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Wilcoxon Signed-Rank Test Calculator

Sample 1 data (comma-separated): Sample 2 data (comma-separated):

Zero Difference Method: Wilcox Pratt Zero split

Continuity Correction: Yes No

Hypothesis

H_{1}

: Two-sided Left-tailed Right-tailed

Significance

α

: 0.05 0.01 0.10

Decimal Places: 2 3 5 8

Clear

Introduction

The Wilcoxon Signed-Rank Test Calculator serves as a non-parametric tool for comparing two related samples or repeated measurements on a single sample. It assesses whether the median of the differences between paired observations $x_{1}$ and $x_{2}$ is zero. This approach is particularly useful when data sets do not meet the strict normality requirements of parametric alternatives.

What this calculator does

This tool processes paired numeric data to determine statistical significance. Users input two sets of data points, select a method for handling zero differences-such as Wilcox, Pratt, or Zero split-and choose a significance level $α$ . The calculator outputs the test statistic $W$ , positive and negative rank sums, a standardised $z$ -score, the resulting $p$ -value, and the effect size $r$ .

Formula used

The calculation determines the mean $μ$ and variance $σ^{2}$ of the rank sums. The test statistic $z$ is derived using a continuity correction to approximate the normal distribution. Here, $n$ represents the number of non-zero pairs, and $t$ denotes the number of observations tied at a specific rank.

μ = \frac{n (n + 1)}{4}

σ^{2} = \frac{n (n + 1) (2 n + 1)}{24} - \sum \frac{t^{3} - t}{48}

How to use this calculator

1. Input the paired data points for Sample 1 and Sample 2 as comma-separated values.
2. Select the preferred method for zero differences and toggle the continuity correction if desired.
3. Choose the alternative hypothesis (two-sided, left-tailed, or right-tailed) and the significance level.
4. Execute the calculation to view the summary table, step-by-step process, and normal distribution chart.

Example calculation

Scenario: A sports analysis study examines the performance scores of ten athletes before and after a specific endurance training programme to determine if the intervention significantly altered their results.

Inputs: Sample 1: $7, 8, 6$ ; Sample 2: $5, 6, 4$ ; $n = 3$ .

Working:

Step 1: $n = 3, W_{plus} = 6$

Step 2: $μ = \frac{3 (3 + 1)}{4} = 3$

Step 3: $σ^{2} = \frac{3 (4) (7)}{24} = 3.5$

Step 4: $z = \frac{6 - 3 - 0.5}{\sqrt{3.5}} \approx 1.336$

Result: $z = 1.34$

Interpretation: The calculated $z$ -score is compared against the critical value to determine if the training had a statistically significant effect.

Summary: The test provides evidence regarding the magnitude and direction of change between the two related conditions.

Understanding the result

The primary output is the $p$ -value; if it is less than the selected $α$ , the null hypothesis is rejected, suggesting a significant difference between pairs. The effect size $r$ quantifies the strength of this difference, while the $z$ -score indicates how many standard deviations the observed sum is from the mean.

Assumptions and limitations

The test assumes that the differences between pairs are independent and come from a continuous distribution. It also requires the distribution of differences to be symmetric around the median. It is limited to paired data with an equal number of observations.

Common mistakes to avoid

A frequent error is inputting samples with unequal sizes, which prevents the calculation of differences. Additionally, misapplying the zero-difference method or neglecting the continuity correction in small samples can lead to inaccurate $p$ -values and flawed statistical conclusions.

Sensitivity and robustness

The calculation is robust against outliers compared to parametric tests, as it relies on ranks rather than raw values. However, it is sensitive to the method used to handle zero differences and ties, which can influence the variance and the resulting significance of the test.

Troubleshooting

If an error occurs, ensure that both data columns contain only numeric values and have identical lengths. If all paired differences result in zero, the test cannot be performed. Check for non-numeric characters or excessive decimal places that may exceed the processing limits of the system.

Frequently asked questions

What is the Wilcox method for zeros?

The Wilcox method excludes any pairs where the difference is zero, effectively reducing the sample size used for the ranking process.

How is effect size calculated?

The effect size $r$ is calculated by dividing the absolute $z$ -score by the square root of the number of valid pairs $n$ .

When should I use a two-sided test?

A two-sided test is appropriate when the research aims to detect a difference in either direction, whereas a one-sided test is used for a specific predicted direction.

Where this calculation is used

In environmental science, this method is used to compare pollutant levels at the same locations across different seasons. Social research often employs it to analyse changes in survey scores before and after an intervention within a single population. Population studies use it to observe shifts in demographic metrics over two distinct time periods. Because it does not assume a normal distribution, it is a staple in academic curricula for non-parametric modelling and probability theory where data symmetry is present but normality is not guaranteed.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.