analysis of variance presentation 317.pptx

Analysis of variance

1. ANO V A Analysis of Variance Presented by : Parminder Kaur Me (CSE) ROLL NO. : 24 317

2. Content INTRODUCTION Assumptions in ANOVA ANOVA Techniques Basic Principle of ANOVA EXAMPLE OF ANOVA

3. INTRODUCTION ▶ ANOVA stands for Analysis of Variance. ▶ It was introduced by Sir R.A Fisher in 1920. ▶ It is a statistical method used to analyse the differences between the means of two or more samples or treatments. ▶ ANOVA is used to test the differences among different groups of data for homogeneity. ▶ The basic principle of ANOVA is to test for differences among the populations by examining amount of variation within each of these samples relative to amount of variation between the samples. ▶ ANOVA is an extension of T test as T-test is not suitable to test for significance of differences among more than two sample means

4. ▶ However, conducting multiple t-tests can lead to severe inflation of the Type I error rate. ▶ ANOVA can be used to test differences among several means for significance without increasing the Type I error rate.

5. Assumptions in ANOVA ▶ Population distribution is normal Example: Suppose a researcher is comparing the average test scores of students from three different schools. If the distribution of test scores in each school follows a bell-shaped curve (normal distribution), this assumption is met. ▶ Samples are random and independent Example: A pharmaceutical company tests the effectiveness of three different painkillers. Patients are randomly assigned to one of the three groups, and their responses are recorded. The assumption is met as long as the patients are selected randomly and their responses do not influence each other. ▶ Homogeneity of sample variance Example: A researcher compares the reaction time of drivers under three different lighting conditions (daylight, dusk, and night). If the variance in reaction times is approximately the same across all three groups, this assumption holds. If one group's reaction times are much more spread out than the others, the assumption is violated.

6. ANOVA TECHNIQUES Two types:- ▶ One- way ANOVA ▶ Two- way ANOVA ▶ One- way ANOVA can only be used when investigating a single factor and single dependent variable.

7. The F- test in One- Way ANOVA ANOVA =Variance Betwee n / ​​ Variance Within

8. ANOVA TABLE (ONE- WAY)

9. EXAMPLE 1

10. Basics ▶ It tests the null hypothesis ▶ H₀:There is no significant difference between the means of all groups.(all groups are same) ▶ H₀=μ₁=,μ₂=,μ₃=….=μĸ ▶ Where μ=group mean ,K=no. of group ALTERNATIVE HYPOTHESIS ▶ HΑ:There are at least two groups means that are statistically significantly different from each other. ▶ H A :μ₁≠μ₂≠μ₃≠…≠μκ

11. CONT… ▶ STEP1: CALCULATE MEAN OF EACH SAMPLE.

12. CONT… ▶ STEP2: CALCULATE GRAND SAMPLE MEAN (mean of sample means)

13. CONT… ▶ STEP3: Calculate sum of square between the samples and sum of square within the samples.

14. CONT… ▶ STEP4: Calculate sum of square (SS) for total variance. ▶ SS(total)= SS Between the samples+ SS within the samples. ▶ SS(total)= 8+24 =32

15. CONT… ▶ STEP5: Setting up ANOVA Table.

16. F TABLE

17. conclusion ▶ The above table shows that calculated value of F is less than the table value of 4.26 at 5% level with d.f being v 1 = 2 and v 2 =9 so we accept the null hypothesis i.e all means are equal .

18. Example 2 Investigate whether different types of exercise programs lead to different weight loss outcomes: Three groups are there Group1 , Group 2 and Group3

19. Continue.. Step 2 :Calculate the means and variances for each group: Overall mean=6.27

20. Continue..

21. Unsolved Problem

22. Thank You