CONTENTSnguyenthong/download... · THE EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN, AND MODE For unimodal frequency curves that are moderately skewed (asymmetrical), we have the empirical

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Associ. Prof. Dr. NGUYEN Thong

INTRODUCTION OF ENVIRONMENTAL DESIGN

Chapter 5: Mean, variance and standard deviation.

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4/19/2016 1

Lecturer: Associ. Prof. Dr. NGUYỄN Thống

E-mail: [email protected] or [email protected]

Web: http://www4.hcmut.edu.vn/~nguyenthong/index

Tél. (08) 38 691 592 - 098 99 66 719

Ho Chi Minh City University of Technology Faculty of Civil Engineering – Department of Water Resources

Engineering & Management




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CONTENTS Chapter 1: Orientation. Evaluation of mathematical skill.

Chapter 2: Taylor series (1). Partial derivatives.

Chapter 3: Taylor series (2). Directional derivatives.

Chapter 4: Gradient vector. Engineering application.


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Chapter 6: Least square method.

Chapter 7: Correlation coefficient.

Chapter 8: Engineering applications.






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Course Goals

• Understand and describe the physical concepts and

mathematical treatment of Taylor series, partial

derivatives, directional derivatives, and gradient

vector.

• Understand and describe the physical concepts and

mathematical treatment of mean, variance, standard

deviation, least square method and correlation

coefficient.

• Apply the obtained skill to fundamental engineering

problems.




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Course Materials

Lecture Notes:

[1 ] M.YUHI AND Y. MAENO. Physical mathematics as an

engineering tool. Nakanishiya Pub. Co., 2004

Reference books:

[1] RAYMOND A. BERNETT et al. Applied Mathematics.

DELLEN PUBLISHING COMPANY, 1989.

[2] ROBERT WREDE, Ph.D et al. Theory and problems of

advanced calculus. Schaum’ouline. McGRAW-HILL, 2002.

[3] JAMES T. McCLAVE et al. Statistics for Businessand

Economics. MAXWELL MACMILLAN INTERNATIONAL

EDITIONS. 1990.



2




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Chapter 5:

Mean

Variance

Standard deviation

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MEAN




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AVERAGES, OR MEASURES OF

CENTRAL TENDENCY

Several types of averages can be defined, the

most common being the arithmetic mean, the

median, the mode, the geometric mean, and

the harmonic mean.

Each has advantages and disadvantages,

depending on the data and the intended

purpose.




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THE ARITHMETIC MEAN

The arithmetic mean, or briefly the mean, of a

set of N numbers X1, X2, X3, . . . , XN is

denoted by and is defined as:

Example: The arithmetic mean of the numbers 8,

3, 5, 12, and 10 is 38/5=7.6

X

N

X

N

X...XXXX

N

1j

j

N321

3




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If the numbers X1, X2, . . . , XK occur f1, f2, . . .

, fK times, respectively (i.e., occur with

frequencies f1, f2, . . . , fK ), the arithmetic

mean is:

Example: If 5, 8, 6, and 2 occur with frequencies 3, 2, 4,

and 1, respectively, the arithmetic mean is:

[3*5+2*8+….]/[3+2+4+1]=5.7

K

1i

i

K

1i

ii

K321

KK332211

f

Xf

f...fff

Xf...XfXfXfX




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THE WEIGHTED ARITHMETIC MEAN

Sometimes we associate with the numbers X1,

X2, . . . , XK certain weighting factors (or

weights) w1, w2, . . . , wK, depending on the

significance or importance attached to the

numbers. In this case,

is called the weighted arithmetic mean

K

1i

i

K

1i

ii

K321

KK332211

w

Xw

w...www

Xw...XwXwXwX




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PROPERTIE

The algebraic sum of the deviations of a

set of numbers from their arithmetic mean

is zero (8,3,5,12,10 with arithmetic mean

7.6).

[(8-7.6)+(3-7.6)+….]=0

The sum of the squares of the deviations

of a set of numbers Xj from any number a is

a minimum if and only if

Xa




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THE MEDIAN

The median of a set of numbers arranged in

order of magnitude (i.e., in an array) is either

the middle value or the arithmetic mean of the

two middle values.

The set of numbers 3, 4, 4, 5, 6, 8, 8, 8, and

10 has median 6

The set of numbers 5, 5, 7, 9, 11, 12, 15, and

18 has median 0.5(9+11)=10.

4




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THE MODE The mode of a set of numbers is that value

which occurs with the greatest frequency; that is,

it is the most common value. The mode may not

exist, and even if it does exist it may not be

unique.

The set 2, 2, 5, 7, 9, 9, 9, 10, 10, 11, 12, and 18 has

mode 9 (unimodal).

The set 3, 5, 8, 10, 12, 15, and 16 has no mode.

The set 2, 3, 4, 4, 4, 5, 5, 7, 7, 7, and 9 has two

modes, 4 and 7, and is called bimodal.




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THE EMPIRICAL RELATION BETWEEN THE MEAN,

MEDIAN, AND MODE

For unimodal frequency curves that are moderately

skewed (asymmetrical), we have the empirical

relation

Mean - mode = 3(mean - median)




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THE GEOMETRIC MEAN G

The geometric mean G of a set of N positive

numbers X1, X2, X3, . . . , XN is the Nth root of

the product of the numbers:

The geometric mean of the numbers 2, 4,

and 8 is:

NN21 x....x.xG

4648.4.2G 33

5




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THE HARMONIC MEAN H

The harmonic mean H of a set of N numbers

X1, X2, X3, . . . , XN is the reciprocal of the

arithmetic mean of the reciprocals of the

numbers:

N

1i i

N

1i i X

1

N

X

1

N

1

1H




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THE RELATION BETWEEN THE ARITHMETIC,

GEOMETRIC, AND HARMONIC MEANS

The geometric mean of a set of positive

numbers X1, X2, . . ., XN is less than or equal

to their arithmetic mean but is greater than or

equal to their harmonic mean. In symbols,

XGH




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THE ROOT MEAN SQUARE

The root mean square (RMS), or quadratic

mean, of a set of numbers X1, X2, . . ., XN is

defined by:

N

XRMS

2

i




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EXERCISES

1. Ten measurements of the diameter of a cylinder

were recorded by a scientist as 3.88, 4.09, 3.92,

3.97, 4.02, 3.95, 4.03, 3.92, 3.98, and 4.06

centimeters (cm). Find the arithmetic mean of the

measurements.

2. A student’s final grades in mathematics, physics,

English and hygiene are, respectively, 82, 86, 90,

and 70. If the respective credits received for these

courses are 3, 5, 3, and 1, determine an appropriate

average grade.

6




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EXERCISES

3. Find the mean, median, and mode for the sets

(a) 3, 5, 2, 6, 5, 9, 5, 2, 8, 6 and (b) 51.6, 48.7,

50.3, 49.5, 48.9.

4. Find (a) the geometric mean and (b) the

arithmetic mean of the numbers 3, 5, 6, 6, 7, 10,

and 12.

5. Find the harmonic mean H of the numbers 3,

5, 6, 6, 7, 10, and 12.




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VARIANCE &

STANDARD DEVIATION




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THE RANGE The range of a set of numbers is the difference

between the largest and smallest numbers in

the set.

EXAMPLE 1. The range of the set 2, 3, 3, 5, 5, 5,

8, 10, 12 is 12 � 2 ¼ 10. Sometimes the range is

given by simply quoting the smallest and largest

numbers; in the above set, for instance, the range

could be indicated as 2 to 12, or 2–12.




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THE MEAN DEVIATION

The mean deviation, or average deviation,

of a set of N numbers X1, X2, . . . , XN is

abbreviated MD and is defined by:

where is the arithmetic mean of the

numbers.

N

XX

MD

N

1j

j

X

7




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Variance (V), Standard deviation (σ)

N

2

i

i 1

x X

VN 1

N

2

i

i 1

x X

VN 1

j

222 2

i i1 1 2 2 j ji 1

n x Xn x X n x X ... n x X

VN 1 N 1

22 2

1 1 2 2 j jV p x X p x X ... p x X

ii

np

N 1

vôùi




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The degree to which numerical data tend to

spread about an average value is called

the dispersion, or variation, of the data:

V, higher more dispersion (more risk)

& vice versa.

0 X X




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Attention:

called ”experience”.

not « bias » (in Excel):

.

N

XxN,1i

2

i

1N

XxN,1i

2

i




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COEFFICIENT OF VARIATION CV

CV higher Research variable values far more dispersed the average value of the variables studied (high risk!)

.

XCV

8




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PARAMETER CALCULATED BY EXCEL

The Functions:

Variance: Var(address of

variable's value)

Standard deviation:

Stdev(address of variable's value)




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STANDARDIZED VARIABLE

The variable that measures the deviation from

the mean in units of the standard deviation is

called a standardized variable, is a

dimensionless quantity (i.e., is independent of

the units used), and is given by:

Properties:

ti dimensionless

Xxt i

i

1;0tit

i

i




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Exercise: Calculating the variance (V) and infer to

value of of standard deviation:

i ni x

i

1 3 14

2 2 11

3 3 12

4 3 7

V=7.80 =2.79 Associ. Prof. Dr. NGUYEN Thong



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i ni x

i

1 4 14

2 2 11

3 3 12

4 2 9

V=3.6 =1.90

9




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33



i ni x

i

1 1 14

2 2 11

3 3 12

4 1 10




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Exercise: The expected profit of two projects are following:

Calculating V1, V2 and standard deviation σ of two projects. Using the mean value and the variance, which project do you will choose? (V1=756 & V2=400). Chọn [1] or [2]?

Project Profit (billion đ) Probability p

[1] 90 0.3

30 0.7

[2] 60 0.5

20 0.5




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NOTE

Using ti max choose project 2.

74,1756

48Xt

1

11

0,2400

40Xt

2

22




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NORMAL

DISTRIBUTION

10




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The normal distribution is the most

widely known and used of all

distributions.

Because the normal distribution

approximates many natural

phenomena so well, it has developed

into a standard of reference for many

probability problems.




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The normal distribution is defined by

2 parameters:

Mean, µ

Standard derviation,

The function of probability density p is:

2

2

2

t

2e

2

1)t(p




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Characteristics of the Normal distribution

Symmetric

Continuous for all values of X between -

∞ and ∞ so that each conceivable interval

of real numbers has a probability other

than zero

-∞ ≤ X ≤ ∞

If we say X ∼ N(µ, σ2) we mean that X

is distributed N(µ, σ2).




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11




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NORMAL DISTRIBUTION N(, )




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Why is the normal distribution useful?

• Many things actually are normally distributed, or

very close to it. For example, height and

intelligence are approximately normally

distributed; measurement errors also often have a

normal distribution

• The normal distribution is easy to work with

mathematically. In many practical cases, the

methods developed using normal theory work

quite well even when the distribution is not

normal.




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Why is the normal distribution useful?

• There is a very strong connection between the

size of a sample N and the extent to which a

sampling distribution approaches the normal

form. Many sampling distributions based on

large N can be approximated by the normal

distribution even though the population

distribution itself is definitely not normal.




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NORMAL DISTRIBUTION N(0,1)

0

-2.5 -1.5 -0.5 0.5 1.5 2.5

p(t)

tt0

with Probability density

(symetric)

S1

S2

t1 t

2

2t

21

p(t) e2

t [ , ]

Mean Stdev,

12




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Property:

- Probability for t between t1 and t2: t1<t<t

2:

- Probability for t be superior t0: t>t0:

p(t)dt 1

2

1

t

1 2 1

t

p(t)dt Pr(t t t ) s

0

0 2

t

p(t)dt Pr(t t ) s




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- Symmetric :

- Property:

So that:

000 t);ttPr()ttPr(

000 t;1)ttPr()ttPr(

000 t;)ttPr(1)ttPr(




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TABLE OF N(0,1)




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BAÛNG TRA HAØM PHAÂN PHOÁI CHUAÅN N(0,1)

0

-2.5 -1.5 -0.5 0.5 1.5 2.5

p(t)

tt0

Vôùi t0 laø giaù trò >=0

Haøm maät ñoä

xaùc suaát

)ttPr( 0

13




Normal distribution t0 0 1 2 3 4 5 6 7 8 9

0 5000 4960 4920 4880 4840 4801 4761 4721 4681 4641

0.1 4602 4562 4522 4483 4443 4404 4364 4325 4686 4247

0.2 4207 4168 4129 4090 4052 4013 3974 3936 3897 3859

0.3 3821 3873 3745 3707 3669 3632 3594 3557 3520 3483

0.4 3446 3409 3372 3336 3300 3264 3228 3192 3156 3121

0.5 3085 3050 3015 2981 2946 2912 2877 2843 2810 2776

0.6 2743 2709 2676 2643 2611 2578 2546 2514 2483 2451

0.7 2420 2389 2358 2327 2296 2266 2236 2206 2217 2148

0.8 2119 2090 2061 2033 2005 1977 1949 1922 1894 1867

0.9 1841 1814 1788 1762 1736 1711 1685 1660 1635 1611

1.0 1587 1562 1539 1515 1492 1469 1446 1423 1401 1379

1.1 1357 1335 1314 1292 1271 1251 1230 1210 1190 1170

1.2 1151 1131 1112 1093 1075 1056 1038 1020 1003 985

1.3 968 951 934 918 901 885 869 853 838 823

1.4 808 793 778 764 749 735 721 708 694 681 Associ. Prof. Dr. NGUYEN Thong



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Probability so that t > t0=0.35

is:

=3632/10000=0.3632

Or, calculating t0 so that the

probability was defined ( has

given).

Example with =0.166 t0=0.97




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EXERCISES




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1. The variable X is distributed

Calculating the probability in the case :

a. x > 1.75

b. x < 1.50

c. 1.50 < x < 1.75

d. Calculating x1 so that Pr(x > x1) = 5%

e. Calculating x2 so that Pr(x < x2) = 5%

)1.0;6.1X(N

14




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2. The variable X is distributed as

N(50;3). Calculating x0 so that:

a.

b.

0 0Pr 50 x x 50 x 90%

%4.95x50xx50Pr 00




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END OF CHAPTER 5

Documents

CONTENTSnguyenthong/download... · THE EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN, AND MODE For unimodal frequency curves that are moderately skewed (asymmetrical), we have the empirical