The Wilcoxon Signed Rank test assumes that the population of interest is both continuous and symmetric (not necessarily normal). Since the mean and median are the same (for symmetrical distribution), the hypothesis tests on the median are the same as the hypothesis test on the mean.
The Wilcoxon test is performed by ranking the non-zero deviations in order of increasing magnitude (that is, the smallest non-zero deviation has a rank of 1 and the largest deviation has a rank of $n$). The ranks of the deviations with positive and negative values are summed.
These sums are used to determine whether or not the deviations are significantly different from zero. Wilcoxon Signed Rank Test is an alternative to the Paired Sample t-test.
Test Statistics: $T^-$: an absolute value of the sum of the negative ranks
Two-tailed Test
$H_0: \mu = \mu_0 \quad$ vs $\quad H_1:\mu \ne \mu_0$
Test Statistics: $min(T^+, T^-)$
Because the underlying population is assumed to be continuous, ties are theoretically impossible, however, in practice ties can exist, especially if the data has only a couple of significant digits.
Two or more deviations having the same magnitude are all given the same average rank. The deviations of zero are theoretically impossible but practically possible. Any deviations of exactly zero are simply thrown out and the value of $n$ is reduced accordingly.
Single Sample Wilcoxon Signed Rank Test
The Wilcoxon Signed Rank Test is important for researchers as it fills a critical gap in statistical analysis.
Non-normal data: Most of the statistical tests, like the dependent samples t-test, assume that the data follows a normal distribution (bell curve). The Wilcoxon Signed Rank Test supersede the assumption of normality, making it ideal for analyzing data that is skewed, ranked, or ordinal (like survey responses on a Likert scale Questions).
Robust against outliers: Outliers (very large or small observations in the data) can significantly skew the results of some statistical tests. The Wilcoxon Signed Rank Test focuses on the ranks of the differences, making it less sensitive to extreme values (outliers) in the data compared to tests that rely on raw numbers.
Focuses on changes within subjects: The Wilcoxon Signed Rank Test is designed for paired data (dependent samples), to look at the same subjects before and after situation (like a treatment) or under two different conditions.
Nonparametric tests are experiments that do not require the underlying population for assumptions. It does not rely on data referring to any particular parametric group of probability distributions. Non parametric Statistical Tools are also called distribution-free tests since they do not have any underlying population.
Table of Contents
Nonparametric tests, also known as distribution-free tests, are statistical methods that do not assume a specific population distribution. Unlike parametric tests, they are flexible and work with ordinal, nominal, or non-normally distributed data. This blog explores when to use nonparametric tests, their advantages, limitations, and the most widely used nonparametric statistical tools in research and data analysis
Nonparametric Tests/ Statistics
The nonparametric tests are helpful when:
Inferences must be made on categorical or ordinal data
The assumption of normality is not appropriate
The sample size is small
Advantages of Non Parametric Statistical Tools
Easy application (does not even need a calculator in many cases)
It can serve as a quick check to determine whether or not further analysis is required
Many assumptions concerning the population of the data source can be relaxed
Can be used to test categorical (yes/ no) data
Can be used to test ordinal (1, 2, 3) data
Disadvantages of Non Parametric Methods
Nonparametric procedures are less efficient than parametric procedures. It means that nonparametric tests require a larger sample size to have the same probability of a Type I error as the equivalent parametric procedure.
Nonparametric procedures often discard helpful information. That is, the magnitudes of the actual data values are lost. As a result, nonparametric procedures are typically less powerful.
That is, they produce conclusions that have a higher probability of being incorrect. Examples of widely used Parametric Tests include the paired and unpaired t-test, Pearson’s product-moment correlation, Analysis of Variance (ANOVA), and multiple regression.
Note: Do not use nonparametric procedures if parametric procedures can be used.
Widely used Non-Parametric Statistical Tools/Tests
Sign Test
Runs Test
Wilcoxon Signed Rank Test
Wilcoxon Rank Sum Test
Spearman’s Rank Correlation
Kruskal Wallis Test
Chi-Square Goodness of Fit Test
Nonparametric tests are crucial tools in statistics because they offer valuable analysis even when the data doesn’t meet the strict assumptions of parametric tests. Non parametric statistical tools/ tests provide a valuable alternative for researchers who encounter data that doesn’t fit the mold of parametric tests. They ensure that valuable insights can still be extracted from the data without compromising the reliability of the analysis.
However, it is essential to note that nonparametric tests can sometimes be less powerful than their corresponding parametric tests. This means non-parametric tests might be less likely to detect a true effect, especially with smaller datasets.
In summary, nonparametric tests are valuable because these kinds of tests offer flexibility in terms of data assumptions and data types. They are particularly useful for small samples, skewed data, and situations where data normality is uncertain. These tests also ensure researchers draw statistically sound conclusions from a wider range of data types and situations. But, it is always a good practice to consider both parametric and non-parametric approaches when appropriate.
Real-World Examples of non parametric Statistical Tools
The non parametric tests are crucial in real-world data where normality, sample size, or measurement scales are limiting factors. The following are some real-world examples of nonparametric statistical tools and how they are applied in different fields:
The non parametric tests are widely used in medicine, social sciences, market research, and quality control, where data is often ordinal, skewed, or categorical.
Mann-Whitney U Test (Wilcoxon Rank-Sum Test): Used to compare two independent groups when data is not normally distributed. For example, a pharmaceutical company tests a new painkiller against a placebo. Patient pain levels (measured on an ordinal scale: mild, moderate, severe) are compared between the two groups. Since the data is not normally distributed, the Mann-Whitney U test is used instead of an independent t-test.
Wilcoxon Signed-Rank Test: Used for comparing paired or matched samples (e.g., before-and-after studies). For example, a fitness trainer measures the weight loss of 15 individuals before and after a 3-month diet program. Since weight loss data may be skewed, the Wilcoxon Signed-Rank Test is used instead of a paired t-test.
Kruskal-Wallis Test: Used for comparing three or more independent groups when ANOVA assumptions are violated. For example, a researcher compares the effectiveness of three different teaching methods (A, B, C) on student exam scores. In case if the scores are not normally distributed, the Kruskal-Wallis test is used instead of one-way ANOVA.
Spearman’s Rank Correlation: Used to measure the strength and direction of a monotonic (but not necessarily linear) relationship. For example, a marketing analyst examines whether social media engagement (likes, shares) correlates with sales rank (ordinal data). Since the relationship may not be linear, Spearman’s correlation should be used instead of Pearson’s.
Chi-Square Test (Goodness-of-Fit & Independence Test): used for testing relationships between categorical variables. For example,
Goodness-of-Fit: A candy company checks if its product colors follow the expected distribution (20% red, 30% blue, etc.) in a sample.
Independence Test: A survey tests if gender (male/female) is independent of voting preference (Candidate X/ Y/ Z).
Friedman Test: Used for comparing multiple related groups (repeated measures). For example, a hospital tests three different blood pressure medications on the same patients over time. Since the data is repeated and non-normal, the Friedman test is used instead of repeated-measures ANOVA.
Sign Test: Used for simple before-after comparison with only direction (increase/decrease) known. For example, a restaurant surveys customers before and after a menu redesign, asking if they are “more satisfied” or “less satisfied.” The Sign Test checks if the change had a significant effect.
McNemar’s Test: Used for analyzing paired nominal data (e.g., yes/ no responses before and after an intervention). For example, a study evaluates whether a training program changes employees’ ability to pass a certification test (pass/fail) before and after training.
Key Decision Factors for Parametric or Non parametric Statistical Tools
The following are key decision factors that may be used for the selection of either parametric or non parametric statistical tools:
Data Type
Parametric: Continuous, normally distributed.
Nonparametric: Ordinal, skewed, small samples, or categorical.
Sample Size
Parametric: Typically requires ≥30 samples (Central Limit Theorem).
Nonparametric: Works with small samples (e.g., n < 20).
Outliers & Skewness
Parametric: Sensitive to outliers; assumes homogeneity of variance.
The post is about Online MCQs on Chi-Square Test Quiz with Answers. The Quiz MCQs on Chi-Square Test cover the topics of attributes, Chi-Square Distribution, Coefficient of Association, Contingency Table, and Hypothesis Testing on Association between attributes, etc. Let us start with MCQs on Chi-Square Test Quiz.
The quiz about Chi-Square Association between attributes.
The relationship/ dependency between the attributes is called association and the measure of degrees of relationship between the attributes is called the coefficient of association. The Chi-Square Statistic is used to test the association between the attributes. The Chi-Square Association is defined as
A population can be divided into two or more mutually exclusive and exhaustive classes according to their characteristics. It is called dichotomy or twofold division if, it is divided into two mutually exclusive classes. A contingency table is a two-way table in which the data is classified according to two attributes, each having two or more levels. A measure of the degree of association between attributes expressed in a contingency table is known as the coefficient of contingency. Pearson’s mean square coefficient of contingency is
\[C=\sqrt{\frac{\chi^2}{n+\chi^2}}\]
MCQs on Chi-Square Test Quiz with Answers
A characteristic which varies in quality from one individual to another is called
The eye colour of 100 men is
Association measures the strength of the relationship between
The presence of an attribute is denoted by
The process of dividing the objects into two mutually exclusive classes is called
There are ———– parameters of Chi-Square distribution.
The parameter of the Chi-Square distribution is ———–.
The value of $\chi^2$ cannot be ———.
The range of $\chi^2$ is
Two attributes $A$ and $B$ are said to be independent if
Two attributes $A$ and $B$ are said to be positively associated if
If $(AB) > \frac{(A)(B)}{n}$ then association is
If $(AB) < \frac{(A)(B)}{n}$ then association between two attributes $A$ and $B$ is
The coefficient of association $Q$ lies between
If $\chi^2_c=5.8$ and $df=1$, we make the following decision ———-.
A contingency table with $r$ rows and $c$ columns is called
A $4 \times 5$ contingency table consists of ———.
If for a contingency table, $df=12$ and the number of rows is 4 then the number of columns will be
For $r\times c$ contingency table, the Chi-Square test has $df=$ ———-.
For the $3\times 3$ contingency table, the degrees of freedom is
Attributes are said to be independent if there is no association between them. Independence means the presence or absence of one attribute does not affect the other. The association is positive if the observed frequency of attributes is greater than the expected frequency and negative association or disassociation (negative association) is if the observed frequency is less than the expected frequency.
