Mining & Mineral Economics

Sampling, Geostatistics & Grade Control

From sample mean and variance to the variogram and ordinary kriging — the statistics that turn drill assays into a defensible grade estimate.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Grade estimation starts with sampling: assays of cores, channels or blastholes. The sample mean estimates grade; the variance/standard deviation quantifies its scatter. Classical statistics assumes samples are independent, but ore grades are spatially correlated — nearby samples are alike. Geostatistics captures this with the variogram

γ (h)

, which rises with separation

h

up to a sill at the range (beyond which samples are uncorrelated); a non-zero intercept is the nugget effect (micro-variability + sampling error). Kriging is the best linear unbiased estimator: it weights surrounding samples using the variogram to estimate a block grade with minimum estimation variance.

The Core Formula Matrix

Sample mean:

\overset{x}{ˉ} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}

Sample variance:

s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \overset{x}{ˉ})^{2}

Experimental variogram:

γ (h) = \frac{1}{2 N ( h )} \sum (x_{i} - x_{i + h})^{2}

Variogram model: nugget

C_{0}

+ sill rise

C

over the range

a

; total sill

= C_{0} + C

The ‘IIT Traps’

⚠
**Use $n - 1$ for the *sample* variance**, not $n$ — the $n$ divisor is the (biased) population variance.
⚠
**The variogram has a factor of $\frac{1}{2}$ ** (it is a *semi*-variogram); forgetting it doubles $γ (h)$ .
⚠
**Nugget is the intercept at $h \to 0$ **, not the value at large $h$ (that is the sill).
⚠
Kriging minimises estimation variance, giving each sample a weight — it is not a simple inverse-distance average.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

In a variogram, the non-zero value of

γ (h)

as the separation distance

h

approaches zero is called the:

Q2MEDIUM2 Marks · NAT

Four channel samples assay

2.0, 4.0, 4.0

and

6.0 %

metal. The sample mean grade is ______ %. (Round off to one decimal place.)

Q3HARD2 Marks · NAT

For the same four samples (

2.0, 4.0, 4.0, 6.0 %

, mean

4.0%

), the sample standard deviation (using the

n - 1

divisor) is ______ %. (Round off to two decimal places.)