1. Elementary Statistical Methods

8/16/2019 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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1. Elementary Statistical Methods

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