First the global mean is calculated from a matrix of three sets each containing three observations. Then the sum of squares is calculated. Lastly, the concept of degree of freedom is explained.
The definition of a chi-square distribution is given. Chi-square is defined as the sum of random normally distributed variables (mean=0, variance=s.d.=1). The number of added squared variables is equal to the degrees of freedom. With more degrees of freedom the probability of larger chi-square values is increased.
The total sum of squares and the total degrees of freedoms are disaggregated by calculating in sample variance and "between" sample variance and their respective degrees of freedoms. It is demonstrated numerically that both these measures add up to the total sum of squares and the total degrees of freedom.