M 2008 Meyer Folleto Statistics and Data Analysis in Geology

7/28/2019 M 2008 Meyer Folleto Statistics and Data Analysis in Geology

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Franz MeyerStatistics & Data Analysis in Geology 1

Dr. Franz J Meyer

Earth and Planetary Remote Sensing,University of Alaska Fairbanks

Statistics and Data Analysis in Geology 6. Normal Distribution

probability plots

central limits theorem


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Normal Distribution


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Normal Distribution

An Enormously Important Distribution

The normal distribution is the most commonly used distribution in statistics

Partly this is due to the fact that the normal distribution is a

reasonable description of

many processes from industrial processes to intelligence test scores

Also, under specific conditions one can assume that sampling distributions are normallydistributed even if the samples are drawn from populations that are not

normally

distributed (this is discussed further when we talk about the Central Limits Theorem)

The normal distribution is also referred to as bell curve

and you see a few examples

below

There are an infinite number of normal

distributions that differ according to their

mean () and variance (2)

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Almost all natural processes follow the normal

distribution

The shape of a Normal distribution corresponds

to a binomial distribution with p = 0.5 (compareto coin toss example of lecture 5)

As N becomes large, the function becomes

continuous and can be represented by the

following equation

it also can be thought of as

for p = 0.5

A normal distribution can be characterized by

only two parameters,

and


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Normal Distribution

The Standard Normal Distribution or Z Distribution

It is often useful to standardize the

variables so that populations can be

compared. Standardization meansthat the mean, , = 0 and thestandard deviation , = 1

Then the equation becomes:

and the curve is expressed in numbers

of standard deviations from the mean


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Normal Distribution


So you convert the normal distribution to the Z distribution by converting the

original values to standard scores, which allows comparison among populations

with different means and variances

Thats interesting as all normal distributions share the following characteristics:

Symmetry

Unimodality

Continuous range from -

to +

A total area under the curve of 1

A common values for the mean, median, and mode

We can make some assumptions about

how the data is distributed within any

normal distribution

About 68% of the data fall within 1

About 95% of all data fall within 2

About 99.5% of all data fall within 3

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Normal Distribution


Standardization of

normal random

variables

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Normal Distribution


For any sample, the way to standardize the data is called Z-transformation.

For every point we calculate a Z-score, which is really a measure ofhow many

standard deviations a point is from the mean.

depending on if you are dealing with a sample

or population. Z scores can be positive or negative.SXXZXZ ii

ii == or


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Normal Distribution


For any sample, the way to standardize the data is called Z-transformation.

For every point we calculate a Z-score, which is really a measure ofhow many

standard deviations a point is from the mean.

depending on if you are dealing with a sample

or population. Z scores can be positive or negative.

Example:

A shell specimen with a value of 12 mm (X = 12) is drawn from a population with

= 10,

= 2. What is that samples Z score?

or the sample is one standard deviation longer than the

mean

What if that same sample is drawn from a population with

= 10,

= 1 (Same mean

different variance)?

In absolute terms the specimen is the same distance from the mean, however relative to

the population as a whole, it is further away (more anomalous).

SXXZXZ ii

ii == or

( ) 12221012 ===Z

( ) 21211012 ===Z


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Normal Distribution


Example cont.:

What if a different specimen (X = 14) is drawn from the population in example 1 with

=

10,

= 2?

So this sample is in the same position relative to the population as that from example 2.

( ) 22421014 ===Z

Z score

4 6 8 10 12 14

16 mm


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Normal Distribution


For each normal distribution, the area

under the curve is equal to 1. That is,

the total probability is equal to 1 (as it

was with the binomial distribution).

Mathematically we can express this as:

For Z-transformed data this is:

+

=1)( dXXf

+

== 12)(21)(

2

dxXdxXf e


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Normal Distribution


Similarly, we can calculate the probability of a sample

as being less than or equal to some preset value Z as

Z

dxX

e

2)(

2

1 2

A different way to represent the

normal distribution is by Cumulative

Probability: They are plots of the

area under the curve versus X.They can be made for any

distribution. These types of plots are

called OGIVE PLOTS, and I will

come back to them later.

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Normal Distribution

For the normal distribution, it is a

pain in the neck to calculate this

integral for every problem that we

are going to do, so tables have

been constructed.

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Normal Distribution


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Normal Distribution

The numbers in the table below

are answers to the question:

What is the Z value

corresponding to a particular

area under the curve?


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Normal Distribution

Example of Cumulated Probability

Grades of chip samples from a body of ore

have a normal distribution with a mean of 12%

() and a standard deviation of 1.6 % ().

(curve to the right helps to visualize the

distribution)

Problem 1: Find the probability of a specimen of

15% or less

Calculate Z score

(15-12)/1.6 = +1.88

The chart on slide 13 gives cumulative probability

from very small (minus infinity) to the value:

+1.88 = 0.97 (we have to interpolate between +1.8

and +1.9)

Make a sketch to see if this makes sense

So the probability of finding a sample with

less than 15% ore is 97%


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Normal Distribution


Problem 2: What is the probability of finding ore

greater than 14%?

Z = (14-12)/1.6 = +1.25

the probability associated with this Z score is

0.895. This is the probability of 14% or less.

The probability of 14% or more is 1

0.895 or

0.105

So the probability of finding a sample morethan 14% ore is 10.5%


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Normal Distribution


Problem 3: What is the probability of finding ore

grade of less than 8%?

Z = (8-12)/1.6 = -2.5

the probability associated with this Z score is

0.0062

So the probability of finding a sample less

than 8% ore is 0.62%, not very likely


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Normal Distribution


Problem 4: What is the probability of a sample

being between 8% and 15%?

Calculate the Z scores for each value:

Z8

= (8-12)/1.6 = -2.5 --> 0.62%

Z15

= (15-12)/1.6 = 1.88 --> 97%

Subtract the smaller from the larger:

97

0.62 = 96.38%, so about 96% or all

samples fall in that range.


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Normal Distribution


area under the curve

1

0.8413 -

0.1587 = 0.6826

68%

2

0.9773 -

0.0228 = 0.9545

95.5%

1.96

= 95%

3

0.9987-

0.00140 = 0.9973

99%


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Normal Distribution

The Central Limits Theorem


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Normal Distribution


If you draw a number of samples from a normal distribution population, we find

that the sample means will form a normal distribution

BUT we don't always know the distribution of the population

Central Limits Theorem:

CLT states that independent of their original statistical distribution, the re-averaged sumof a sufficiently large number of identically distributed independent

random variables

will

be approximately normally distributed.

In other words, ifsufficiently large sets

of random samples are taken from any

population, and the means are calculated for those samples, then these sample

means

will tend to be normally distributed.


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Normal Distribution


Central Limits Theorem:

Again in other words: if we take all possible samples of size n from any population with a mean of

and a standard deviation of, the distribution of sample means will have:

mean of also written as

Standard deviation of means,

This is also called the standard error of the mean, se

will be normally distribution when the parent population is normal

will approach a normal distribution as N approaches infinity regardless of the distribution of the

parent population.

=X =XX

nsX

=


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Normal Distribution


1 2 4 25


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Normal Distribution


Some animated examples:

Uniform distribution:

Log-normal distribution:

Parabolic distributions:

http://www.

statisticalengineering.com/central_limit_the

orem.htm


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This means, if we average enough we can always reduce data of unknownstatistics to data of known properties.

Practically, we can use our Z-statistic

useful when we want to infer something from single values taken from a normalpopulation (Xi

drawn from population)

and adapt it for CLT for a sample of size N drawn from a population with known

mean and standard deviation.

You can see that equation (1) is the same as (2) if n = 1 (a single sample)

So both equations are just more specific forms of the general equation

se

is the standard deviation of means =

Normal Distribution


= i

i

XZ

n

XZ/1

=

es

XZ

=

n/1

(1)

(2)


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Normal Distribution


For the example from earlier:

A sample with a value of 14% (X = 14) is

drawn from a population with

= 12,

=

1.6. What is the probability of finding a

single sample equal to or greater than 14%

ore? First calculate that samples Z score.

Or the probability of finding one such

sample or greater was about 10.5%.

25.16.12

6.11214 ===Z


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Normal Distribution


For the example from earlier:

Now, what if we selected 4 samples (n = 4)

and the mean of those specimens was

14%?

And the probability of finding four such

specimens is less, in fact it is only 0.62%!!!

5.28.0

2

)2/1(6.1

2

4/16.1

1214==

=

=Z

M 2008 Meyer Folleto Statistics and Data Analysis in Geology

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