One sample Wilcoxon signed-rank test overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

One sample Wilcoxon signed-rank test	Goodness of fit test	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent variable	Independent variable
None	None
Dependent variable	Dependent variable
One of ordinal level	One categorical with $J$ independent groups ($J \geqslant 2$)
Null hypothesis	Null hypothesis
H₀: $m = m_0$ Here $m$ is the population median, and $m_0$ is the population median according to the null hypothesis.	H₀: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$ or equivalently H₀: the probability of drawing an observation from condition 1 is $\pi_1$, the probability of drawing an observation from condition 2 is $\pi_2$, $\ldots$, the probability of drawing an observation from condition $J$ is $\pi_J$
Alternative hypothesis	Alternative hypothesis
H₁ two sided: $m \neq m_0$ H₁ right sided: $m > m_0$ H₁ left sided: $m < m_0$	H₁: the population proportions are not all as specified under the null hypothesis or equivalently H₁: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
Assumptions	Assumptions
The population distribution of the scores is symmetric Sample is a simple random sample from the population. That is, observations are independent of one another	Sample size is large enough for $X^2$ to be approximately chi-squared distributed. Rule of thumb: all $J$ expected cell counts are 5 or more Sample is a simple random sample from the population. That is, observations are independent of one another
Test statistic	Test statistic
Two different types of test statistics can be used, but both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic. In order to compute each of the test statistics, follow the steps below: For each subject, compute the sign of the difference score $\mbox{sign}_d = \mbox{sgn}(\mbox{score} - m_0)$. The sign is 1 if the difference is larger than zero, -1 if the diffence is smaller than zero, and 0 if the difference is equal to zero. For each subject, compute the absolute value of the difference score $\|\mbox{score} - m_0\|$. Exclude subjects with a difference score of zero. This leaves us with a remaining number of difference scores equal to $N_r$. Assign ranks $R_d$ to the $N_r$ remaining absolute difference scores. The smallest absolute difference score corresponds to a rank score of 1, and the largest absolute difference score corresponds to a rank score of $N_r$. If there are ties, assign them the average of the ranks they occupy. Then compute the test statistic: $W_1 = \sum\, R_d^{+}$ or $W_1 = \sum\, R_d^{-}$ That is, sum all ranks corresponding to a positive difference or sum all ranks corresponding to a negative difference. Theoratically, both definitions will result in the same test outcome. However: Tables with critical values for $W_1$ are usually based on the smaller of $\sum\, R_d^{+}$ and $\sum\, R_d^{-}$. So if you are using such a table, pick the smaller one. If you are using the normal approximation to find the $p$ value, it makes things most straightforward if you use $W_1 = \sum\, R_d^{+}$ (if you use $W_1 = \sum\, R_d^{-}$, the right and left sided alternative hypotheses 'flip'). $W_2 = \sum\, \mbox{sign}_d \times R_d$ That is, for each remaining difference score, multiply the rank of the absolute difference score by the sign of the difference score, and then sum all of the products.	$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$ Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells.
Sampling distribution of $W_1$ and of $W_2$ if H₀ were true	Sampling distribution of $X^2$ if H₀ were true
Sampling distribution of $W_1$: If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated.	Approximately the chi-squared distribution with $J - 1$ degrees of freedom
Significant?	Significant?
For large samples, the table for standard normal probabilities can be used: Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$	Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
Example context	Example context
Is the median mental health score of office workers different from $m_0 = 50$?	Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$?
SPSS	SPSS
Specify the measurement level of your variable on the Variable View tab, in the column named Measure. Then go to: Analyze > Nonparametric Tests > One Sample... On the Objective tab, choose Customize Analysis On the Fields tab, specify the variable for which you want to compute the Wilcoxon signed-rank test On the Settings tab, choose Customize tests and check the box for 'Compare median to hypothesized (Wilcoxon signed-rank test)'. Fill in your $m_0$ in the box next to Hypothesized median Click Run Double click on the output table to see the full results	Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square... Put your categorical variable in the box below Test Variable List Fill in the population proportions / probabilities according to $H_0$ in the box below Expected Values. If $H_0$ states that they are all equal, just pick 'All categories equal' (default)
Jamovi	Jamovi
T-Tests > One Sample T-Test Put your variable in the box below Dependent Variables Under Tests, select Wilcoxon rank Under Hypothesis, fill in the value for $m_0$ in the box next to Test Value, and select your alternative hypothesis	Frequencies > N Outcomes - $\chi^2$ Goodness of fit Put your categorical variable in the box below Variable Click on Expected Proportions and fill in the population proportions / probabilities according to $H_0$ in the boxes below Ratio. If $H_0$ states that they are all equal, you can leave the ratios equal to the default values (1)
Practice questions	Practice questions

One sample Wilcoxon signed-rank test - overview