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Non parametric test in statistics is a set of practices of statistical analysis that do not require any data for the assumptions. Due to this cause, they are sometimes specified as distribution-free tests. Further, non-parametric tests serve as another option to parametric tests such as T-test or ANOVA that can be engaged only if the underlying data meets certain criteria and assumptions. In some cases, even if the data lacks to meet the vital assumptions but the sample size of the data is vast enough, we can still apply the parametric tests rather than non-parametric tests.
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Key Takeaways: Parametric tests, Nonparametric test, nominal variable, interval variable, Mean, Median, Standard deviation, binomial distribution
Also read: Isosceles Triangle Theorems
What is a Non-Parametric Test?
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Non-parametric tests are mathematical practices that are used in statistical hypothesis testing. This method is taken into account when the data is unsymmetrical and the assumptions for the underlying populations are not required.
- In general terms, if the given population is unsure or when data is not distributed normally, in this case, non-parametric tests are used.
- Usually, a non-parametric test is used when the data is non-continuous and consists of a large sample size.
- It is not dependent on any data referred to any particular parametric group of probability distributions.
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Non-parametric T-test
At any occasion, when a few assumptions made in the given population are unsure, we use non-parametric tests. When the data is on an ordinary level of measurement or not distributed normally, we take the use of non-parametric tests for analysis. The basic concept is to use a parametric T-test for normally distributed data and a non-parametric test for crooked data.
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Non-parametric paired T-test
The paired sample T-test is used to meet two means scores, and these scores come from the same group. Pair sample T-test is used when variables are not dependent and have two stages, and those stages are repetitive measures.
Difference Between Parametric And Non-Parametric Test
The differences between parametric and non-parametric tests is that the parametric test assumes the fundamental statistical distributions in the data given while the non-parametric test does not rely on any distribution of the data.
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Continuity and Differentiability Detailed Video Explanation:
Non-Parametric Test Formula
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Non-Parametric Test Formula is given in Kruskal-Wallis H-Test, we use the formula to evaluate the results. The formula is expressed as;
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Methods For Non-Parametric Test
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There are various types of non-parametric tests such as:
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1-sample sign Test
This test is used to evaluate the median of a population followed by comparing it to a reference value or target value.
- Left tailed test- H0: median ≥ Hypothesized value x; H1: median < x
- Right tailed test- H0: median ≤ Hypothesized value x; H1: median >x
- Two tailed test- H0: median = Hypothesized value x; H1: median ≠ x
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1-sample Wilcoxon Signed Rank Test
This test is similar to that of 1-sample sign Test but the data is assumed to come from an equal distribution.
- Null hypothesis, H0: median difference should be zero.
- Test statistics: the test statistics W, is expressed as the smaller of +W or -W.
- Where +W or -W are the sums of the positive and the negative ranks of the distinct scores.
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Freidman Test
Freidman tests inspect the difference between groups with ordinal and non-independent variables.
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Kruskal-Wallis Test
This test helps in estimating whether two or more medians are distinct. The ranks of the data points are used in calculations rather than the data point themselves.
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Mann-Kendall trend Test
This test examines the trends in time-series data.
- The idea behind the Mann-Kendall trend test is that, if a trend is present, the sign values will tend to increase or decrease constantly.
- Every value is compared to the preceding value in time series which is expressed as total of,
n(n-1)/2 pairs of data,
where, n is the number of observations in the set.
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Mann-Whitney Test
This test determines the difference between two non-dependant groups on a condition that the dependant variables will either be regular or ordinal. If r1 and r2 are the sum of the ranks in group 1 and group 2 respectively, then the test statistic ‘U’ is given as;
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Mood’s Median Test
This test is used in place of sign test when we have two independent samples. The test statistic used is the chi-square test statistic, given as
Where, O – observed Frequency
E – Expected Frequency
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Spearman Rank Correlation
This test is used to find the correlation between two data sets. Spearman Rank Correlation is expressed as,
Where, d – difference between ranks
d2 – square of a difference.
Advantages and Disadvantages Of Non-Parametric Test
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The advantages and disadvantages of a non-parametric test are as follows:
| ADVANTAGES | DISADVANTAGES |
|---|---|
| The primary advantage of non-parametric test is that they will have extra statistical power if the assumptions for the parametric tests are breached. | The basic disadvantage in nonparametric is that these are less strong than parametric test if the assumptions have not been broken. |
| The non-parametric test consists of short calculations which are easy to grasp. | In case of nonparametric the calculations by hand are hard to do. |
| There are multiple assumptions in nonparametric test than parametric. | The packages of computer software do not include critical value tables for many other non-parametric tests. |
| It is applicable to all types of data like nominal variable, interval variable, etc irrespective of small sample sizes or large ones. | The results of nonparametric tests may or may not be true as it is based on distribution free data. |
Applications Of Non-Parametric Test
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The circumstances where non-parametric tests are used are:
- When parametric tests are not content.
- While testing the hypothesis, it does not have any distribution.
- For swift data analysis.
- When data available is unscaled.
Also read: First Order Differential Equation
Things to Remember
- Non-parametric tests are not dependant on any distribution; therefore, it is a kind of robust test and possess broader range of situations.
- Parametric test is completely statistically data driven and possess more chances of accuracy.
- In case of parametric “mean value” is taken into consideration while in non-parametric “median value” is taken into account.
- The U-test is also known as Mann-Whitney Wilcoxon Test.
- Parameters for using the normal distribution consists of Mean and Standard deviation.
- The non-parametric tests are also stated as distribution-free tests as they don’t include fixed parameters to be followed.
Also read: Difference between Sequence and Series
Sample Questions
Ques. Solve:
(a) Evaluate the continuity of the function ‘f’ which is f(x) = 2x + 3 at x=1. (2 marks)
(b) Check-out whether the function ‘f’ given as f(x) = x2 is continuous at x = 0. (2 marks)
Ans: a. First make sure that the function is stated at the given point x=1 and hence the value of x = 5. Then evaluate the limit of the function at x = 1.
limx→1fx = limx→12x+3 = 2.1+3 =5.
Hence, limx→1fx = 5= f1.
Therefore, f(x) is continuous at x=1.
- It is necessary to make sure that function at the given point x=0 and its value is 0.
Then calculate the limit of the function at x= 0.
limx→0fx = limx→02x→1=02=0.
Thus, limx→0fx = 0 = f(0)
So, ‘f’ is continuous at x = 0.
Ques. Mention different sort of nonparametric tests (3 marks)
Ans: There are four different types of nonparametric tests:
- Sign Test
- Wilcoxon signed-ranked Test
- Kruskal Wallis Test
- Mann Whitney U test
Ques. What is the typical application of the Chi-square test? (3 marks)
Ans: The common application of Chi—Square test are as listed below:
- As an alternative test to find the importance of the difference in two or more proportions:
- To compare the values of two binomial specimen if they are small-scale;
- To collate the frequencies of two multinomial samples.
- It is used as a test of goodness of fit.
- As a test of bond between two events in Multinomial or Binomial samples.
Ques. What are the pros and cons of nonparametric tests? (3 marks)
Ans: The pros of nonparametric tests are given below as:
- They are easy to grasp.
- They are not difficult to calculate.
- Assumption of distribution is not required.
- It is applicable for all the kind of data.
The cons of nonparametric tests are given as:
- It is not as efficient as parametric test.
- The results acquired may or may not be accurate as the distribution is free.
Ques. What are the implementation of nonparametric tests? (3 marks)
Ans: Nonparametric tests can be implemented as:
- Nonparametric test is taken in account when the data fails to satisfy the conditions that are needed to be met by parametric test.
- In hydrogeological problems, nonparametric tests are very useful.
- Nonparametric tests are used for rapid data analysis.
- It is used for testing hypotheses when there is no distribution available.
- Nonparametric is also used in the case of unscaled data.
- The nonparametric tests as Kruskal-Wallis and U test are used to judge the difference between two medians or to non-dependant groups on a condition that the dependant variables will either be ordinal or regular.
Ques. List down the primary assumptions of the Chi square test? (3 marks)
Ans: In statistics, Chi square tests are usually used for testing bond between categorical variables. The basic Chi square assumptions are as follows:
- The data given in the Chi square must not be in percentage form but in form of frequency.
- The variables or the categories must be mutually exclusive.
- The gist must contribute the data to only one cell in the x2.
- For the two distinct groups that are related, we take the use of two different tests. Hence, the study groups must be independent.
Ques. Draw simplified flow diagrams concluding some of the factors that should be taken into account when choosing a hypothesis test. (5 marks)
Ans: 
Summary of consideration for one sample hypothesis test.
Summary of consideration for two sample hypothesis tests.
Ques. Define in brief:
(a) What is Binomial distribution? (1 mark)
(b) State Mean and Variance of Binomial distribution. (2 marks)
(c) Express formula of Binomial distribution. (2 marks)
Ans: a. Binomial distribution
Binomial distribution is stated as the probability of having only two possible outcomes they are success or failure. The term Binomial distribution is also known as Bernoulli distribution. This forms as one of the most vital bases of strategic calculations.
- Mean and variance
If ‘x’ is the probability of success and ‘b’ is the probability of failure in a binomial distribution, then the number of successes in ‘n’ trials means the mean value of the binomial distribution is
E(X) = µ = n. a
The variance of binomial distribution
is V(X) = σ2 = n. a. b
- Formula for binomial distribution:
Px = nCx. px. q n-x
Where,
P – binomial probability
x - number of times for a specific outcome withing n trials
nCx – number of combinations
p - probability of success in single trial
q - probability of failure in one trial
n- number of trials
Ques. What is a Nonparametric test. Explain in points? (3 marks)
Ans: The nonparametric test can be explained as,
- Non-parametric test is the mathematical process used in statistical hypothesis testing which does not make the assumptions about the frequency distribution of variables that are to be evaluated.
- The Non-parametric test is used when there is unsorted data.
- It comprises of techniques that does not depend upon data pertaining to any specific distribution.
- The characteristics and number of parameters are pretty tangible and not defined.
- The primary rule is to use a parametric test for normally distributed information and non-parametric test for skewed data.
Ques. When to use the non-parametric test and parametric test? (2 marks)
Ans: The points below show the use of non-parametric test and parametric test:
- If the mean of the data is more precisely and up to the point presented with the centre of distribution, and the sample size is large enough, one can use parametric test.
- Non-parametric test is used when data presented is not normal or to analyse data when the distribution assumptions of more ordinary procedures are not contented.
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