Lecture (3)

Lecture (3)Lecture (3)

Description of Central Description of Central TendencyTendency

Hydrological Records Hydrological Records

Population vs. Sample Notation

Population Vs Sample

World People Arabs

Infinite Record (i.e. very long)

selected year

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.

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

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

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

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

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

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

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

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

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


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.

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.

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.

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.

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)

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.

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

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.

Mode in a Histogram

1. Mode(s)2. Median3. Mean

0

2

46

810

1214

16

1820

1 2 3 4 5 6 7

Four Rules of Summation

naaaa ...

n

naan

i

1


nn XXXaaXaXaX ...... 2121

n

ii

n

ii XaaX

11


naXXX

aXaXaX

n

n

)...(

)(...)()(

21

21

naXaXn

ii

n

ii

11

)(


)...()...(

)(...)()(

2121

2211

nn

nn

YYYXXX

YXYXYX

n

ii

n

ii

n

iii YXYX

111

)(

Excel Application

• See Excel

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)

Geometric Mean

• Use GEOMEAN to calculate the geometric meanCell =GEOMEAN (number1, number2,

etc.)

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

Lecture (3)

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