lecture (3) description of central tendency. hydrological records
TRANSCRIPT
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Lecture (3)Lecture (3)
Description of Central Description of Central TendencyTendency
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Hydrological Records 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 AveragesDifferent Types of Means or Averages
ArithmeticGeometricHarmonicQuadratic
Consider 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 MeanArithmetic Mean
1 2 31
1 1 2 2 3 31
1
1 1...
1 1...
n
n ii
k
k k i iki
ii
X = X X X X Xn n
and
x = f x f x f x f x f xn
f
x X
X
tX1
X2 Xi Xn
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The Short Cut MethodThe Short Cut Method
1
1
k
i i ji
k
ii
j
f x x
f
x x
Assume the mean is <x>=xjCalculate the deviation from the assumed mean, (xi-xj)
1
n
i ji
j
X X
n
X X
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Geometric MeanGeometric Mean
1 2 2 1
1 2 31
1 2 31
. . ...
. . ...
k
ik ii
nn n
g n ii
nff ff f fng n i
i
X = X X X X X
and
x = x x x x x
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Geometric Mean (cont.)Geometric Mean (cont.)
1
1
1 2 31
1log
log
1 1 2 2 3 3
1
1
1log
log
1 1log log log log ... log log
1log log log log ... log
1log log
n
ig i
k
i ig i
n
g n ii
XX n
g
g k nk
ii
k
g i ii
f xx n
g
X X X X X Xn n
X e e
and
x f x f x f x f xf
x f xn
x e e
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Harmonic MeanHarmonic Mean
11 2 3
1 2 3
31 2
11 2 3
1 1 1 1 1...
...
...
h n
in i
kh k
k i
ik i
n nX =
X X X X X
and
f f f f nx =
f f ff fx x x x x
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Quadratic Mean (Mean Square Value)Quadratic Mean (Mean Square Value)
2 2 2 21 2 3 2
1
2 2 2 21 1 2 2 3 3 2
11 2 3
1
... 1
... 1
...
nn
q ii
kk k
q i ikik
ii
X X X XX = X
n n
and
f x f x f x f xx = f x
f f f ff
X
tX1
X2 Xi Xn
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General Formula General Formula
1
1
For ungrouped data
1
For grouped data
1
nr
ri
i
kr
ri i
i
M Xn
m f xn
, for 1
, for -1
, for 2
, for 0
h h
q q
g g
M X m x r
M X m x r
M X m x r
M X m x r
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Applications and Limitations Applications and Limitations
0
20
40
60
80
100
120
140
1601-1-1978
1-3-1978
1-5-1978
1-7-1978
1-9-1978
1-11-1978
1-1-1979
1-3-1979
1-5-1979
1-7-1979
1-9-1979
1-11-1979
Series1
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Applications and Limitations (Cont.) Applications and Limitations (Cont.)
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Applications and Limitations (Cont.) Applications and Limitations (Cont.)
Flow parallel to the layers
Flow perpendicular to the layers
1
1h n
i i
nK
K
1
1 n
a ii
K Kn
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Applications and Limitations (Cont.) Applications and Limitations (Cont.)
22
1 1
1 1n n
i ii i
Y X Xn n
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 [(n+1)/2]th 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 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.
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
x
F(x)
f(x)
F(x) or f(x)
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Percentiles, Quartiles and Median Percentiles, Quartiles and Median (Cont.)(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 Graphical Representation
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
x
F(x)
f(x)
F(x) or f(x)
X_0.25
X_0.50 X_0.75
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ModeModeThe 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. Median3. Mean
0
2
46
810
1214
16
1820
1 2 3 4 5 6 7
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Four Rules of Summation
naaaa ...
n
naan
i
1
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Four Rules of Summation
nn XXXaaXaXaX ...... 2121
n
ii
n
ii XaaX
11
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Four Rules of Summation
naXXX
aXaXaX
n
n
)...(
)(...)()(
21
21
naXaXn
ii
n
ii
11
)(
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Four Rules of Summation
)...()...(
)(...)()(
2121
2211
nn
nn
YYYXXX
YXYXYX
n
ii
n
ii
n
iii YXYX
111
)(
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Excel Application
• See Excel
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Mean, Median, Mode
• Use AVERAGE or AVERAGEA to calculate the arithmetic meanCell =AVERAGE(number1, number2, etc.)
• Use MEDIAN to return the middle numberCell =MEDIAN(number1, number2, etc)
• Use MODE to return the most common valueCell =MODE(number1, number2, etc)
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Geometric Mean
• Use GEOMEAN to calculate the geometric meanCell =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