Geomath Lect12 Stats

8/10/2019 Geomath Lect12 Stats

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Probability and Statistics in Geology

Probability and statistics are an important aspect of Earth Science. Understanding the details, population of a data sample How rounded are these pebbles ? Where did they come from ? How likely is an earthquake here in Northridge ?



Probability and Statistics in Geology

Statistics Histograms Probability Error Analysis Regression ] Discuss ne t !ee"



Are Statistics Always Right ?Can They be Misleading ?

#oss a coin $ times....%Heads or tails & % 'hat is most unli"ely & 'hat is more li"ely &

.....si tails.....( heads and ( tails

So is H#H#H# more li"ely than ###### & ...)o, both aree*ually unli"ely+



Are Statistics Always Right ?Can They be Misleading ?

#he result ( heads and ( tails is more li"ely only because #here are many combinations !here this can occur

e.g. HH#H##, or H#H##H, or HHH###...- et/s try it...



What is a Statistic ?

0s this a statistic &

1 0n 2345, the oil refining capacity of 6elgium!as (7.$ million tonnes per year%

#his is actually, 8ust a fact 9 not a statistic



What is a Statistic ?

:onsider a pebbly beach Ho! could you determine the composition, mass, length,

shape of these particular pebbles & 'ould these si;es be the same on e<ery beach &



What is a Statistic ? - Specimen

et/s pic" up a pebble and loo" at it 9 this is a specimen #his pebble could probably gi<e us the composition

but !ould it be inclusi<e of all the pebbles & 0s it typical & Ho! could be impro<e this specimen &



What is a Statistic ? - Sample

'e could pic" up 255 pebbles, this is a sample from the beach #his should gi<e you a much better idea of your beach roc"s :ould !e do any better &



What is a Statistic ? - Population

=r !e could sample A the pebbles on the beach+ #his is the population of all pebbles )o! measure the composition, si;e, shape, of each 0s this a realistic plan &



What is a Statistic ? - Population

Specimen > =ne ob8ect Sample > A subset number of ob8ects Population > All the ob8ects

#hese terms are often misused in science and literature.



Faults in Southern California

Abo<e is a map of faults found in southern :alifornia 0f !e 8ust study the San ?acinto fault,!hat is this called statistically &0f !e study the system, San ?acinto, Elsinor, and San Andreas

!hat is this called statistcally &



So What is a Statistic ?

0s the a<erage mass of a pebble a statistic & #his depends on !hether this a<erage is determine

@rom a sample of pebbles or the total population ...

0f !e ta"e the a<erage of the total population 9 this considered a parameter and is no! a simple fact

#he a<erage of a sample , ho!e<er, is a statistic .



So What is a Statistic ?

A statistic is an attempt to estimate the a<erage mass

of all the pebbles by calculating the a<erage mass of some of the pebbles

Statistics are generally based on a sample of the population



lection Polls

Polling !uestion" # Who did the best $ob in the debate ?%=bama BCc:ain (5C

Estimates of <oter intentions obtained before an election are statistics ...a sample of the population



lection Polls=bama ($c:ain 2$7

#he final result of an election, ho!e<er, is an election parameter #he final result is a fact , a measure of the entire <oting population

=bama $$, 7,7(5c:ain ,(B(,$42



&ac' to the Pebbly &each Average, Mean, and Median

PebbleF ass g-2 (4B7 ( 3( (3

B ($B77B

$ 7 54 (4

(4$3 ((525 (25

#he typical mass of pebbles on a particular beach can be described by the mean , same as the a<erage -w

w = ()* Σ w ii + (

*

#he mean is the 1total mass of the sample% di<ided by #he 1number of pebbles% G 'hat is mean of these pebbles &




PebbleF ass g-2 777 7 5( (25B ((5

($B$ (4B4 (4$

(43 ( 325 (322 (33

Another !ay of finding the typical mass of pebbles

is to use the median <alue. Median means 1middle% and is the !eight of the middle Pebble if all are lined up ranked - from lightest to hea<iest. ou must ha<e an odd number of pebbles to get the median

0n the abo<e e ample, pebble F$ has a mass of (4B g !hich gi<es the median <alue of this pebble sample




'ill the median al!ays be the same as the mean &

'ith an e<en number of pebbles 255-, you can a<erage #he 5 th and 2 st pebbles.

PebbleF ass g-2 777 7 5( (25B ((5

($B$ (4B4 (4$

(43 ( 325 (322 (33



&ac' to the Pebbly &each- Dispersion

'hat about other aspects of the distribution of pebbles & Ho! can !e tell if the pebbles are similar in si;e i.e. !ell or poorly sorted-

PebbleF ass g-2 777 7 5( (25B ((5

($B$ (4B4 (4$

(43 ( 325 (322 (33

'e could gi<e the total range of si;es 9 "no!n as the dispersion 6ut ho! much does this tell us about all the sample pebbles &



&ac' to the Pebbly &each- Dispersion

#he hea<iest and lightest pebbles may not be 1typical% =ne !ay to get an accurate measure of ho! similar your

Pebbles are is to use the mean s*uare of the standard deviation

PebbleF ass g-2 777 7 5( (25B ((5

($B

$ (4B4 (4$

(43 ( 325 (322 (33

σ 7 I mass G w-7

#his measures the de<iation from the mean 9 also "no!n as the varianceG the bar indicates the a<erage of all calculations

#he standard deviation is thes*uare root of this <alue.



&ac' to the Pebbly &each- Dispersion PebbleF ass g-

2 777 7 5( (25B ((5

($B

$ (4B4 (4$(4

3 ( 325 (322 (33

σ 7 I mass G w-7

'hy do !e s*uare this difference & Some !ill be negati<e, !e 8ust !ant the de<iation of each @rom the a<erage <alue. 0f σ 7 is small 9 then the masses are similar and well sorted 0f σ 7 is large 9 then the masses are !idely <aryingand are poorly sorted



,isuali ing .istribution of .ata

Ho! can you display graphically the distribution of a large

number of pebbles &

'hich si;es occur most often & 'hich are fairly rare &



,isuali ing .istribution of .ata" /istogra0

A histogram displays the pebble mass count in bins 25 bins sho!n 'e first count the number of occurences frequency - in each binand list them in a table called the frequency distribution #hen plot this frequency as a bar chart against mass

Pebble mass (g)

F r e q u e n c y

Range(g) Number

200-235 1236-260 3261-285 7286-315 9316-335 16336-365 22366-385 19386-415 14416-435 6436-465 2

Frequency !s"r!bu"!#n



/istogra0s in Matlab 1or 2cta3e4

#o plot histograms in atlab>

JJ I 755>7 > 55 C set bin range and increment, here 7 JJ y I pebblefile >,7- C read column 7 of file of pebble masses JJ hist y, - C plots histogram sho!n abo<e

for data y- and bins -

Pebble mass (g)

F r e q u e n c y

PebbleF ass g-2 777 7 5

( (25B ((5($B

$ (4B4 (4$

(43 ( 325 (322 (33

$#un" #% all &ebbles



,isuali ing .istribution of .ata'ar!ne se!sm!c s"u y *eerara"ne e" al+ 2007

'e/re interested in earth*ua"e paths !hich come from e<ery possible a;imuth !ithin ($5 o the back azimuth -.

Ho! can !e graphically represent the distribution of cyclical data or direction &



,isuali ing .istribution of .ata" Rose .iagra0s

A rose diagram is li"e plottinga histogram on a polar graph.

#he direction is represented by#he angle around the plot and#he frequency is proportional#o distance from the center.

Here fre*uency ranges from

5 to $ and an angle of (5o

is themost fre*uent occuring $ times.

A list of fault dip angles could be plotted in this !ay.



Plotting Rose .iagra0s in Matlab 1or 2cta3e4

#o plot rose diagrams in atlab>

JJ dip I faultdipfile >,2- C reads first column of data input JJ dipradians I dip.Kpi.L2 5 C con<erts angles to radians JJ bins I 255 C specify the number of bins JJ rose dipradians,bins- C plot the rose diagram



Probability

'hat is Probability &

0f 0 measure a large number of data points, ho! often do 0 obtain a particular result &

Pebble mass (g)

F r e q u

e n c y

@or the pebbles masses measured here, the most probably mass is ( 5 grams

#his mass <alue occurs in77 fre*uency- out of 255 cases

or 77C of the time.

#hus the estimated probability of pic"ing up a pebble in this area !ith a mass of ( 5 grams is 77C.



Probability



Pebble mass (g)

F r e q u

e n c y



or 77C of the time.




Probability" What is *or0al ?

ou can compare your data distribution to theoretical estimates

#he most common distribution used is a normal distribution als"no!n as a Gaussian distribution.

Pebble mass (g)

P r # b a b ! l ! " y

Range(g) Number Pr#bab!l!"y

200-235 1 +01236-260 3 +03

261-285 7 +07286-315 9 +09316-335 16 +16336-365 22 +22366-385 19 +19386-415 14 +14416-435 6 +06436-465 2 +02

Frequency !s"r!bu"!#n , Pr#bab!l!"y



Gaussian .istribution

P x - I e MG x G x -7L7σ 7]

s*rt 7 πσ 7-

#he Naussian distribution is !ritten as abo<e and describes the relati<e probability of obtaining the <alue, x .

Here σ is the standard de<iation and x is the average of all x



x

P(x)


#his is a Naussian distribution for 5mean I .5 and σ I 7.5 ou are more li"ely to obtain a <alue bet!een BG$ !here the

graph is high And less li"ely to obtain a <alue bet!een 2G7, or 3G25

P x - I e MG x G x -7L7σ 7]

s*rt 7 πσ 7




'e can *uantify this by loo"ing at the area under the cur<e, thetotal area under the cur<e is 2.5

#he area under the cur<e bet!een 2 G 7 is sho!n in gray.

#his area is much smaller than the dar" gray bloc" bet!een B G 4.

x

P(x)




#he area under the cur<e bet!een (G4 is 5.$ ( and is termed 2.5this is "no!n as the $ C confidence limit-

#he area under the cur<e bet!een 2G3 is 5.3 B and is termed 2.0 σthis is "no!n as the 3 C confidence limit-

x

P ( x ) 2.5 σ

7.5 σ

#o *uantify these1areas% !e use

established <aluesfor multiples of the standard deviation from the mean



6inear Regression"/ow to Fit a 6ine to Scattered .ata

)o! that !e/<e learned

statistical analysis of asingle varia le

'e can also considerstatistical analysis of tworelated varia les .

'e may be able toappro imate thisrelationship by a straightline.

Ho! do !e find thisline & 'hich line is best &

Pebble diameter

D i s t a n c e

f r o m s h

o r e , m

-




#his gi<es you thede<iation of one pointfrom the line.#o obtain the mean

s*uare de<iation, !e ta"ethe a<erage of y for allpoints

'e calculate this usingthe same e*uation forstandard de<iation !hich!e used before.

Pebble diameter

D i s t a n c e

f r o m s h

o r e , m

-

∆ y

σ 7 I ∆ ! G∆ ! -7 #he line !ith the smallest σ !ill ha<e

the est fit to the data



(4


)o! that !e/<e learnedstatistical analysis of asingle varia le

'e can also considerstatistical analysis of tworelated varia les .

'e may be able toappro imate thisrelationship by a straightline.

Ho! do !e find thisline & 'hich line is best &

Pebble diameter

D i s t a n c e f r o m s h o r e

, m -



(E


#he line dra! to theright is one possibility.

Ho! can !e determine!hether this line is betterthan another 9 in a*uantitati<e !ay &

Pebble diameter

D i s t a n c e f r o m s h o r e

, m -

∆ y

'e can calculate themean square deviation byloo"ing the distance each

point is from the predictedline

#he de<ation of one point is sho!n by y and is estimated in the1y direction% only.

Geomath Lect12 Stats

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