*2.4.3 One-way ANOVA and their notation*

When there is just one explanatory variable, we refer to the analysis of variance as a one-way ANOVA.

Here is a key to symbols you may see as you read through this section.

k = the number of groups/populations/

xi j = the jth response sampled from the ith group/population.

xi = the sample mean of responses from the ith group <sup>¼</sup> <sup>1</sup> ni P<sup>n</sup> j¼i xij

si = the sample standard deviation from the ith group <sup>¼</sup> <sup>1</sup>*=*ð Þ ni � <sup>1</sup> <sup>P</sup>ni <sup>j</sup>¼<sup>1</sup>ð Þ xij‐xi <sup>2</sup>

n = the total sample <sup>¼</sup> <sup>P</sup><sup>k</sup> <sup>i</sup>¼<sup>0</sup>xi

x = the mean of all responses ¼ 1*=*n P*:* ij xij

#### *2.4.4 Parting the total variability*

Viewed as one sample one might measure the total amount of variability among observations by summing the squares of the differences between each xi j and x: Sources of variability:

1.SST (stands for the sum of squares total)Pni j¼1*:* Pni <sup>j</sup>¼<sup>1</sup>ð Þ xij � <sup>x</sup> <sup>2</sup>

2. Sum of Square Group between group

$$\mathbf{SSG} = \sum\_{i=0}^{k} \mathbf{ni(xij - x)}^{2}$$

Sum of Square Group within groups means

$$\mathbf{\color{red}{3.8SE}} = \sum\_{j=1}^{\text{ni}} \text{.} \sum\_{j=1}^{\text{ni}} (\mathbf{x}\mathbf{i}\mathbf{j} - \mathbf{x})\mathbf{2} = \sum\_{i=1}^{k} (\mathbf{n}\mathbf{i} - \mathbf{1})\mathbf{s}\_i^2$$

It is the case that SST = SSG + SSE.


**Table 4.**

*Analysis of Variance Computations (ANOVA).*
