It was a
computer with geostatistical software that created a phantom gold resource out of bogus
grades and barren rock. What geostatistical software cannot possibly create is a
single degree of freedom. Geostatisticians deem degrees
of freedom as redundant as the variance of the distance-weighted
average. Agterberg made this variance
vanish but the concept of degrees of freedom is more resilient to tampering.
here to find out how geostatisticians fumbled degrees
of freedom. Peruse tables in textbooks on mathematical
statistics where all sort of values are listed as a function
of degrees of freedom and figure out why they went missing
in textbooks on geostatistics. Just remember that degrees
of freedom are awarded to measured values but not to functionally
dependent (calculated!) values. Degrees of freedom are
positive integers for sets of measured values with equal
weights but positive irrationals for sets of measured
values with variable weights.
software is a powerful tool to compute statistics for
a wide range of applications in mineral exploration, mining,
processing, smelting and refining. Spreadsheet templates
are simple to set up and easy to modify to suit either
smaller or larger sets of primary data. Only three types
of templates were required to compute 95% confidence limits
for content and grade of a gold reserve defined by more
than 300 boreholes.
G Wells predicted that statistical thinking would become
as important as the ability to read and write. Would Wells
have embraced the nouveau pseudo science of interpolation
without justification with the same unbridled passion
as the world's mining industry did? After all, Wells was
familiar with Fisher's work long before Fisher became
Sir Ronald A Fisher in 1952.
Some caution is in order! Knowledge of mathematical statistics
is empowering but addictive. Studying mathematical statistics
and playing with spreadsheet templates may be the beginning
of an irreversible process!
False and true tests for bias
A test program for bias between ash contents of Pennsylvania anthracite shipments to the Port of Rotterdam shows that interpolation between measured values makes no sense.
and the kriging game
Start step-out drilling to create phantom ore, compute
three lines of kriged boreholes between two lines of salted
boreholes, get an illusion of spatial dependence between
kriged boreholes in barren rock, and apply analysis of
variance to show that salted boreholes do not even display
and the bonanza borehole
Bre-X's salters worked overtime on core samples from Borehole
SEZ198 but Fisher's forbidden F-test proved that Bre-X's
glory hole was as barren as Busang's gold phantom resource.
and the kriging game
Play with Clark's coordinates, find out what would
happen if each distance-weighted average-cum-kriged
estimate were to have its own variance, figure out how missing
variances of distance-weighted averages could possibly
converge on the Central Limit Theorem but don't waste
too much time while trying to enter the infinite set of
all possible coordinates.
Platinum Palladium Concentrate
Interleaved sampling protocols make it possible to compute
confidence limits for PGM contents and grades of concentrate
production at the smelter of Stillwater Mining Company.
Uncorrected sampling variogram
Corrected sampling variogram
A 2000-Year Global Temperature Reconstruction Based on Non-Treering Series, Loehle, C, Energy & Environment, Vol 18, Nov 2007.
Cerattepe bulk sampling program for gold
sampling program for silver
Cerattepe correlation-regression statistics
A paper on this project was submitted at the request of and rejected by JMG's Editor-in-Chief simply because he wanted his degrees of freedom
in my paper. So I suggested that he should review the applicable
ISO standard method.
for spatial dependence
Testing ordered sets of standard uniform random numbers
for spatial dependence is an exercise in futility but
counting degrees of freedom is not. This information was
sent to MG's Editor-in-Chief in 1998.
Confidence limits for central value of skewed gold assays
Although this large set of gold assays departs significantly from the normal distribution, its is possible to compute confidence limits for its central value.
ore zone of borehole in massive sulfide
Set of fifty ore zones
of boreholes in massive sulfide
Confidence limits for density- and length-weighted
average grades of core samples from a single ore zone
and from a set of fifty (50) ore zones in the same massive
A paper entitled
'Borehole Statistics with Spreadsheet Software'
explains how to obtain a unique 'fingerprint' for each
borehole. The stats for DDH9801 and DDH9802 show that
not all core samples have the same length.
Matheron's 1954 correction
Matheron derived the weighted average lead and silver grades of core samples with variable lengths. He didn't derive the variances of his weighted averages. Neither did I because he didn't report primary data. Find out what I did with Matheron's variances of metal grades and his central values.
Matheron's 1967 Thesis
Matheron showed two (2) sets of eleven (11) data each but he did not apply Fisher's F-test to the variance of each set and the first variance terms of the ordered sets. He did not test for spatial dependence or count degrees of freedom. Neither did his disciples.
A - Interleaved sampling protocol, Sample selection
B - Interleaved sampling protocol, Sample preparation
C - Measures for variability and precision, Mined ore
D - Measures for variability and precision, Mill feed
E - Student's t-test for paired data
These appendices are part of 'Sampling
in Mineral Processing'