Chi-Square and Analysis of Variance

Chi-Square characteristics, test, computational procedure, formula, ANOVA, and their difference.

1. CHI-SQUARE AND ANALYSIS OF VARIANCE 2   1

2. Content Layout Chi-Square Characteristics of Chi-Square Chi-square Test Computational Procedure – Chi Square Test Chi Square – Test Of Independence Formulae To Be Used Contingency Table Analysis Of Variance (ANOVA) Assumptions Steps in Analysis Of Variance (ANOVA) Computational Procedure In ANOVA (One Way) Difference Between Chi Square & ANOVA 𝜒2 2

3. Chi-square ( 𝜒 2) The  Chi Square  statistic is commonly used for testing relationships between categorical variables. The null hypothesis of the  Chi - Square  test is that no relationship exists on the categorical variables in the population; they are independent. 2   3

4. CHARACTERISTICS OF CHI SQUARE Every Chi square distribution extends indefinitely to right from zero. It is skewed to right As degree of freedom increases, Chi square curve become more bell shaped and approaches normal distribution . Its mean is degree of freedom Its variance is twice degree of freedom 2   4

5. CHI SQUARE ( Χ2 TEST) Chi Square Test deals with analysis of categorical data in terms of frequencies / proportions / percentages . It is primarily of three types: Test of Homogeneity : To determine whether different population are similar w.r.t some characteristics. Test of Independence: Tests whether the characteristics of the elements of the same population are related or independent. Test of Goodness of Fit: To determine whether there is a significant difference between an observed frequency distribution and theoretical probability distribution. 2   5

6. COMPUTATIONAL PROCEDURE – CHI SQUARE TEST Formulate Null & Alternative Hypothesis State type of test Select LOS Compute expected frequencies assuming H0 to be true. Compute χ2 calculated value using 𝜒 2 cal = Extract 𝜒2 crit value from table Compare 𝜒2 cal & 𝜒2 crit and make decision   2   6

7. CHI SQUARE – TEST OF INDEPENDENCE FORMULAE TO BE USED Computation of expected frequency Fe = (RT x CT) / GT where RT = Row Total, CT = Column Total, GT = Grand Total Computation of degree of freedom Degree of freedom= (r – 1) (c – 1 ) r =No. of rows, c=No. of column 2   7

8. CONTIGENCY TABLE A table having R rows and C columns. Each row corresponds to a level of one variable, each column to a level of another variable. Entries in the body of the table are the frequencies with which each variable combination occurred. 2   8

9. ANALYSIS OF VARIANCE (ANOVA) Analysis of variance ( ANOVA ) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample.  ANOVA  was developed by statistician and evolutionary biologist Ronald Fisher. 9

10. ANALYSIS OF VARIANCE (ANOVA) It enables us to test for the significance of the differences among more than two sample means . Using ANOVA, we will be able to make inferences about whether our samples are drawn from population having the same mean. Examples : Comparing the mileage of five different brands of cars Testing which of the four different training methods produces the fastest learning record Comparing the average salary of three different companies In each of these cases, we would compare the means of more than two sample means. F-Distribution is used to analyze certain situations 2   10

11. ASSUMPTIONS Populations are normally distributed Samples are random and independent Population Variances are equal. 2   11

12. STEPS IN ANALYSIS OF VARIANCE Determine one estimate of the population variance from the variance among the sample means. Determine second estimate of the population variance from the variance within the sample means. Compare these two estimates. If they are approximately equal in value, accept the null hypotheses. 2   12

13. COMPUTATIONAL PROCEDURE IN ANOVA (ONE WAY ) Define Null & Alternative Hypothesis Select Significance Level Calculate Sum of all observations: T = Ʃx 𝑖 Calculate correction factor: CF = T2 / nT where nT = sample size Calculate Sum of squares total, SST = Σ(Σ𝑥𝑖2) − CF Calculate Sum of squares between columns, SSB = Σ((Σ𝑥𝑖)2/𝑛𝑖) − CF Calculate Sum of squares within columns, SSW = SST - SSB Calculate = w OR = w where s=variance Calculate Mean of squares between groups, MSB = SSB / (k – 1) where k = no. of samples Calculate Mean of squares within groups, MSW = SSW / ( nT – k) Calculate Fcal = MSB / MSW Calculate Fcrit = F( dfnum , dfden , α) where dfnum = k – 1, dfden = nT – k Compare Fcal & Fcrit and make your statistical & managerial decisions   2   13

14. DIFFERENCE BETWEEN CHI SQUARE & ANOVA Chi Square (χ2 Test ) It enables us to test whether more than two population proportions can be considered equal Anova (F Test) Analysis of Variance ( ANOVA) enables us to test whether more than two population means can be considered equal. 14

15. THANK YOU!!! 15