# 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** (MSS_{B}) ” and ” **Mean Sum Square WITHIN Variance** (MSS_{W}) “.

**#** 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**

H_{0} : There is no significant difference between the means of all groups. In other words ( all groups are same ).

**H _{0} : μ_{1} = μ_{2} = μ_{3} = ……. = μ_{k}**

Where μ : group mean & k : number of groups

**2. The Alternative Hypothesis**

H_{A} : There are at least two groups mean that are statistically significantly different from each other.

**H _{A} : μ_{1} ≠ μ_{2} ≠ μ_{3}…… ≠ μ_{k}**

## 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 ).

**σ _{1} = σ_{2} = σ_{3} = …… = σ_{k}**

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 *Σ*X : Total sum of population & N : No. of population

**STEP 2 : Total Sum of Square ( SS _{T} )**

**STEP 3 : Between Sum of Squares ( SS _{B} ) or Sum of Square Between Groups**

**STEP 4 : Within Sum of Square ( SS _{W} ) or Sum of Squares within Groups**

From SS_{T} = SS_{B} + SS_{W}

**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.