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 ?
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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 next week
Are Statistics Always Right ? Can They be Misleading ?
Toss a coin 6 times....”Heads or tails ? ” What is most unlikely ? What is more likely ?
.....six tails.....3 heads and 3 tails
So is HTHTHT more likely than TTTTTT ? ...No, both areequally unlikely!
Are Statistics Always Right ? Can They be Misleading ?
The result 3 heads and 3 tails is more likely only because There are many combinations where this can occur (e.g. HHTHTT, or HTHTTH, or HHHTTT...) Let's try it...
What is a Statistic ?
Is this a statistic ?
“ In 1970, the oil refining capacity of Belgium was 32.6 million tonnes per year”
This is actually, just a fact – not a statistic
What is a Statistic ?
Consider a pebbly beach How could you determine the composition, mass, length, shape of these particular pebbles ? Would these sizes be the same on every beach ?
What is a Statistic ? - Specimen
Let's pick up a pebble and look at it – this is a specimen This pebble could probably give us the composition but would it be inclusive of all the pebbles ? Is it typical ? How could be improve this specimen ?
What is a Statistic ? - Sample
We could pick up 100 pebbles, this is a sample from the beach This should give you a much better idea of your beach rocks Could we do any better ?
What is a Statistic ? - Population
Or we could sample ALL the pebbles on the beach! This is the population of all pebbles Now measure the composition, size, shape, of each Is this a realistic plan ?
What is a Statistic ? - Population
Specimen: One object Sample: A subset number of objects Population: All the objects
These terms are often misused in science and literature.
Faults in Southern California
Above is a map of faults found in southern California If we just study the San Jacinto fault, what is this called statistically ?If we study the system, San Jacinto, Elsinor, and San Andreas what is this called statistcally ?
So What is a Statistic ?
Is the average mass of a pebble a statistic ? This depends on whether this average is determine From a sample of pebbles or the total population...
If we take the average of the total population – this considered a parameter and is now a simple fact The average of a sample, however, is a statistic.
So What is a Statistic ?
A statistic is an attempt to estimate the average mass of all the pebbles by calculating the average mass of some of the pebbles
Statistics are generally based on a sample of the population
Election Polls
Polling question: “ Who did the best job in the debate ?”Obama 54%McCain 30%
Estimates of voter intentions obtained before an election are statistics...a sample of the population
Election PollsObama 365McCain 162
The final result of an election, however, is an election parameter The final result is a fact, a measure of the entire voting population
Another way of finding the typical mass of pebbles is to use the median value. Median means “middle” and is the weight of the middle Pebble if all are lined up (ranked) from lightest to heaviest. You must have an odd number of pebbles to get the median
In the above example, pebble #6 has a mass of 374 g which gives the median value of this pebble sample
Back to the Pebbly BeachAverage, Mean, and Median
Will the median always be the same as the mean ?
With an even number of pebbles (100), you can average The 50th and 51st pebbles.
We could give the total range of sizes – known as the dispersion But how much does this tell us about all the sample pebbles ?
Back to the Pebbly Beach- Dispersion
The heaviest and lightest pebbles may not be “typical” One way to get an accurate measure of how similar your Pebbles are is to use the mean square of the standard deviation
Why do we square this difference ? Some will be negative, we just want the deviation of each From the average value. If 2 is small – then the masses are similar and well sorted If 2 is large – then the masses are widely varying and are poorly sorted
Visualizing Distribution of Data
How can you display graphically the distribution of a large number of pebbles ?
Which sizes occur most often ? Which are fairly rare ?
Visualizing Distribution of Data: Histogram
A histogram displays the pebble mass count in bins (10 bins shown) We first count the number of occurences (frequency) in each bin and list them in a table called the frequency distribution Then plot this frequency as a bar chart against mass
>> x = 200:25:500 % set bin range and increment, here 25 >> y = pebblefile(:,2) % read column 2 of file of pebble masses >> hist(y,x) % plots histogram shown above
Visualizing Distribution of DataMarine seismic study, Weeraratne et al., 2007
We're interested in earthquake paths which come from every possible azimuth within 360o (the back azimuth).
How can we graphically represent the distribution of cyclical data or direction ?
Visualizing Distribution of Data: Rose Diagrams
A rose diagram is like plottinga histogram on a polar graph.
The direction is represented byThe angle around the plot andThe frequency is proportionalTo distance from the center.
Here frequency ranges from0 to 6 and an angle of 30o is the most frequent occuring 6 times.
A list of fault dip angles could be plotted in this way.
Plotting Rose Diagrams in Matlab (or Octave)
To plot rose diagrams in Matlab:
>> dip = faultdipfile(:,1) % reads first column of data input >> dipradians = dip.*pi./180 % converts angles to radians >> bins = 100 % specify the number of bins >> rose(dipradians,bins) % plot the rose diagram
Probability What is Probability ?
If I measure a large number of data points, how often do I obtain a particular result ?
Pebble mass (g)
Fre
quen
cy
For the pebbles masses measured here, the most probably mass is 350 grams
This mass value occurs in 22 (frequency) out of 100 cases or 22% of the time.
Thus the estimated probability of picking up a pebble in this area with a mass of 350 grams is 22%.
Probability What is Probability ?
If I measure a large number of data points, how often do I obtain a particular result ?
Pebble mass (g)
Fre
quen
cy
For the pebbles masses measured here, the most probably mass is 350 grams
This mass value occurs in 22 (frequency) out of 100 cases or 22% of the time.
Thus the estimated probability of picking up a pebble in this area with a mass of 350 grams is 22%.
Probability What is Probability ?
If I measure a large number of data points, how often do I obtain a particular result ?
Pebble mass (g)
Fre
quen
cy
For the pebbles masses measured here, the most probably mass is 350 grams
This mass value occurs in 22 (frequency) out of 100 cases or 22% of the time.
Thus the estimated probability of picking up a pebble in this area with a mass of 350 grams is 22%.
Probability
We can then add another column to the data which shows the probability for each bin size