Mathematics & Geostatistics

Statistics & Geostatistics

Mean, variance and the semivariogram — the quantitative backbone GATE GG leans on for assay data, grade estimation and geophysical signal handling.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Geological data are samples of a spatially-varying field. Descriptive statistics (mean, variance) summarise the spread of assay or measurement values, while geostatistics adds the spatial dimension: the semivariogram

γ (h)

measures how dissimilar two samples become as the separation (lag)

h

grows. Its nugget captures micro-scale variance and measurement error, the sill is the plateau variance, and the range is the lag beyond which samples are no longer correlated. Kriging uses this model to interpolate grades as a best linear unbiased estimator.

The Core Formula Matrix

Arithmetic mean:

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

Population variance:

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

Coefficient of variation:

C V = σ / \overset{x}{ˉ}

(high

C V

⇒ erratic, nuggety grades).

Experimental semivariogram:

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

Spherical model: rises from the nugget

C_{0}

to the sill

C_{0} + C

over the range

a

The ‘IIT Traps’

⚠
Range vs sill. The *range* is a distance (lag), the *sill* is a variance — never interchange them.
⚠
Nugget ≠ zero. A finite intercept at $h = 0$ reflects real micro-variability plus sampling error, not a mistake.
⚠
Skewed grades. Gold/precious-metal grades are log-normal; use the geometric mean or log-transform before kriging, not the raw arithmetic mean.

📚 Standard references

Mining Geostatistics — A.G. Journel & C.J. Huijbregts
An Introduction to Applied Geostatistics — Isaaks & Srivastava

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · NAT

The arithmetic mean of the assay values

{2, 4, 6, 8, 10}

g/t is _____ g/t.

g/t

Q2MEDIUM2 Marks · NAT

For the data

{4, 6, 8}

, the population variance is _____.

Q3HARD2 Marks · MCQ

On a spherical semivariogram, the lag distance at which the model first reaches its sill is the