Correlations

Let us consider $n$ pairs of measurements $(y_{i},z_{i}),$ $i=1,/ldots,n,$ (e.g. Cu and Pb grades of ore, or porosity and permeability of sand stone) and denote corresponding the means and deviations by $/bar{y}, /bar{z}, s_{y} /; /mbox{and} /; s_{z}$. The corresponding random variables are $Y$ and $Z$ with $m_{Y}=EY, /; m_{Z}=EZ, /; /sigma^2_Y= Var(Y)$ and $/sigma _Z^2=Var(Z)$. Dependences of these variables may be measured by the covariance

From $n$ measurements it is estimated by

As relative quantity it is more advantageous to use the correlation

respectively the correlation coefficient

The values of $/rho$, respectively, $r$ are between $-1$ und $+1$.

Example 2.2: Considered the dependency of grade of iron $Z$ (in %) ``kieseliger Hämatiterze'' with the density $Y$ $(g/cm^{3})$ and supposed measured values (H. Bottke, Bergbauwiss. 10, 1963, 377):

Table 2.6: Density and Grade of Iron.

$y_{i}$	2.8	2.9	3.0	3.1	3.2	3.2	3.2	3.3	3.4
$z_{i}$	27	23	30	28	30	32	34	33	30

units <8cm,4.5mm> x from 2.7 to 3.5, y from 22 to 35 left shiftedto x=2.7 label Z,(in %) ticks short numbered from 23 to 35 by 1 / bottom shiftedto y=22 label $Y$ $(in$ $g/cm^3$) ticks short numbered from 2.8 to 3.5 by .1 / $/circ$ at 2.8 27 $/circ$ at 2.9 23 $/circ$ at 3.0 30 $/circ$ at 3.1 28 $/circ$ at 3.2 30 $/circ$ at 3.2 32 $/circ$ at 3.2 34 $/circ$ at 3.3 33 $/circ$ at 3.4 30 2.7 24.6 3.4 33.05 /

We compute the quantities:
$/bar{y}=/frac{1}{n} /sum y_{i}=3.12$
$/bar{z}=29.67$
$s_y^2=/frac{1}{n-1}/sum(y_{i}-/bar{y})^{2}=.0369$
$s_z^2=/frac{1}{n-1}/sum(z_{i}-/bar{z})^{2}=11.25$
$s_{yz}=/frac{1}{n-1}/sum(y_{i}-/bar{y})(z_{i}-/bar{z})=.4458$
$r_{yz}=s_{yz}/(s_{y}s_{z})=.69.$

Example 2.3: The scattergram in Figure 2.6 illustrates the dependency (the correlation) between the two variables copper and nickel using the data of Table 2.7. The used computer program (Dutter, 1992[]) computes, beside the usual statistics as mean value and variance, estimates of the regression line and its confidence bands between the two variables. The model of the line nickel over copper is

and the model copper over nickel

The correlation coefficient is computed to be

Figure 2.6: Scattergram of Copper and Nickel Data.