Non Parametric Test

A non parametric test is a type of statistical hypothesis test that does not depend on assumptions about the population’s distribution. Since these tests are distribution-free, they are useful when data do not meet the requirements of parametric tests or when the sample size is small.
 

For example, a teacher wants to compare the performance of students taught using Method A and Method B. The class has very few students, and their scores are not normally distributed. Instead of using a t-test, the teacher can use a Mann–Whitney U test, a non-parametric test, to compare the two groups safely.

A non parametric test should be used when the assumptions required for parametric tests are not satisfied. Specifically, you should consider using a non-parametric test in these situations:

Skewed Data: If the data are not normally distributed, the mean may not accurately reflect the data. Parametric tests may give misleading results, so non-parametric tests are preferred.
 

Small Sample Size: When the dataset is very small, it is difficult to verify the distribution. Non parametric tests are suitable for analyzing such data.

Categorical or Ranked Data: If the data is nominal (names, labels) or ordinal (ranked), parametric tests cannot be applied. Non parametric tests can handle these types of data effectively.

Parametric and Non parametric are the two types of hypothesis testing. Here, we will discuss the key differences between the two methods:

The statistical tests that do not rely on a specific distribution for the data are non parametric tests. They are handy for ordinal data, small sample sizes, or data that violate parametric assumptions. The types of non parametric tests are: 

• Mann–Whitney U Test
  
  The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is used to compare two independent groups.
  
  Null Hypothesis (H₀): The two populations are equal.
  
  Test statistics: Calculate \(U = \min(U_1, U_2) \) where
  
  \(U_1 = n_1 n_2 + \frac{n_1 (n_1 + 1)}{2} - R_1 \)
  
  
  \(U_2 = n_1 n_2 + \frac{n_2 (n_2 + 1)}{2} - R_2 \)
  
  Decision Rule: Reject the null hypothesis H0 if U < critical value. 

• Wilcoxon Signed-Rank Test
  
  The Wilcoxon Signed-Rank Test is used to compare two dependent ordinal samples, similar to the paired t-test. Assumes a symmetric distribution. 
  
  Null Hypothesis (H₀): The median difference is 0. 
  
  Test statistics (W): Test statistics is the smaller of the sum of positive and negative ranks
  
  Decision Rule: Reject \(H_0 \) if W < critical value. 

• Sign Test 
  
  The Sign Test is a simple method to compare paired samples. It counts the number of positive and negative differences and is used when the data do not meet the assumptions of other paired tests.
  
  Null Hypothesis (H₀): The median difference is 0.
  
  Test statistics: Smaller of the number of positive and negative differences.
  
  Decision Rule: Reject \(H_0\) if the test statistic< critical value.  

• Kruskal–Wallis Test 
  
  The Kruskal-Wallis test is a statistical method for comparing the medians of three or more independent groups. 
  
  Null Hypothesis (H₀): In this test, all group medians are equal. 
  
  Test statistics: \(H = \frac{12}{N(N+1)} \sum \frac{R_j^2}{n_j} - 3(N+1) \)  
  
  Decision Rule: Reject H₀ if H > critical value.

Non parametric tests are statistical methods that do not assume a specific data distribution. They are useful for ordinal data or when normality assumptions are violated. The Kruskal-Wallis H-test is a popular non-parametric test used to compare the medians of three or more independent groups. Its formula is:

\(H = \frac{12}{N(N+1)} \sum \frac{R_j^2}{n_j} - 3(N+1) \)

Where H is the Kruskal-Wallis test 

N is the total number of observations across all groups

\(n_i\) is the number of observations in group i

\(R_i\) is the sum of ranks for observations in group i

Non parametric tests are statistical methods that do not assume any specific distribution of data. They are especially useful for ordinal or ranked data and are widely applied in social sciences, education, and psychology. Here are the advantages and disadvantages of non parametric tests. 

Mastering non parametric tests helps you analyze data that doesn't follow normal distribution. These tips make it easier to choose the right test and interpret results accurately.

• Focus on understanding when to use non-parametric tests instead of parametric ones.
   
• Learn common tests like Mann-Whitney, Wilcoxon, and Kruskal-Wallis for different scenarios.
   
• Practice interpreting data in ranks or categories rather than raw numerical values.
   
• Use visual tools like box plots to understand data distribution and differences.
   
• Solve varied real-world problems to strengthen your grasp of non-parametric methods.
   

• Parents can help children understand non-parametric ideas by encouraging simple ranking tasks at home, such as ordering objects by size, height, or preference.
   

• To introduce non parametric concepts, teachers can use real-life examples like survey results or classroom feedback. 
   

• Teachers can incorporate visual aids such as charts, box plots, diagrams, and flashcards to help students recognize data patterns.

Non parametric tests are useful when data doesn't fit normal distribution assumptions. They help analyze ranked, categorical, or skewed data across various fields effectively.

Healthcare research: Used to compare patient recovery rates when the data doesn't follow a normal distribution.
 

Marketing analysis: Helps businesses analyze customer satisfaction surveys where responses are in ranks or categories.
 

Education: Applied to compare student performance based on grades or rankings rather than exact scores.
 

Finance: Used to study stock market trends and investment returns when data is not normally distributed.
 

Psychology: Helps analyze behavioral or emotional response data collected through ordinal scales or observations.