# CONJUGATE VARIANCE

Variance calculation inevitable results in a "common" area that has to be attributed in some way.

##### ALLOCATION ON TWO DIMENSIONS

When variance is calculated on two dimensions - price and volume - the common area that must be attributed can be depicted as a rectangle. The area, represented in yellow below, represents the intersection of the volume and of the price variance effect.

One method is simply to attribute all this common area either to volume or to price variance. This reflects a widespread but somewhat arbitrary practice.

'Mparanza avoids this issue by allocating the common area equally between price and volume variance.

The formula we use is the following

##### ON THREE DIMENSIONS

Things get more complicated when variance is calculated on three dimensions -for instance price, volume and margin rate"- to identify the factors behind margin change.

The common rectangle area becomes a tri-dimensional box, or rather a collection of boxes in the tridimensional space.

We can visualize the initial situation as a blue Duplo parallelepiped.

The simple, one-dimensional, variances are represented as the green, yellow and red blocks. For simplicity, we are showing positive variances :-).

We can visualize the pairwise common areas as the orange, grey and white blocks. These blocks need each to be divided in two, into triangular-based prisms. The volume of each prism must be attributed to the corresponding variance dimension.

We can visualize the threesome common area as the pink block. This volume needs to be divided in three equal slices, and each slice needs to be attributed to a variance dimension.

It can be difficult to visualize a cube divided by three, so here you go.

We use this approach to calculate three-dimensional variance in the options 2,3,6 above. Option 4, being four-dimensional, would be too complicated to calculate with attributions, so we use a simpler, somewhat volume-variance-biased, formula.

A huge thanks to Ludovico Ruggeri Laderchi for the explanation, Duplo pictures and formulas.