Deviation vector

A deviation vector d measures how different the results for a variable are from the mean for that variable. A vector is a matrix with one column. The deviation vectors are calculated as follows,

The apostrophe symbol means transposition: reversing the length/width orientation of the matrix. Multiplying a deviation vector by its transpose gives us several important results. First of all, is proportional the the square of the length of the deviation vector according to this formula,

The length of the deviation vector is longer when there is more variability, and shorter when there is less variability. In other words, it is proportional to the standard deviation , as shown below,

Multiplying a deviation vector by its own transpose gives us a measure related to its variance or, in other words, its standard deviation. Multiplying one variable's deviation vector by the transpose of another variable's deviation vector gives us a measure of the covariance between the two variables, and is related to the angle between the deviation vectors. These can be combined into a variance and covariance matrix.

And a few relationships between these results and other important calculations,