Non-parametric Tests in statistics chi-square and kolmogorov

Non-parametric tests are statistical methods used when data do not necessarily fit a normal distribution or when the assumptions required for parametric tests cannot be met. Two widely used non-parametric tests are the Chi-Square test and the Kolmogorov-Smirnov test.

1. CHI-SQUARE GOODNESS OF FIT TEST & KOLMOGOROV-SMIRNOV ONE STATISTIC TEST Team Members: Anjali Prajapati – 901 Akshada Vaze – 906 Harshvardhan More – 908 Iqra Ansari – 910 Shruti More – 927 Samiksha Lad – 942 Date:07/03/2024

2. INTRODUCTION

3. CHI-SQUARE GOODNESS OF FIT TEST The Chi-square goodness of fit test is a statistical hypothesis test used to determine whether a variable is likely to come from a specified distribution or not. It is often used to evaluate whether sample data is representative of the full population.

4. ASSUMPTIONS

5. EXAMPLE Suppose we have a class of 50 students, and the national survey indicates following distribution of favourite subjects :Maths 40%,Science 30% ,English 20%, History10%, After surveying our class we find the following distribution: Maths 25 students ,Science 15 students ,English 5 students, History 5 students. Test the hypothesis that distribution of favourite subjects in class is same as national distribution. Hypothesis: H0: Distribution of favourite subjects in class is same as national distribution. H1: Distribution of favourite subjects in class is not same as national distribution. SOLUTION:

6. EXAMPLE Formula :   χ 2 =  ∑(Oi-Ei) 2 /Ei Oi Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei Maths 25 0.40*50=20 25 1.25 Science 15 0.30*50=15 English 5 0.20*50=10 25 2.5 History 5 0.10*50=5 Total 50 50

7. EXAMPLE χ 2 =  ∑(Oi-Ei) 2 /Ei Summing the values we get; Chi -Square calculated as: χ 2 cal = 1.25 + 0 + 2.5 +0 = 3.75 Chi-Square tabulated as: χ 2 tab = χ 2 (n-1), α = χ 2 (4-1), 0.05 = χ 2 3,0.05 χ 2 tab = 7.815 Decision Criteria: χ 2 cal < χ 2 tab 3.75<7.815 Accept H0. Conclusion: Distribution of favourite subjects in class is same as national distribution.

8. REAL LIFE APPLICATION Artificial Intelligence and Game Design Customer Behavior Analysis in E-commerce Cryptocurrency Analysis

9. ADVANTAGE & DISADVANTAGE Simple to Implement Non-parametric Flexible Provides Statistical Significance Identifies Patterns and Trends Sensitive to Sample Size Assumption of Independence Limited to Categorical Data Does Not Provide Directionality Interpretation Challenges

10. KOLMOGOROV-SMIRNOV TEST

11. WHEN TO USE? Parametric Test Non-Parametric Test

12. KOLMOGOROV-SMIRNOV TEST

13. ONE SAMPLE TEST

14. ONE SAMPLE TEST Acceptance Criteria:  If calculated value is less than critical value accept null hypothesis. Rejection Criteria:  If calculated value is greater than table value reject null hypothesis.

15. EXAMPLE In a study done from various streams of a college 60 students, with equal number of students drawn from each stream, are we interviewed and their intention to join the Drama Club of college was noted. B.Sc. B.A. B.Com M.A. M.Com No. in each class 5 9 11 16 19 It was expected that 12 students from each class would join the Drama Club. Using the K-S test to find if there is any difference among student classes with regard to their intention of joining the Drama Club.

16. EXAMPLE Solution: Hypothesis: H : There is no difference among students of different streams with respect to their intention of joining the drama club. VS H 1 : There is no difference among students of different streams with respect to their intention of joining the drama club. We develop the cumulative frequencies for observed and theoretical distributions.

17. EXAMPLE

18. EXAMPLE Since the calculated value is greater than the critical value, hence we reject the null hypothesis and conclude that there is a difference among students of different streams in their intention of joining the Club.

19. ADVANTAGES DISADVANTAGES

20. REAL LIFE APPLICATION

21. CONCLUSION Chi-Square Goodness of Fit Test: The chi-square goodness-of-fit test is a valuable tool for assessing the agreement between observed and expected frequencies in categorical data. A low p-value indicates significant disparities, prompting a rejection of the null hypothesis. Conversely, a high p-value suggests a good fit, supporting the validity of the proposed distribution or model. This test is especially applicable when analyzing nominal variables in various fields, providing a concise measure of statistical significance.

22. CONCLUSION Kolmogorov- smirnov Test (One-sample) : The Kolmogorov-Smirnov (KS) test is widely used for comparing data distributions, assessing goodness-of- fit, and evaluating model performance. It finds application across diverse fields such as machine learning, statistical analysis . Its nonparametric nature and simplicity make it suitable for analyzing complex or non-normal data. Caution is necessary due to its sensitivity to sample size, outliers, and focus on continuous distributions.

23. REFERENCES https://dataaspirant.com/kolmogorov-smirnov-test/ https://online.stat.psu.edu/stat415/book/export/html/838 https://dataaspirant.com/kolmogorov-smirnov-test/ https://datatab.net/tutorial/parametric-and-non-parametric-tests https://statology.org/chi-square-test-real-life-examples/ https://www.tutorialspoint.com/statistics/kolmogorov_smirnov_test.htm

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