# Wilcoxon Signed-Rank Test in Excel

The Wilcoxon Signed-Rank Test is often used as a nonparametric substitute for the two-sample paired t test. The two-sample paired t-test is used to determine whether there is a difference in paired data such as before and after data pairs. The Wilcoxon Signed-Rank Test can be substituted for the two-sample paired t-test when the differences between the data pairs are not normally distributed, which is required by the two-sample paired t-test.

The two-sample paired t-test has a Null Hypothesis which states that the mean difference between the paired data points is zero. The Wilcoxon Signed-Rank Test has a Null Hypothesis which states that the median difference between the paired data points is zero.

The Wilcoxon Signed-Rank Test requires that differences be taken between data points and therefore can only be used to analyze interval or ratio data. The Wilcoxon Signed-Rank Test cannot be used to analyze ordinal or nominal data because the differences between these types of data points have no specific value.

The Sign Test would be a viable substitute for the two-sample paired t-test if the data are ordinal. With ordinal data, differences matter but the differences in values are meaningless. An example of ordinal data is a customer service rating scale from 1 to 10. Interval data has meaningful differences between data points but the zero point is arbitrarily chosen. Temperature is an example of interval data. Ratio data has meaningful differences between data points and the zero point indicates that there is none of the variable. Height and weight are ratio data. The Wilcoxon Signed-Rank Test can only be used with interval or ratio data, but not with when ordinal data is being analyzed.

## Comparison of the Wilcoxon Signed-Rank Test to Similar Tests

1) The two-sample paired t-test is used to determine whether the mean difference between data pairs is zero. Its Null Hypothesis states that this mean difference is zero. the two-sample paired t-test can be used only when the differences between data points is meaningful (interval and ratio data have meaningful differences between data points but ordinal and nominal data do not) and the differences between the paired data points is normally distributed. The Null Hypothesis states that this difference is zero and it is therefore the difference between the data pairs is the distributed variable in this t-test. The test statistic in this t-test is a calculated from the average value of differences.

2) The Wilcoxon Signed-Rank Test is used to determine whether the median difference between data pairs is zero when the differences are not normally distributed. Its Null Hypothesis states that this median difference is zero. A major difference between the Sign Test and Wilcoxon Signed-Rank Test is that the Sign Test can be used to analyze ordinal data. With ordinal data, differences matter but the differences in values are meaningless. An example of ordinal data is a customer service rating scale from 1 to 10. The Wilcoxon Signed Rank Test cannot be used to analyze ordinal data because the Wilcoxon Signed Rank Test calculates the differences between data points to establish a median difference. The WIlcoxon Signed-Rank Test can only be used to analyze interval data or ratio data. Interval data has meaningful differences between data points but the zero point is arbitrarily chosen. Temperature is an example of interval data. Ratio data has meaningful differences between data points and the zero point indicates that there is none of the variable. Height and weight are ratio data. The Wilcoxon Signed Rank Test also requires that the differences be symmetrically distributed around the median. This Sign Test does not require this. A quick histogram of the differences will show whether they are symmetrically distributed around the median.

3) The Sign Test is used to determine if there is an equal number of differences between paired data in each direction. The Sign Test is used when the differences are not normally distributed or the data is ordinal. Its Null Hypothesis states that the difference in the number of differences in both directions is zero. The Sign Test can also be used when the differences are not symmetrically distributed about the median, as required by the Wilcoxon Signed-Rank Test. The Sign Test requires fewer assumptions than the two-sample paired t-test and the Wilcoxon Signed-Rank Test but lacks the statistical power of either. Unlike the Wilcoxon Signed-Rank Test, the SignTest can be used if the data is ordinal.

## The Difference Between Performing the Wilcoxon Signed-Rank Test on Large Samples and on Small Samples

The Wilcoxon Signed-Rank Test is uses slightly different procedures for analyzing for small samples than for large samples. Small samples mean sample groups with less than 10 pairs of sample data. For small samples, the calculated test statistic W is compared directly to values on a chart to determine whether or not there is enough evidence to reject the Null Hypothesis, which states that median difference between the paired data points is zero, i.e., there is no difference between before and after data.

When sample size exceeds 10, the distribution of the test statistic variable W can be approximated by the normal distribution. We can then calculate a Z Score and, equivalently, a p value to determine whether or not there is enough evidence to reject the Null Hypothesis, which states that median difference between the paired data points is zero, i.e., there is no difference between the before and after data points.

## Assumptions of the Wilcoxon Signed-Rank Test

1) Data are paired and come from the same population.

2) Each data pair must be randomly chosen and independent of any other pair. The two data points of the data pair can be associated with each other, e.g., when Before and After data is being analyzed, but the data pairs must be independent of each other.

3) The data are measured on an interval or ratio scale because differences between data points must be specific and measurable. Ordinal and nominal data do not have specific measurable distances between data points. These differences do not have to be normally distributed, as required by the two-sample paired t-test that the Wilcoxon Signed-Rank Test often replaces.

4) The distribution of the differences should be relatively symmetrical about the median. We see if this is the case by creating histograms of differences for each sample group.

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