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Non-parametric Methods · SIGN-P / WSR-P

Sample Size for Comparing Paired Observations

Plan one set of independent paired observations using the direction of paired differences.

1. Planning setup

The sign test uses only the direction of nonzero differences.

D = Observation 2 − Observation 1

A positive difference means Observation 2 is larger than Observation 1.

2. Information obtained from previous study / literature

p(D > 0 | D ≠ 0), among nonzero paired differences.

Zeros are excluded from the sign statistic and inflate the complete-pair target.

3. Information decided by the researcher

Tests whether the probability of a positive difference differs from 50%.

4. Results

Enter the hypothesis-test assumptions and select Calculate.

Formula, assumptions, limitations, and methodological note

Example clinical scenario

A rehabilitation study records knee-pain scores in the same patients before and after physiotherapy. Because paired changes may be skewed, the researcher plans a paired sign test or Wilcoxon signed-rank analysis.

Formula and methodological note

The paired sign statistic counts positive nonzero differences, S = Σ I(Dᵢ > 0), where D = Observation 2 − Observation 1. Under the null, S follows Binomial(n, 0.5); the calculator enumerates exact rejection regions for each integer n.

Pairs must be independent of other pairs. Exact zero differences are excluded, and anticipated zeros and incomplete pairs are inflated sequentially after the power search. Realised power may be lower if zeros or incomplete pairs exceed their planning values.

The Normal-shift converter uses p = Φ(δ/SD) only to translate a paired difference and SD into a planning sign probability. The exact sign test itself does not require Normal differences.

The Wilcoxon signed-rank option uses the Al-Sunduqchi–Guenther distribution-adjusted paired-t approximation described in PASS. It converts the candidate signed-rank sample to an effective paired-t sample using a working-distribution factor and evaluates approximate power with the noncentral t distribution. This is not exact signed-rank power.

The signed-rank approximation assumes continuous, symmetric paired differences under a location shift. Skewness, multimodality, heaping, zeros, or tied absolute differences can make it unreliable; use simulation or specialist software when these features are material.

These tools support research planning and do not replace statistical consultation.

Method references

  1. Hollander M, Wolfe DA, Chicken E. Nonparametric Statistical Methods. 3rd ed. Wiley; 2014.
  2. Conover WJ. Practical Nonparametric Statistics. 3rd ed. Wiley; 1999.
  3. Al-Sunduqchi MS. Determining the Appropriate Sample Size for Inferences Based on the Wilcoxon Statistics [doctoral dissertation]. University of Wyoming; 1990.
  4. NCSS. PASS: Paired Wilcoxon Signed-Rank Tests. PASS Sample Size Software documentation. Procedure documentation.
  5. Lenth RV. Algorithm AS 243: Cumulative distribution function of the non-central t distribution. Applied Statistics. 1989;38(1):185–189. doi:10.2307/2347693.

Suggested citation

Yaacob, Najib Majdi. “Sample Size for Comparing Paired Observations (SIGN-P / WSR-P).” Web calculator. NajibMY, 2026. Accessed July 29, 2026. https://najibmy.netlify.app/tools/sample-size/comparing-paired-observations/.