Wilcoxon Signed-Rank Test Table: Critical Values & Use

Introduction

Have you ever calculated a Wilcoxon test statistic but felt unsure what to do next?

A Wilcoxon Signed-Rank Test Table helps you find critical values and make sense of your test results. This nonparametric test is useful when comparing paired or dependent observations. It is often used when your data does not meet the assumptions of a paired t-test.

For beginners in the USA, understanding the table can make statistical analysis much easier. You do not need advanced statistics knowledge to use it correctly.

In this guide, you will learn how to use a Wilcoxon signed-rank test table step by step. We will explain sample size, significance levels, test statistics, and critical values. You will also see a complete worked example from start to finish.

By the end, you will know how to read the table and determine whether your result is statistically significant.

What Is the Wilcoxon Signed-Rank Test?

The Wilcoxon Signed-Rank Test is a nonparametric statistical test for comparing two related sets of measurements. It focuses on the differences between paired observations rather than treating them as independent data.

This test is useful when the same people, objects, or subjects are measured twice. For example, you might compare blood pressure before and after a treatment. Each person’s two measurements form a matched pair.

The test ranks the absolute differences between paired observations. It then considers the direction of those differences. This helps determine whether the typical difference between the two measurements is statistically meaningful.

When Is the Wilcoxon Signed-Rank Test Used?

Use the test when you have paired or dependent observations. Common situations include:

• Before-and-after measurements from the same participants
• Matched subjects in two conditions
• Repeated measurements from the same individual
• Paired experimental observations

The Wilcoxon test can be useful when the paired differences do not meet the normality assumption required by a paired t-test. However, the test still relies on assumptions about the paired differences and their measurement scale.

Wilcoxon Signed-Rank Test vs. Paired t-Test

The paired t-test compares the mean difference between paired observations. The Wilcoxon signed-rank test instead uses the ranks of the differences.

The paired t-test generally requires normally distributed differences. The Wilcoxon test provides a nonparametric alternative when that assumption is not reasonable.

Pro Tip: Check your study design first. The fact that data are non-normal alone does not automatically mean the Wilcoxon test is the best choice.

What Is a Wilcoxon Signed-Rank Test Table?

A Wilcoxon signed-rank test table lists critical values used to evaluate the results of a Wilcoxon signed-rank test. It helps you decide whether your calculated test statistic provides enough evidence to reject the null hypothesis.

The table mainly depends on sample size and the chosen significance level. You calculate the test statistic from your paired data, then compare it with the appropriate critical value.

Key Parts of a Wilcoxon Signed-Rank Test Table

• Sample size: This is the number of non-zero paired differences used in the test. Zero differences are typically excluded from the ranking process.
• Significance level (α): This sets the threshold for statistical significance. Common choices include 0.05 and 0.01.
• Critical value: This is the reference value obtained from the table. It depends on the sample size and significance level.
• Test statistic: This is calculated from your sample data. Depending on the table’s convention, it may use the smaller of the positive and negative rank sums.

To use the table correctly, you must first identify the correct sample size and significance level. Then, compare your calculated statistic with the matching critical value.

Pro Tip: Always check whether your reference table uses a one-sided or two-sided test. Different tables may also use different test-statistic conventions.

Wilcoxon Signed-Rank Test Critical Values Table

The table below gives Wilcoxon signed-rank critical values for common sample sizes and significance levels. It uses the smaller rank sum, W = min(W⁺, W⁻), as the test statistic.

For a two-tailed test, use the two-tailed columns. For a one-tailed test, use the one-tailed columns. Reject the null hypothesis when your calculated W is less than or equal to the critical value.

Two-Tailed Wilcoxon Signed-Rank Critical Values

Sample Size (n)	α = 0.10	α = 0.05	α = 0.02	α = 0.01
5	0	—	—	—
6	2	0	—	—
7	3	2	—	—
8	5	3	0	—
9	8	5	1	—
10	10	8	3	—
11	13	10	5	—
12	17	13	7	—
13	21	17	9	—
14	25	21	12	—
15	30	25	15	—
16	35	29	19	—
17	41	34	23	—
18	47	40	27	—
19	53	46	32	—
20	60	52	37	—
21	67	58	42	—
22	75	65	48	—
23	83	73	54	—
24	91	81	61	—
25	100	89	68	—
26	110	98	75	—
27	119	107	83	—
28	130	116	91	—
29	140	126	100	—
30	151	137	109	—

How to use it: Find your effective sample size in the first column. Then select your significance level. The value where they meet is your critical value.

Important: The sample size is the number of non-zero paired differences included in the ranking. Zero differences are excluded. Ties require appropriate rank handling.

Table note: A dash (—) indicates that the exact critical-value table does not provide a rejection value at that sample size and significance level. Different published tables may use different conventions, so always confirm the statistic definition before comparing values.

How to Use the Wilcoxon Signed-Rank Test Table

Knowing how to calculate the Wilcoxon statistic is only half the process. You also need the correct Wilcoxon signed-rank test table to evaluate your result.

Follow these steps to use the table correctly.

1. Determine the Sample Size

Count the paired observations with non-zero differences. Exclude pairs where the difference equals zero.

This effective sample size determines which row you should use in the table.

2. Choose the Significance Level

Select your significance level, represented by α. Common choices include 0.05 and 0.01.

Your choice should match the significance level stated in your statistical test.

3. Calculate the Wilcoxon Test Statistic

Calculate the differences between each pair and rank their absolute values. Assign signs based on the direction of each difference.

Then calculate the rank sums according to your table’s convention. Many tables use W = min(W⁺, W⁻).

4. Find the Corresponding Critical Value

Locate your sample size in the table’s row. Next, find your chosen significance level in the appropriate column.

The value where they intersect is your critical value.

5. Compare the Two Values

Compare your calculated test statistic with the critical value.

When using a table based on the smaller rank sum, a statistic less than or equal to the critical value indicates rejection of the null hypothesis.

6. Determine the Result

If the test statistic meets the rejection rule, the result is statistically significant at your chosen α level.

Otherwise, you fail to reject the null hypothesis. This does not prove that the null hypothesis is true.

Pro Tip: Check whether your table uses one-tailed or two-tailed critical values before making the comparison.

How to Read a Wilcoxon Signed-Rank Test Table

Reading a Wilcoxon signed-rank test table becomes straightforward once you know which row and column to use.

Start by identifying your effective sample size. This is usually the number of non-zero paired differences remaining after excluding zero differences. Find this number in the table’s Sample Size (n) column.

Next, identify your chosen significance level (α). For example, you might use α = 0.05. Find that value across the table’s header. Make sure you select the correct section for a one-tailed or two-tailed test.

Follow the row for your sample size across to the column for your significance level. The number at their intersection is the critical value.

For example, suppose your effective sample size is 10 and your selected two-tailed significance level is 0.05. Using the table above, the critical value is 8.

If your calculated W statistic is 8 or smaller, you reject the null hypothesis under this table convention. If W is greater than 8, you fail to reject the null hypothesis.

Always confirm the table’s definition of W before interpreting the result. Different references can use different conventions for reporting Wilcoxon statistics.

Wilcoxon Signed-Rank Test Example

Suppose a researcher wants to know whether a training program changes test scores. Eight participants take a test before and after training.

Participant	Before	After	Difference
1	70	71	+1
2	65	67	+2
3	80	77	−3
4	72	76	+4
5	68	73	+5
6	75	81	+6
7	82	89	+7
8	77	85	+8

There are no zero differences, so the effective sample size is n = 8.

Step 1: Rank the Absolute Differences

Ignore the signs and rank the absolute differences from smallest to largest.

Difference	Absolute Difference	Rank	Signed Rank
+1	1	1	+1
+2	2	2	+2
−3	3	3	−3
+4	4	4	+4
+5	5	5	+5
+6	6	6	+6
+7	7	7	+7
+8	8	8	+8

The sum of positive ranks is:

W⁺ = 1 + 2 + 4 + 5 + 6 + 7 + 8 = 33

The sum of negative ranks is:

W⁻ = 3

Using the smaller-rank-sum convention:

W = min(W⁺, W⁻) = 3

Step 2: Find the Critical Value

Suppose we use a two-tailed test with α = 0.05.

Find n = 8 in the critical-values table. Then move to the α = 0.05 column.

The critical value is 3.

Step 3: Compare the Values

Our calculated statistic is W = 3.

The critical value is 3.

Because W ≤ critical value, we reject the null hypothesis at the 0.05 significance level.

Pro Tip: Always state your test direction, significance level, sample size, and statistic convention before interpreting the result.

Wilcoxon Test Statistic vs. Critical Value

The test statistic and critical value both help determine the outcome of a Wilcoxon signed-rank test. However, they come from different parts of the analysis.

Term	Meaning
Test statistic	Calculated from the paired sample data
Critical value	Obtained from the appropriate reference table
p-value	Probability-based measure used to assess statistical significance

The test statistic comes directly from your observed differences and their ranks. Its calculation depends on the convention used by your statistical method.

The critical value comes from a reference table. It depends on factors such as the effective sample size, significance level, and whether the test is one-tailed or two-tailed.

You compare the test statistic with the critical value when using the critical-value approach. With the p-value approach, you compare the p-value with your selected significance level instead.

For example, if a table uses the smaller rank sum, a statistic at or below its critical value leads to rejection of the null hypothesis.

Important: Do not compare a test statistic with a critical value from a different table convention.

Wilcoxon Signed-Rank Test vs. Paired t-Test

Both tests can analyze paired or dependent observations, such as before-and-after measurements. However, they use different approaches to evaluate the differences.

Feature	Wilcoxon Signed-Rank Test	Paired t-Test
Type of data	Paired quantitative or ordinal data	Paired quantitative data
Main approach	Uses ranked differences	Uses mean differences
Key assumption	Paired differences should have an appropriate symmetric distribution for the usual signed-rank interpretation	Paired differences should be approximately normally distributed
Outliers	Often less sensitive than the paired t-test	Can be more sensitive to extreme differences
Best suited for	Data that does not reasonably meet paired t-test assumptions	Data with approximately normal paired differences

The paired t-test is often appropriate when the differences are reasonably close to normal and the mean difference is the quantity of interest.

The Wilcoxon signed-rank test can be useful when the normality assumption for the paired differences is questionable. It ranks the differences instead of relying directly on their measured values.

Neither test is automatically better. Your data type, study design, distribution, and research question should guide the choice.

Common Mistakes

Using a Wilcoxon signed-rank test table looks simple, but small errors can change your conclusion. Most mistakes happen when choosing the sample size, significance level, or test direction.

1. Using the Wrong Sample Size

The sample size for the Wilcoxon signed-rank test is based on the non-zero paired differences used for ranking. Do not automatically use the total number of observations.

For example, if you have 12 pairs but two differences equal zero, your effective sample size is n = 10. Use the row for 10, not 12.

2. Choosing the Wrong Significance Level

Your critical value depends on the selected significance level (α). Common choices include 0.05 and 0.01.

Always match the table column to the α value specified in your hypothesis test. Using the wrong column can produce an incorrect decision.

3. Confusing the Test Statistic With the Critical Value

The test statistic comes from your data. The critical value comes from the reference table.

Do not treat these values as interchangeable. First calculate your statistic, then compare it with the appropriate critical value.

4. Handling Zero Differences or Ties Incorrectly

Zero differences are normally excluded before ranking, which changes the effective sample size.

Tied absolute differences require average ranks rather than assigning the same sequential rank. Follow the ranking method required by your statistical procedure.

5. Ignoring One-Sided vs. Two-Sided Testing

A one-sided test and a two-sided test can use different critical values. Always determine your test direction before looking up the value.

Expert Tip: Write down n, α, test direction, and your statistic convention before opening the critical-value table. This simple checklist prevents many lookup errors.

Conclusion

The Wilcoxon signed-rank test table makes it easier to evaluate paired data and interpret test results. You learned how to identify sample size, choose significance levels, find critical values, and compare them with your test statistic.

You also saw how zero differences, ties, and test direction can affect your table lookup. Understanding these details helps you avoid common statistical errors and make more confident decisions.

If you’re new to nonparametric testing, take your time with each step. Always confirm your table’s statistical convention before interpreting the result.

Ready to apply what you learned? Explore our related statistical calculators and guides for more practical examples. You can also share this guide with someone learning statistics.