Raw data (i.e., Raw score matrix) and calculate mean vectors and standard deviations
Variable

Individuals 
Σ 
Mean 
σ^{2}

σ


x1_raw

1 
2 
1 
5 
4 
13 
2.6 
3.3

1.8166

x2_raw

2 
3 
2 
4 
4 
15 
3.0 

x3_raw

3 
4 
3 
5 
4 
19 
3.8 
Obtain a listing of deviations from the mean vector (i.e., deviation matrix)
Individuals


Deviations from Variable Means 
x1

1.6





x2







x3






Matrix multiplication  this procedure calculates the crossproduct of the original matrix of differences (A) with a transposed copy (A^{'}) of itself (i.e., one in which rows and columns have been inverted). This results in a new matrix with dimensions of 3 (i.e., rows of A) x 3 (i.e., (columns of A^{'})

* 

= 

Arrange values into Deviation SSCP (Sums of Squares and Cross Products) matrix and compare with the values obtained in the correlation worksheet. This procedure calculates the sums of squares along the diagonal and the sums of Cross product in the lower triangle off the diagonal. All products are calculated as length vector 1 * length of vector 2 * sine of angle between vector 1 and 2. As the different variables are orthogonal to each other (i.e., the angle between them is 90^{o}), the angle term of the product formula therefore is always 1 (i.e., sine(90^{o}) = 1) and can thus be dropped from these calculations. Note there are three different, mathematically equivalent techniques to calculate the deviation SSCP. The symbol X in all tables below refers to the differences from the mean and not the raw variable measures.
Symbols 
Example Numbers 

Σ X_{1}^{2}



13.2 


Σ X_{1}X_{2}

Σ X_{2}^{2}





Σ X_{1}X_{3}

Σ X_{2}X_{3}

Σ X_{3}^{2}




VarianceCovariance Matrix. This procedure calculates the variance (i.e., MS) for each variable along the diagonal and the covariances the lower triangle off the diagonal. Recall that variance estimates refer to a sum of squares divided by its degrees of freedom
Symbols 
Example Numbers 

σ^{2}_{1}=Σ(X_{1}^{2})/(N1)



3.3 


Σ(X_{1}X_{2})/(N1)

σ^{2}_{2}=Σ(X_{2}^{2})/(N1)





Σ(X_{1}X_{3})/(N1)

Σ(X_{2}X_{3})/(N1)

σ^{2}_{3}=Σ(X_{3}^{2})/(N1)




Correlation Matrix. i.e., Standardized variance/covariance matrix  Divide each entry by two standard deviations (σ), one for each relevant row and column.
Symbols 
Example Numbers 


x1

x2

x3

x1

x2

x3

x1

r_{1}_{1}
= Σ(X_{1}^{2})/((N1)σ_{1}σ_{1})



1 


x2

r_{12} = Σ(X_{1}X_{2})/((N1)σ_{1}σ_{2})

r_{22} = Σ(X_{2}^{2})/((N1)σ_{2}σ_{2})





x3

r_{13} = Σ(X_{1}X_{3})/((N1)σ_{1}σ_{3})

r_{23} = Σ(X_{2}X_{3})/((N1)σ_{2}σ_{3})

r_{33} = Σ(X_{3}^{2})/((N1)σ_{3}σ_{3})




Matrix of Coefficients of Determination (r^{2})  Note: Do not use in Correlation analysis as it assumes a level of causality of the independent over the dependent variable. Square each entry from above.
Symbols 
Example Numbers 

x1

x2

x3

x1

x2

x3


x1

r^{2}_{11} = (Σ(X_{1}^{2})/((N1)σ_{1}σ_{1}))^{2} 


x2

r^{2}_{12} = (Σ(X_{1}X_{2})/((N1)σ_{1}σ_{2}))^{2}  r^{2}_{22} = (Σ(X_{2}^{2})/((N1)σ_{2}σ_{2}))^{2}  
x3

r^{2}_{13} = (Σ(X_{1}X_{3})/((N1)σ_{1}σ_{3}))^{2}  r^{2}_{23} = (Σ(X_{2}X_{3})/((N1)σ_{2}σ_{3}))^{2}  r^{2}_{33} = (Σ(X_{3}^{2})/((N1)σ_{3}σ_{3}))^{2} 
If you have difficulties in performing these steps, review some of the basic concepts in matrix algebra.
Rerun all calculations from above with data converted to standard normal deviates (i.e., standardized score matrix) where you subtract the variable's mean from each value and divide it by the variable's standard deviation.
Variable

Individuals 
Σ

σ^{2}

σ


x1_stand

0.8808 







x2_stand









x3_stand









 continue with calculations from section 1.