Dispersion Variance

In selection problems, it is of little use to know the mean grade of a working stope if we don't have some measure of the dispersion (or variability) of the grades of production size units within the stope.

Let $V$ be a production stope centered on point ${/boma x}$ and divided into $N$ equally sized production units $v({/boma x}_i)$ centered on points ${/boma x}_i$:

Let $z_{v}(/boma{x}_{i})$ and $z_{V}(/boma{x})$ be mean values over the corresponding volumes, namely

units <1.0cm,1.0cm> x from 0.0 to 1.6, y from 0.0 to 1.6

4 4 x from -1.0 to 2.0, y from 0.0 to 2.0 $v$ at 0.2 1.4 $V$ at -0.4 1.4

To each of the $N$ positions ${/boma x}_i$ of the units $v({/boma x}_i)$ inside stope $V$ corresponds a deviation $[z_{v}(/boma{x}_{i}) - z_{V}(/boma{x})]$. The dispersion of the $N$ grades $z_v({/boma x}_i)$ about their mean value $z_V({/boma x})$ can be characterized by the mean square deviation:

$z_{v}(/boma{x}_{i})$ can of course be used to produce a histogram which should give an indication on the distribution of the random variable. We interpret $s^{2}(/boma{x})$ as realization of a random variable

Essentially, it represents the empirical variance, however, dispersed in space. Its expectation we call theoretical dispersion variance

$S^{2}(/boma{x})$ also depends on space $/boma{x}$, however, $/sigma_D^2 (v/V)$ assuming stationarity of second order, does not.

Let us now get the size of $v$ smaller and smaller and correspondingly $N$ larger and larger, the sums will tend to integrals, and we get

from where it follows, after exchanging the integral and expectation,

$$/displaystyle /sigma_D^2 (v/V) /textstyle = /displaystyle /frac{1}{V} /int_{V(/boma{x})} E[Z_{v}(/boma{y})-Z_{V}(/boma{x})]^{2}d/boma{y}$$

$$/textstyle = /displaystyle /frac{1}{V} /int_{V(/boma{x})} /sigma_E^2 (v(/boma{y})/V(/boma{x}))d/boma{y}/ / / .$$ (5.4)

The dispersion variance shows up to be the mean value of the estimation variance over the volume $V$.