**Subject :**Math -3

**Unit :**Statistical Techniques - II

## Analysis of Variance

**Analysis of Variance:**

Analysis of variance basically is an arithmetical method by which we split up the total variability into component variations ascribable to different sources of causes. In the words of **Yule and Kendall**, The analysis of variance is essentially a procedure for testing the difference between different groups of data for homogeneity. While defining analysis of variance, **Ronal A. Fisher wrote**, The separation of the variance ascribable to one group of causes from the variance ascribable to other groups. In simple words, Analysis of Variance is a statistical technique, with help of which total variation, is partitioned into variation caused by each set of independent factors and homogeneity of several means is tested.

**Components of Total Variability:**

In general (say one way classification) total variability is partitioned into two parts that is:

Total Variability = Variability between samples Variability within samples.

Or

Total Variation = Variation between samples Variation within samples.

**Assumptions of Analysis of Variance:**

The analysis of variance is based on certain assumptions as given below :

1. Normality of the Distribution : The population for each sample must be normally distributed with mean μ and unknown variance σ ^{2}.

2. Independence of Samples : All the sample observations must be selected randomly. The total variation of the various sources of variation should be additive.

3. Additivity : The total variation of the various sources of variation should be additive.

4. Equal variances (but unknown) : The populations from which the n samples say are drawn have means μ_{1}, μ_{2}, ..., μn and unknown variance σ ^{2} _{1} = σ ^{2}_{ 2 }= ...... = σ ^{2} _{n} = σ ^{ 2}.

5. The error components are independent and have mean 0 and variance σ ^{2}.

The tests of significance performed in the analysis of variance are meaningful under its assumptions.

**Analysis of Variance Table:**

**Remark**: There are three methods, to calculate mean and variances :

(**i**) Direct Method

(**ii**) Indirect Method

(**iii**) Step deviation Method or Coding Method (change of origin or / and scale)