ANOVA

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.