Stochastic Modelling and Geostatistics

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Lecture (3)
Description of Central Tendency
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Hydrological Records
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Population vs. Sample Notation
Population Vs Sample
World People Arabs
Infinite Record (i.e. very long) selected year
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Different Types of Means or Averages
Arithmetic Geometric Harmonic Quadratic Conside
r a sample of n observations, X1, X2, , Xi, ,
Xn which can be grouped into k classes with
class marks x1,x2,, xi,, xk with corresponding
absolute frequencies, f1,f2,,fi,,fk.
 
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Arithmetic Mean
X
X1
Xn
Xi
X2
t
 
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The Short Cut Method
Assume the mean is ltxgtxj Calculate the deviation
from the assumed mean, (xi-xj)
 
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Geometric Mean
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Geometric Mean (cont.)
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Harmonic Mean
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Quadratic Mean (Mean Square Value)
X
X1
Xn
Xi
X2
t
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General Formula
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Applications and Limitations
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Applications and Limitations (Cont.)
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Applications and Limitations (Cont.)
Flow parallel to the layers
Flow perpendicular to the layers
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Applications and Limitations (Cont.)
Quadratic mean describes dispersion, spread or
scatter around the mean, and is known as the
standard deviation from the mean.
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Median

• Any value M for which at least 50 of all
  observations are at or above M and at least 50
  are at or below M.

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Median Estimation

• Order all observations from smallest to largest.
• If the number of observations is odd, it is the
  middle object, namely the (n1)/2th
  observation.
• For n 61, it is the 31st
• If the number of observations is even then, to
  get a unique value, take the average of the
  (n/2)th and the (n/2 1)th observation.
• For 60, it is the average of the 30th and the
  31st observation.

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The median has nice properties

• Easy to understand
• (½ data above, ½ data below)
• Resistant measure of central tendency (location)
  not affected by extreme (unusual) observations.

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Percentiles and Quartiles
In the cumulative distribution diagram, the range
is from 0 to 100. If this range is divided into
a hundred equal parts. The projection of these
parts on the x-axis are percentiles and denoted
by, X_0.01, X_0.02,, X_0.99.
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Percentiles, Quartiles and Median (Cont.)
The 25th and 75th percentiles correspond to the
first and third quartiles. Median (Xm) it is
the second quartile, X_0.50, divides the set of
observations into two numerically equal
groups. Median geometrically is the value that
divides the frequency histogram into two parts
having equal areas.
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Graphical Representation
X_0.50
X_0.75
X_0.25
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Mode
The mode is the variate that corresponds to the
largest ordinate of a frequency curve.
Frequency distributions can be described
as Uni-modal, bi-model, multi-model if it has
one, two or more modes.
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Mode in a Histogram

1. Mode(s)
2. Median
3. Mean

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Four Rules of Summation
n
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Four Rules of Summation
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Four Rules of Summation
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Four Rules of Summation
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Excel Application

• See Excel

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Mean, Median, Mode

• Use AVERAGE or AVERAGEA to calculate the
  arithmetic mean
• Cell AVERAGE(number1, number2, etc.)
• Use MEDIAN to return the middle number
• Cell MEDIAN(number1, number2, etc)
• Use MODE to return the most common value
• Cell MODE(number1, number2, etc)

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Geometric Mean

• Use GEOMEAN to calculate the geometric mean
• Cell GEOMEAN (number1, number2, etc.)

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Percentiles and Quartiles

• Use PERCENTILE to return the kth percentile of a
  data set
• Cell PERCENTILE(array, percentile)
• Percentile argument is a value between 0 and 1
• Use QUARTILE to return the given quartile of a
  data set
• Cell QUARTILE(array, quart)
• Quart is 1, 2, 3 or 4
• IQR Q3-Q1
• May return different values to statistical package