NonParametric Test
Nonparametric tests are experiments which do not require the underlying population for assumptions. It does not rely on any data referring to any particular parametric group of probability distributions. Nonparametric methods are also called distributionfree tests since they do not have any underlying population.
Definition
Nonparametric tests are the mathematical methods used in statistical hypothesis testing, which do not make assumptions about the frequency distribution of variables that are to be evaluated. The nonparametric experiment is used when there are skewed data and it comprises techniques that do not depend on data pertaining to any particular distribution.
The word nonparametric does not mean that these models do not have any parameters. The fact is, the characteristics and number of parameters are pretty flexible and not predefined. Therefore, these models are called distributionfree models.
Applications of NonParametric Test
The conditions when nonparametric tests are used are listed below:
 When parametric tests are not satisfied.
 When testing hypothesis does not have any distribution.
 For quick data analysis.
 When unscaled data is available.
Advantages and Disadvantages of NonParametric test
Advantages 
Disadvantages 


Sign Test Statistics
The sign test is conducted under the following conditions.
 When we need to compare paired data
 The paired data obtained from similar conditions
 No assumptions made about the original population
Sign Test is merely based on the signs (+ or ) of the deviations xy and not on their magnitudes. This test is applicable when zero differences or tie between the paired observations cannot occur. If zero differences or tie occur, then they will be eliminated from the analysis and the number of paired observations counted is reduced. This method can be used to examine individual data also.
Sign Test Assumption
Let (x_{1}y_{1}),(x_{2},y_{2}),….,(x_{n},y_{n}) be paired observations and d_{i }= x_{i}y_{i} are the differences between the observations, where i=1,2,….,n.
 The value of d_{i} can be positive or negative and all those values are independent.
 Each d_{i} comes from the same continuous population.
 The values x_{i} and y_{i} represent the order, so that the comparisons “greater than”, “less than”, and “equal to” are meaningful.
Now we will take here the Null hypothesis, H_{0} : p=½ = 0.5 and Alternate hypothesis, H_{1}:q ≠ ½=0.5.
Let us consider the number of positive signs in d_{i} be m. Therefore,
p = m/n and q = n – (m/n)
Case 1(n < 30)
If either np or nq is less than 5, then we can use binomial approximation.
where m is the number of positive deviations.
If P > α then we accept the null hypothesis else we reject it.
Case 2 (n < 30):
If both np or nq > 5, then we use a normal distribution for approximation.
The limits of the approval region are given by (pz_{p}σ_{p},p+z_{p}σ_{p}), where z_{p} is the value obtained from the standard normal table with α level of significance. If the value of α is not given, we consider it as, α = 0.05.
If ‘p’ lies within (pz_{p}σ_{p},p+z_{p}σ_{p}) we accept the hypothesis, else we reject it.
Case 3 (n ≥ 30)
If n ≥ 30, then we can find Mean = np and standard deviation = npq
And, a = number of negative observations
KruskalWallis Htest
Kruskal Wallis H test is used to test whether two or more populations are identical. In this test, the null hypothesis is H_{0}:μ_{1 }= μ_{2} = γ_{3} (when there are three populations) and the alternative hypothesis is H1:μ_{1}≠μ_{2}≠γ_{3}. In the KruskalWallis test, we first evaluate the ranks of the observation lists in the samples and then determine the rank sums for each sample. To calculate the test result, we use the below formula:
where,
n is the total number of observations in all samples,
m is the number of samples,
n_{i} represents the number of observations in i_{th} sample,
R_{i} denotes the rank sum of i_{th} sample.
Here, we have to use the x^{2} distribution with m – 1, degree of freedom (df) and α level of significance to determine the critical value. If the estimated value is less than x^{2}, then the null hypothesis is accepted, else rejected.
NonParametric TTest
Whenever a few assumptions in the given population are uncertain, we use nonparametric tests, which are also considered parametric counterparts. When data are not distributed normally or when they are on an ordinal level of measurement, we have to use nonparametric tests for analysis. The basic rule is to use a parametric ttest for normally distributed data and a nonparametric test for skewed data.
NonParametric Paired tTest
The paired sample ttest is used to match two means scores and these scores come from the same group. Pair samples ttest is used when variables are independent and have two levels and those levels are repeated measures.