1. Elementary Statistical Methods
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Elementary statisticalElementary statistical
methodsmethodsDr.Dr. Arifudin Arifudin IdrusIdrus
Department of Geological EngineeringDepartment of Geological Engineering
GadjahGadjah MadaMada UniversityUniversity
EE--mail:mail: [email protected]@ugm.ac.id
GEOSTATISTICS COURSEGEOSTATISTICS COURSE
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Summary of formulaeSummary of formulae MeanMean
Variance Variance
Standard deviationStandard deviation
Coefficient of variationCoefficient of variation
CV =CV =
==
N
x x i∑=
1)(
2
2
−−= ∑ N x xS i
2S S =
MEAN
DEVIATION STANDARD
X
S
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Basic statisticsBasic statistics
DATA DATA
Mining data is usually presented in the formMining data is usually presented in the form
of drill hole data. A typical set of drillof drill hole data. A typical set of drill
hole data is usually of the basic form.hole data is usually of the basic form.
Data is of 2 types:Data is of 2 types:
unun--groupedgrouped
groupedgrouped
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UnUn--grouped data:grouped data:
Means that the data has not been ordered.Means that the data has not been ordered.
A frequency tally is one way of ordering un A frequency tally is one way of ordering un--grouped data.grouped data.
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Frequency polygonFrequency polygon
A plot of frequency A plot of frequency
against grade oragainst grade orrelative frequencyrelative frequencyagainst is calledagainst is calledfrequency polygonfrequency polygon
If the midpoints of theIf the midpoints of therectangles are joined,rectangles are joined,
we generate awe generate afrequency polygon.frequency polygon.
f
g
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Cumulative polygonCumulative polygonIf the cumulativeIf the cumulative
frequency or cumulativefrequency or cumulativeprecentprecent is plottedis plotted
against grade aagainst grade a
cumulative polygon iscumulative polygon isdeveloped.developed.
Cummulative frequency
Cummulative precent
g
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Some important summation resultsSome important summation results
∑∑ ∑ =− + +=+n
i
i
n
i
n
i
iii y x y x11 1
)(
∑∑ ∑ =− + −=−n
i
i
n
i
n
i
iii y x y x11 1
)(
∑ ∑− =n
i
ii xk kx1
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Double summationDouble summation
∑∑==
n
j
m
i 11
)()(
)(
242322141312
2
1
432
4
2
2
1
x x x x x x
x x x x
i
iii
j
ij
i
+++++=
= ∑∑∑===
∑∑=
n
ii
equalto1
eg
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DATA HANDLINGDATA HANDLING In analyzing data, methods must be used forIn analyzing data, methods must be used for
handling both grouped and unhandling both grouped and un--grouped datagrouped data The meanThe mean (grouped data)(grouped data)
x =x =
The medianThe median
the median is the midpoint of an assay ofthe median is the midpoint of an assay ofdata or it is the points above which anddata or it is the points above which andbelow which 50% of the scores fall.below which 50% of the scores fall.
nn
ΣΣfxfx
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Summary of some useful statisticalSummary of some useful statistical
formulaeformulae
unun--grouped datagrouped data
mean =mean =
variance (N large)variance (N large)
N
xi∑
22
2
2
2
2
2
2
1
)(
⎥
⎦
⎤⎢
⎣
⎡−=
−=
−
−=
∑∑
∑
∑
N
x
N
xS
x N xS
N
x xS
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Summary of some useful statisticalSummary of some useful statistical
formulaeformulae
Grouped dataGrouped data
MeanMean
Variance Variance
j
j
j x f N
∑= 1
1
/)
)(
1
1
2
11
2
12
2
1
2
−
−=
−
−
=
∑ ∑
∑=
N
N x f x f S
x x f
N
S
i
j
n
j
j
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CHOOSING THE BIN WIDTH OF A GAUSSIAN HISTOGRAM
= mean= mean
σσ = standard= standard deviationdeviationN N = number= number of valuesof values
being plottedbeing plotted
∆ ∆ xx = Bin= Bin widthwidth
X
⎟ ⎠
⎞⎜⎝
⎛ ⋅=∆ N x
20σ
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tandard deviation
A key parameter to be used when A key parameter to be used whendescribing the complexity of grade isdescribing the complexity of grade is
the coefficient of variation.the coefficient of variation.
CV =CV =
* CV is a function of volume * CV is a function of volume
MEANMEAN
STANDARD DEVIATIONSTANDARD DEVIATION
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Diagrammatic only, not to scale
L E S S
D I F F I C U
L T G R A
D E E S T I M A T E M
O R E
D I F F I C U L T
HOMOGENEITY LOWHIGH
H I G H P R O
P O R T I O N O F
O R E M I N E R A L L O W
NO DEPOSITS-CANNOT BE BOTHHIGH PROPORTION & HIGHLY VARIABLE
POSSIBLE LOW GRADEFINE GRAINED Au, Sn
EC
Fe
p
B
PC
Ni
SSn
VSn
V
E – EvaporateC – CoalFe – Bedded Iron OreP – Phospate
B – BauxitePb – Zn – Stratiform NickelNi – Stratiform NickelSSn – Stratiform TinPC – Porphyry Coppers
Vsn – Tin Veins V – Gold, Silver VeinsU - Uranium
(Mary Kathleen)Pb - ZnU
(CV
IncreasingMajorElements)
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COEFFICIENT OF VARIATION FOR ESSAY DATA FROMSELECTED ORE DEPOSITS
1.151.15
GetchellGetchell
mine, Nevadamine, Nevada
1.811.81Manganese depositManganese deposit
Phillipsburg district MontanaPhillipsburg district Montana
1.561.56Mt. View tungstenMt. View tungsten
prospect,prospect, HyderHyder , Alaska , Alaska
0.370.37MouatMouat MineMine
Stillwater andStillwater and
SweetgrassSweetgrass Countries, Mont.Countries, Mont.
1.551.55ShamvaShamva , , ShouternShoutern RhodesiaRhodesia
1.121.121.231.231.071.071.241.242137 vein2137 vein
FresnilloFresnillo minemine
ZacatecasZacatecas , Mexico , Mexico
0.850.850.570.571.121.122.242.24Brown veinBrown vein
Frisco Mine, Chihuahua, MexicoFrisco Mine, Chihuahua, Mexico
ZINCZINCLEADLEADSILVER SILVER GOLDGOLDMINEMINE
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COEFFICIENT OF VARIATION FOR SOME
TYPES OF GOLD DEPOSITS
1.0
1.2
1.5
2.0
3.0
4.0
PLACER
VEIN LIKE
CARLIN
LARGE FINE GRAINDEPOSITS:HOMESTAKEGETCHELL
MT. CHARLOTTE
(1m samples)
CV = SD/mean
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Simple distribution conceptsSimple distribution conceptsConsider the following histograms based on theConsider the following histograms based on the
normal distribution:normal distribution:
Frequency
Mean
Measure
Measure
ean
Frequency
Measureean
Frequency
A B
A
B
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Some important notes:Some important notes: Both histograms have the same meanBoth histograms have the same mean
gradegrade
Histograms B has a greater variance thanHistograms B has a greater variance than
histogram A histogram A
The area under both curves is 1The area under both curves is 1
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EksponentialEksponential distributiondistribution Y = f (x) = Y = f (x) =
MeanMean θθ
Variance Variance θθ22
Standard deviationStandard deviation θθ
C.V. = = 1.0C.V. = = 1.0
θ
θ
/1 xe
−
θ
θ
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To solve graphicallyTo solve graphically
( )( )
( )( )θ θ
θ θ
θ
θ
1ln
1ln
.,.
1lnln
)( /1
+−
=
−=
== −
x y
ei
x y
so
e x f y x Or Y = mx + B
Where Y = relative frequency/width of intervalx = grade
Plot ln Y vs xThe plot will give a line with:1. Negative slope2. Y intercept = 1/θ
3. Slope = -1/θ
Hence the mean and standard deviation, θ, of the exponential data can be foundFrom the Y intercept (i.e., x=0) or from the slope of the line.
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LOGNORMAL DISTRIBUTIONLOGNORMAL DISTRIBUTION
2/
2
2/
2
2
β
β α
γ µ
γ β
α
µ
e
e
=∴
==
=
= +
Mean of the logarithms of raw data
variance of the logarithms of raw data
medianα e=
Note: median corresponds to the 50th percentile of thecumulative frequency of the data
BUT
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To calculate β, the standard deviation of the logaritms, use the factthat the logarithms of the raw data are distributed normally
Mean130 130 x
For a normal distribution, 1 standard deviation is contained between the 16th
Dan 15th percentiles, and 1 standard deviation is contained between the 50th
And 84th percentiles.
HANCE:
[ ]
[ ]1684
16505084
lnln5.0
)ln(ln)ln(ln2/1
x x
x x x x
−=
−+−= β
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SichelSichel’ ’ ss estimatorestimator
2, β γ µ nF ⋅=
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For exampleFor example…….. Page 21Page 21
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