# One sample Wilcoxon signed-rank test - overview

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One sample Wilcoxon signed-rank test | Cochran's Q test | Mann-Whitney-Wilcoxon test |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent/grouping variable | |

None | One within subject factor ($\geq 2$ related groups) | One categorical with 2 independent groups | |

Dependent variable | Dependent variable | Dependent variable | |

One of ordinal level | One categorical with 2 independent groups | One of ordinal level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: $m = m_0$
Here $m$ is the population median, and $m_0$ is the population median according to the null hypothesis. | H_{0}: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
- H
_{0}: the population median for group 1 is equal to the population median for group 2
Formulation 1: - H
_{0}: the population scores in group 1 are not systematically higher or lower than the population scores in group 2
- H
_{0}: P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1} two sided: $m \neq m_0$H _{1} right sided: $m > m_0$H _{1} left sided: $m < m_0$
| H_{1}: not all population proportions are equal | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
- H
_{1}two sided: the population median for group 1 is not equal to the population median for group 2 - H
_{1}right sided: the population median for group 1 is larger than the population median for group 2 - H
_{1}left sided: the population median for group 1 is smaller than the population median for group 2
Formulation 1: - H
_{1}two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2 - H
_{1}right sided: the population scores in group 1 are systematically higher than the population scores in group 2 - H
_{1}left sided: the population scores in group 1 are systematically lower than the population scores in group 2
- H
_{1}two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1) - H
_{1}right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1) - H
_{1}left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)
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Assumptions | 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 of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
| - Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
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Test statistic | 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.
- $W_1 = \sum\, R_d^{+}$
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.
| If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
- $W$ = sum of ranks in group 1
- $U = W - \dfrac{n_1(n_1 + 1)}{2}$
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | |

Sampling distribution of $W_1$ and of $W_2$ if H_{0} were true | Sampling distribution of $Q$ if H_{0} were true | Sampling distribution of $W$ and of $U$ if H_{0} 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. | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | |

Significant? | 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 $\alpha$
- 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 $\alpha$
- 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 $\alpha$
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
- 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 $\alpha$
| 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 $\alpha$
- 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 $\alpha$
- 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 $\alpha$
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n.a. | Equivalent to | Equivalent to | |

- | Friedman test, with a categorical dependent variable consisting of two independent groups. | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | |

Example context | Example context | Example context | |

Is the median mental health score of office workers different from $m_0 = 50$? | Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | Do men tend to score higher on social economic status than women? | |

SPSS | 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 > K Related Samples...
- Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables
- Under Test Type, select Cochran's Q test
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
- Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
- Continue and click OK
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Jamovi | 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
| Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman - Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
| T-Tests > Independent Samples T-Test
- Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
- Under Tests, select Mann-Whitney U
- Under Hypothesis, select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |