positive changes
W⁻	Sum of negative ranks	Σ Rank(\|D_i\|) for D_i < 0	Reflects magnitude of negative changes
W	Test statistic	min(W⁺, W⁻)	Compared against W_critical from this table
W_critical	Critical value	Lookup(n, α, tail)	Decision threshold from this table
μ_W	Mean of W under H₀	n(n + 1) / 4	Large-sample normal approximation
σ_W	Std. error of W	√[n(n+1)(2n+1) / 24]	Denominator of Z approximation
Z	Z-score approximation	(W − μ_W) / σ_W	Used when n > 20 (compare to Z-table)
t_j	Ties in group j	Count of tied \|D_i\| values per group	Applied in variance correction formula

Wilcoxon Signed-Rank Test: Key Facts & Figures

1945

Year Frank Wilcoxon published the original test in Biometrics Bulletin

n = 5

Minimum effective sample size — below this, significance at α = 0.05 (two-tailed) is mathematically impossible

95.5%

Asymptotic relative efficiency (ARE) vs paired t-test on normally distributed data (3/π)

n > 20

Threshold at which the normal Z-score approximation becomes appropriate

Alpha level columns in this table: α = 0.10, 0.05, 0.02, 0.01 (two-tailed)

Wilcoxon Signed-Rank Table PDF — Free Download

Download a free printable Wilcoxon critical value table formatted for exams, research labs, and classroom use. All versions cover n = 5 to 50 at standard significance levels.

Two-Tailed Critical Values α = 0.10, 0.05, 0.02, 0.01 · n 5–50 · PDF One-Tailed Critical Values α = 0.05, 0.025, 0.01, 0.005 · n 5–50 · PDF Exam Reference Card Table + formulas + decision rule · PDF

Sources & Further Reading

The critical values in this table are derived from the exact distribution of the Wilcoxon signed-rank statistic as described in the original publication and validated against peer-reviewed statistical references:

Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80–83. https://doi.org/10.2307/3001968 — The original paper introducing both the rank-sum and signed-rank tests. The signed-rank procedure for paired data appears on page 81.

NIST/SEMATECH (2012). e-Handbook of Statistical Methods — Wilcoxon Signed-Rank Test. National Institute of Standards and Technology. itl.nist.gov — U.S. government reference covering the exact distribution, normal approximation, tie handling, and a fully worked example.

Conover, W. J. (1999). Practical Nonparametric Statistics (3rd ed.). Wiley. Chapter 5: The Wilcoxon Signed Ranks Test — Graduate-level treatment covering exact tables, asymptotic efficiency, and power comparisons. Source of the ARE = 3/π result widely cited in the literature.

Penn State STAT 415: Introduction to Mathematical Statistics. Non-parametric Tests — Wilcoxon Signed Rank Test. Pennsylvania State University. online.stat.psu.edu — Open-access course materials with step-by-step examples and discussion of assumptions. Freely available to all students and researchers.

Lehmann, E. L., & D'Abrera, H. J. M. (2006). Nonparametrics: Statistical Methods Based on Ranks. Springer. — Classic reference text from UC Berkeley establishing the theoretical foundations of rank-based tests including the Wilcoxon signed-rank procedure. Source for the ARE = 3/π ≈ 0.955 asymptotic efficiency result.

Harvard Chan Biostats (2021). Nonparametric Methods — Wilcoxon Signed-Rank Test. Harvard T.H. Chan School of Public Health, Department of Biostatistics. hsph.harvard.edu — Course notes from Harvard's biostatistics department detailing applications in clinical and epidemiological research.

Frequently Asked Questions About the Wilcoxon Signed-Rank Table

What is the Wilcoxon signed-rank critical value table?

The Wilcoxon signed-rank table lists the maximum value of the W statistic (the smaller of W⁺ or W⁻) at which you can still reject the null hypothesis. For a given n and α, if your calculated W is less than or equal to the tabled value, the paired difference is statistically significant. A dash (—) in a cell means rejection is impossible at that level for that n.

Why do I subtract zero differences from n?

A difference of exactly zero contains no information about direction. Assigning it a rank would distort both W⁺ and W⁻ without reflecting a real positive or negative change. Excluding zero-difference pairs is therefore standard practice, not a data-manipulation choice. Your table lookup must use the reduced n — not the original count — or the critical value will be incorrect.

When do I use one-tailed vs two-tailed critical values?

Use two-tailed values (the default) when your research question is "Is there any difference between the groups?" Use one-tailed only when your pre-registered hypothesis specifies a direction — for example, "The intervention will reduce scores." One-tailed critical values are larger (less conservative) than two-tailed at the same α, so selecting one-tailed post hoc to get a significant result constitutes p-hacking.

What does it mean when the table shows a dash (—)?

A dash means no critical value exists at that combination of n and α. With very small n, even assigning all ranks to one side does not produce a small enough W to reach the required probability threshold. The minimum n for any significance at two-tailed α = 0.05 is n = 6. At two-tailed α = 0.01, you need n = 8. This is a mathematical constraint of the discrete rank distribution, not a limitation of your data.

What is the difference between W and T in the Wilcoxon test?

These are two notations for the same thing. Wilcoxon's original 1945 paper used T. Modern textbooks — including most APA-formatted publications — have largely shifted to W. Both equal min(W⁺, W⁻), and both use the exact same critical value table. If your software reports "T = 12" and the table lists "W_critical = 15" at your n and α, the comparison is valid: 12 ≤ 15 → reject H₀.

How do I handle tied differences in the ranking step?

When two or more |D_i| values are equal, assign each the midrank — the average of the ranks those positions would have received. For example, if the 4th and 5th smallest absolute differences are tied, both receive rank 4.5. The sign associated with each original difference is carried over to its midrank. Midranks still sum to n(n+1)/2, so the verification check still applies.

Is the Wilcoxon signed-rank test appropriate for Likert scale data?

Yes — in most research contexts, the Wilcoxon signed-rank test is preferable to the paired t-test for Likert-scale responses. Likert data is ordinally scaled (the intervals between response categories are not guaranteed to be equal), violating the interval-scale assumption of the t-test. The Wilcoxon test requires only that the data can be ranked, making it well-suited for 5-point and 7-point Likert items in psychology, education, and survey research.

What is the Wilcoxon signed-rank test's null hypothesis?

H₀ states that the median of the paired differences is zero — meaning the two conditions produce symmetrically distributed responses with no systematic shift. The alternative H₁ (two-tailed) is that the median difference is not zero. The test does not assume a specific shape for the distribution of differences, only that the distribution is symmetric around the median.

Can I download the Wilcoxon signed-rank critical value table as a PDF?

Yes — three free PDF versions are available in the download section above. The two-tailed PDF covers α = 0.10, 0.05, 0.02, and 0.01 for n = 5 to 50. The one-tailed PDF covers α = 0.05, 0.025, 0.01, and 0.005 for the same n range. The exam reference card includes the complete table plus the decision rule and normal approximation formulas.

Related Statistical Tables & Resources

📊

Mann-Whitney U Table Non-parametric test for independent samples

🔢

Paired Samples t-Test Parametric alternative when normality holds

📋

T-Distribution Table Critical values for t-test and confidence intervals

📋

Z-Table (Standard Normal) Used for the Wilcoxon normal approximation (n > 20)

🎓

Hypothesis Testing Guide Complete guide with test types and worked examples

🏠

Statistics Fundamentals Your complete statistics reference — tables, calculators & guides

How to Use This Table

1Compute D_i = X_i − Y_i; drop zeros

2Rank |D_i| smallest → largest; use midranks for ties

3Compute W = min(W⁺, W⁻)

4Find row for effective n & your α column

5If W ≤ W_critical → Reject H₀

Quick Reference (Two-Tailed α = 0.05)

n = 8 W_crit = 3

n = 10 W_crit = 8

n = 15 W_crit = 25

n = 20 W_crit = 52

n = 25 W_crit = 89

n = 30 W_crit = 137

Key Formulas

W = min(W⁺, W⁻)
Verify: W⁺ + W⁻ = n(n+1)/2
Z ≈ (W − n(n+1)/4)
/ √(n(n+1)(2n+1)/24)

Decision Rule

W ≤ W_critical
Reject H₀ — significant difference

W > W_critical
Fail to reject H₀ — no significant difference

Statistics Fundamentals

Tables, calculators & complete guides

More Tables

→ Mann-Whitney U Table (independent samples) → T-Distribution Table → Z-Table → Chi-Square Table → Pearson Correlation Table

Understanding the Wilcoxon Signed-Rank Table

Why Critical Values Increase With n

As n grows, the maximum possible rank sum n(n+1)/2 grows much faster than the critical value. Large-n studies have greater power to detect small effects because moderate imbalances between W⁺ and W⁻ become meaningful — whereas in small samples only near-total dominance of one sign reaches significance. This is the same power-sample size relationship that governs all frequentist tests.

Why the Inclusive Decision Rule Matters

The critical value is an inclusive upper bound on W for rejection. If W equals the tabled value exactly, you reject H₀ — not fail to reject. This is because the distribution of W under H₀ is discrete: the rejection region is defined as W ≤ W_critical to ensure the Type I error rate does not exceed α, even though the exact probability at that value may be slightly below α.

Wilcoxon vs Mann-Whitney U: A Common Confusion

Both tests were published by Wilcoxon (1945) and both use ranks — but they answer different questions. The signed-rank test is for paired (dependent) data: the same subjects measured twice, or matched pairs. The Mann-Whitney U test is for two independent groups. Using the wrong test — or the wrong table — produces invalid inferences, so always confirm that your data structure matches the test you have chosen.