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Wilcoxon Signed-Rank Test Research Paper


Wilcoxon signed-rank test is applicable while weighing against corresponding samples to evaluate whether there is a variation in their ranked population averages. In other words, Wilcoxon signed-rank tests are applied in samples that do not meet the requirements of the parametric tests. In most cases, the Wilcoxon signed-rank is used in the circumstances that the population is not normally distributed (Martinez, 2007).

The Wilcoxon signed-rank can be used in the place of student paired t-test particularly when the sample means are small. The assumptions made while using the Wilcoxon signed-rank test enables it to be more accurate than the dependent variables t-test particularly when the sample means are small. However, when the paired sample means are large, the t-test for non-dependent variable would be appropriate.

The definition of Wilcoxon test

The Wilcoxon signed-rank test refers to a non-parametric statistical theory that is very significant in carrying out tests of two related models as well as repeated dimensions on individual samples to establish whether there are variations in their populace mean ranks (Gravetter & Wallnau, 2009). Further, the test is also useful in assessing the differences existing between the population mean ranks of matched samples.

Moreover, the analysis plays an alternative means of assessment to the paired Student’s test for corresponding pairs as well as the t-test for independent samples in the event that the populace is not normally distributed. In carrying out the Wilcoxon test, the statistics from the corresponding population are paired off. The test also applies random sampling of the independent pairs (Gravetter & Wallnau, 2009).

Moreover, an ordinal scale is vital in measuring the statistics following a normal distribution. In essence, the hypothesis testing of non-parametric data is essential in assessing records that can be placed in a given order but lack the statistical figures. In fact, the test is invaluable in analyzing clientele fulfillment (Gravetter & Wallnau, 2009).

The invention of the Wilcoxon test

Frank Wilcoxon, an American statistician developed the test in nineteen forty-five. The Wilcoxon test was put forward together with rank-sum test in order to examine two independent variables. In fact, frank Wilcoxon proposed that in the circumstances where two sample populations to be tested is small then the W-test is suitable since it is more accurate than the paired t-test (Gravetter & Wallnau, 2009).

Later statisticians approved the test for non-parametric variables. In other words, the non-parametric variables are dependent sample population that does not meet the parametric tests. Consequently, the test was later given the name Wilcoxon T-test or simply as T. However, the name was lat er changed to W or t-test for non-independent population samples (Gravetter & Wallnau, 2009).

When the Wilcoxon test is applicable

The application of the Wilcoxon test is motivated by the improbability relating to the assumptions of normality in the t-test. For instance, Wilcoxon test is applicable in the situations that involve matched pairs as well as repeated dimension of similar items. Further, the Wilcoxon test is applied in the case where there are improper mean variations in the t-test (Gravetter & Wallnau, 2009).

In addition, the test remains indispensable in the circumstances when the distributions do not obey the normal essential suppositions. For example, in the normal parametric statistical procedures, the assumption is that the data will follow the uniform distributions. However, in the actual circumstances, the data do not follow the uniform distribution. The test is also used when the variables are at least ordinal.

In essence, the test is applied when similar participants carry out both situations for the study. In other words, the test is suitable for scrutiny of information that originates from recurring dimensional designs involving two situations. Further, the test is used when the statistics do not convene to the necessities for a parametric test (Gravetter & Wallnau, 2009).

In this regard, the test is suitable when the data are not distributed normally as well as when there are variations in marking the values of both the variances. In addition, for the Wilcoxon test to be carried out, the statistics must be measured on an ordinal scale.

The sense behind conducting the test involves ranking the available figures to produce two order sums for individual situations. The methodical variation between the circumstances leads to placing the conditions with higher orders in one situation and the ones with lower ranks on the other situation (Jackson, 2009). Further, if the ranks are equivalent, then the orders are distributed in a similar array. The test is also significant in carrying out a single sample.

For example, in conducting assessment on shopping centers, the use of Wilcoxon test is highly appropriate. In other words, in the study of several shops, the researcher would suppose that the respondents are probable to back up or object to questions put to them with an equal likelihood of half. In addition, when there is availability of various unrelated samples that require comparisons, the application of the Wilcoxon rank test is vital.

When not to use the Wilcoxon test

As indicated above the Wilcoxon test is used in the situations where the sample population is small. In the situations where the sampled population sizes are large, the Wilcoxon text cannot be applied (Cleves, 2008).

In other words, while studying two population samples that are not identical or corresponding and the samples are large then Wilcoxon test is not applicable. Instead, unpaired t-test is used. The reason why the unpaired t-test is not preferable is that type 1 error is likely to be yielded particularly in the circumstances when the population sample size is small (Cleves, 2008).

In other words, the unpaired t-test is preferable in the situations when the sample population size is large. However, when the distribution is skewed and the sample population is large, then Wilcoxon can still be applied (Martinez, 2007). In essence, Wilcoxon test yield better results when the paired population sample being studied is small. When paired sample sizes are large, then the Wilcoxon cannot be applied and instead the single t-test is used to test the variables.

According to Jackson (2009), Wilcoxon test can be used in the place of the paired student t-test. Wilcoxon test is appropriate in the evaluation of data that are derived from the repeated measures. In addition, in the circumstances that the derived data does not meet the parametric test requirements or when the data are not distributed normally, then the Wilcoxon test is applied.

The test assumptions

For the Wilcoxon test to be appropriate, several assumptions are applied (Kirk, 2006). The first assumption is that the data must be matched and have to be drawn from the corresponding populace. Secondly, independent and random pairing must be attained. The final assumption is that the data has to be ordinal. However, the normal distribution of data is necessary but not a must requirement (Kirk, 2006).

The Wilcoxon test formula

As indicated, the Wilcoxon test assumes a formula that test the median of sample sizes of the paired numbers. The sample size should be small. The hypothesis is whether there are variations in the medians of the sample pairs (Martinez, 2007). In other words, the unacceptable premise tests whether there is no existing deviations in the paired sample middle measure while the acceptable premise tests otherwise.

To derive the formula N is understood to be the size of the sample or the amount of paired samples. Therefore, the total data sample equals to 2N. If i=1, ….., N and x1i and x2i is to represent the measurements, then Wilcoxon test (W) ={∑[sgn(x2i-x1i)*Ri]}. In the formula, sgn represent the sign function while Ri represent the ranks (Martinez, 2007). As indicated, the hypothesis to be tested using the formula is as follows

  • H0: there are no median differences between the pairs
  • H1: median differences exist


In most cases, the Wilcoxon tests will be appropriate when the section the population to be tested is small and the distribution is not normal. The Wilcoxon tests contrasts the ordinary tests where the population is large and the distribution is average. However, most of the statistical tests do not behave in the normal distribution. In fact, in Wilcoxon tests, the data is not normally distributed and the sample sizes are small.

In addition, Wilcoxon tests are used in evaluating opinionated data as well as data that cannot be accurately measured. In other words, the measure is based on the nominal scale. Though data from opinions are normally difficult to measure, they are put in numerical scales that make it easier to be evaluated. In these kinds of data, there is no assumption that there is normal distribution. Therefore, Wilcoxon test would be appropriate.


Cleves, M. A. (2008). An introduction to survival analysis using stata. New York, NY: Stata Press.

Gravetter, F. J. & Wallnau, L. B. (2009). Statistics for the behavioral sciences. Belmont, CA: Cengage Learning.

Jackson, S. L. (2009). Statistics plain and simple. Belmont, CA: Cengage Learning.

Kirk, R. E. (2006). Statistics: An introduction. Belmont, CA: Cengage Learning.

Martinez, R. (2007). Diagnostics for choosing between Log-rank and Wilcoxon tests. Michigan, MI: ProQuest.

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