ONE WAY ANOVA
# The Analysis of Variance ( ANOVA ) is used to determine whether there is any statistical significant difference between the means of three or more independent ( Unrelated ) groups.
# Developed by R.A. Fisher in 1920.
# Also known as F-test Which is based on F-distribution (G.W. Snedecor ).
# It Compares the means between the group and significantly different from each other.
# It determines whether all groups are taken from common population or not.
# Anova is Ratio between ” Mean Sum Squares BETWEEN Variance (MSSB) ” and ” Mean Sum Square WITHIN Variance (MSSW) “.
# The variation among the observations of each specific group is called its Internal Variation and the totality of the internal variation is called Variability within groups.
# The totality of variations from one group to another i.e. variation due to group is called Variability between groups.
What Anova Test/Find
1. The Null Hypothesis
H0 : There is no significant difference between the means of all groups. In other words ( all groups are same ).
H0 : μ1 = μ2 = μ3 = ……. = μk
Where μ : group mean & k : number of groups
2. The Alternative Hypothesis
HA : There are at least two groups mean that are statistically significantly different from each other.
HA : μ1 ≠ μ2 ≠ μ3…… ≠ μk
Assumptions of ANOVA
1). Random Selection : – Samples are randomly selected.
2). Normal Distribution : – Independent variable should be normally distributed.
3). Homogeneity of Variances : – All sub-population have the same variance ( Homoscedastic ).
σ1 = σ2 = σ3 = …… = σk
4). Additivity of Variances : – Total Variance should be equal to sum of between variance & within variance.
Steps to solve Anova
STEP 1 : Correction Factor ( C )
Here ΣX : Total sum of population & N : No. of population
STEP 2 : Total Sum of Square ( SST )
STEP 3 : Between Sum of Squares ( SSB ) or Sum of Square Between Groups
STEP 4 : Within Sum of Square ( SSW ) or Sum of Squares within Groups
From SST = SSB + SSW
STEP 5 : Preparation of Anova Table
STEP 6 : Degree of Freedom ( df )
STEP 7 : Variance Estimates or Mean Sum of Squares
STEP 8 : F-Ratio
STEP 9 : Interpretation
- F(3,9) = F- Ratio
- 3 = df for variance b/w groups
- 9 = df for variance within groups
- 18.16 = Value of F-Ratio
- F Value with df(3,9)
- at 0.05 = 4.46
- at 0.01 = 8.02
The obtained F value is greater than the table value at 0.01 level of significance. The Null Hypothesis is Rejected at 0.01 level of significance. ( More instruction Check Notes below )
Note 1 : Level of significance/ error is may be 0.01 or 0.05 in question. You have to find F value according to level of significance mention in question.
Note 2 :
* If F value less than level of significance ( P value ) then Null Hypothesis is Accepted.
* If F value is greater than level of significance ( P value ) then Null Hypothesis is Rejected.
- STATEMENT Eg :
# At the α = 0.05 level of significance, there is not enough evidence to conclude that there is a difference in the average pollution indexes for the two areas. ( If F value is less )
# At the α = 0.05 level of significance, there exists enough evidence to
conclude that there is an effect due to door color. ( If F value is Greater )
Note 1 : Both statements which are mentioned above are example you have to write statement according to question asked but in similar manner.