One-sample tests are not used too often, but are useful to compare a set of values to a given default value. For example, one might ask if a set of five-point Likert scores are significantly different from a “default” or “neutral” score of 3. Another use might be to compare a current set of values to a previously published value.
• One-sample data
• Data are ordinal, interval, or ratio
• To be a test of the median, the data need to be relatively symmetrical about their median.
If the distributions of the data are symmetric:
• Null hypothesis: The median of the data is not different than the default value.
• Alternative hypothesis (two-sided): The median of the data is different than the default value.
If the distributions of the data are not symmetric:
• Null hypothesis: The data are not stochastically different from the default value.
• Alternative hypothesis (two-sided): The data are stochastically different from the default value.
Reporting significant results as e.g. “Likert scores were significantly different from a neutral value of 3” is acceptable.
Notes on name of test
The names used for the one-sample Wilcoxon signed-rank test and similar tests can be confusing. “Sign test” may be used, although properly the sign test is a different test. Both “signed-rank test” and “sign test” are sometimes used to refer to either one-sample or two-sample tests.
The best advice is to use a name specific to the test being used.
Other notes and alternative tests
Some authors recommend this test only in cases where the data are symmetric. It is my understanding that this requirement is only for the test to be considered a test of the median.
If data are not symmetrical, the sign test can be used as an alternative that is specific for the median.
The packages used in this chapter include:
The following commands will install these packages if they are not already installed:
This example will re-visit the Maggie Simpson data from the Descriptive Statistics for Likert Data chapter.
The example answers the question, “Are Maggie’s scores significantly different from a ‘neutral’ score of 3?”
The test will be conducted with the wilcox.test function, which produces a p-value for the hypothesis, as well a pseudo-median and confidence interval.
Note that the bar plot shows that the data are relatively symmetrical in distribution.
Speaker Rater Likert
'Maggie Simpson' 1 3
'Maggie Simpson' 2 4
'Maggie Simpson' 3 5
'Maggie Simpson' 4 4
'Maggie Simpson' 5 4
'Maggie Simpson' 6 4
'Maggie Simpson' 7 4
'Maggie Simpson' 8 3
'Maggie Simpson' 9 2
'Maggie Simpson' 10 5
Data = read.table(textConnection(Input),header=TRUE)
### Create a new variable which is the likert scores as an ordered factor
Data$Likert.f = factor(Data$Likert,
ordered = TRUE)
### Check the data frame
### Remove unnecessary objects
Note that the variable we want to count is Likert.f, which is a factor variable. Counts for Likert.f are cross tabulated over values of Speaker. The prop.table function translates a table into proportions. The margin=1 option indicates that the proportions are calculated for each row.
xtabs( ~ Speaker + Likert.f,
data = Data)
Speaker 2 3 4 5
Maggie Simpson 1 2 5 2
XT = xtabs( ~ Speaker + Likert.f,
data = Data)
margin = 1)
Speaker 2 3 4 5
Maggie Simpson 0.1 0.2 0.5 0.2
XT = xtabs(~ Likert.f,
Summarize(Likert ~ Speaker,
Speaker n mean sd min Q1 median Q3 max percZero
1 Maggie Simpson 10 3.8 0.919 2 3.25 4 4 5 0
In the wilcox.test function, the mu option indicates the value of the default value to compare to. In this example Data$Likert is the one-sample set of values on which to conduct the test. For the meaning of other options, see ?wilcox.test.
Wilcoxon signed rank test with continuity correction
V = 32.5, p-value = 0.04007
alternative hypothesis: true location is not equal to 3
### Note p-value in the output above
### You will get the "cannot compute exact p-value with ties" error
### You can ignore this, or use the exact=FALSE option.
95 percent confidence interval:
### Note that the output will also produce a pseudo-median value
### and a confidence interval if the conf.int=TRUE option is used.
I am not aware of any established effect size statistic for the one-sample Wilcoxon signed-rank test. However, using a statistic analogous to the r used in the Mann–Whitney test may make sense.
The following interpretation is based on my personal intuition. It is not intended to be universal.
0.10 – < 0.40
0.40 – < 0.60
1. Considering Maggie Simpson’s data,
a. What was her median score?
b. What were the first and third quartiles for her scores?
c. Are the data reasonably symmetric about their median?
d. Based on this, what is null hypothesis of the test?
e. According to the one-sample Wilcoxon signed-rank test, are her
scores significantly different from a neutral score of 3?
f. Is the confidence interval output from the test useful in answering the previous question?
g. Overall, how would you summarize her results? Be sure to address the practical implication of her scores compared with a neutral score of 3.
h. Do these results reflect what you would expect from looking
at the bar plot?
2. Brian Griffin wants to assess the education level of students in his course on creative writing for adults. He wants to know the median education level of his class, and if the education level of his class is different from the typical Bachelor’s level.
Brian used the following table to code his data.
Code Abbreviation Level
1 < HS Less than high school
2 HS High school
3 BA Bachelor’s
4 MA Master’s
5 PhD Doctorate
The following are his course data.
Instructor Student Education
'Brian Griffin' a 3
'Brian Griffin' b 2
'Brian Griffin' c 3
'Brian Griffin' d 3
'Brian Griffin' e 3
'Brian Griffin' f 3
'Brian Griffin' g 4
'Brian Griffin' h 5
'Brian Griffin' i 3
'Brian Griffin' j 4
'Brian Griffin' k 3
'Brian Griffin' l 2
For each of the following, answer the question, and show
the output from the analyses you used to answer the question.
a. What was the median education level? (Be sure to report the education level, not just the numeric code!)
b. Are the data reasonably symmetric about their median?
c. Based on this, what is the null hypothesis of the test?
d. According to the one-sample Wilcoxon signed-rank test, are the
education levels significantly different from a typical level of Bachelor’s?
e. Is the confidence interval output from the test useful in answering the previous question?
f. Overall, how would you summarize the results? Be sure to address the practical implications.
g. Plot Brian’s data in a way that helps you visualize the data.
h. Do the results reflect what you would expect from looking at the plot?