Partition of Sums of Squares in ANOVA (I)

To illustrate the idea of Least Squares it is convenient to organize the data in a table with each treatment a row in the table (but note that this is not the format in which you have to organize your data!).

We first can compute means per rows (treatments) and across rows (overall):

For example, treatment means (rows) for i=1 and i=2 are:

μ ?_1=Σ(y_1j )/4=2.128

μ ?_2=Σ(y_2j )/4=1.316

Overall mean: μ ?=Σ(y_ij )/(7×4)=2.040

We can calculate the total sum of squares (TSS) as the sum of the differences of each cell with respect to the overall mean and take the square of each of them:

TSS=∑_ij?(y_ij-μ ? )^2 =(2.387-2.040)^2+(2.453-2.040)^2+…+(2.544-2.040)^2=7.577