International Journal of Scientific and Statistical Computing (IJSSC) Volume: 2 Issue: 1

---

INTERNATIONAL JOURNAL OF SCIENTIFIC
AND STATISTICAL COMPUTING (IJSSC)






VOLUME 2, ISSUE 1, 2011

EDITED BY
DR. NABEEL TAHIR








ISSN (Online): 2180-1339
International Journal of Scientific and Statistical Computing (IJSSC) is published both in
traditional paper form and in Internet. This journal is published at the website
http://www.cscjournals.org, maintained by Computer Science Journals (CSC Journals), Malaysia.


IJSSC Journal is a part of CSC Publishers
Computer Science Journals
http://www.cscjournals.org



INTERNATIONAL JOURNAL OF SCIENTIFIC AND STATISTICAL
COMPUTING (IJSSC)

---

Book: Volume 2, Issue 1, May 2011
Publishing Date: 31- 05 - 2011
ISSN (Online): 2180 -1339

This work is subjected to copyright. All rights are reserved whether the whole or
part of the material is concerned, specifically the rights of translation, reprinting,
re-use of illusions, recitation, broadcasting, reproduction on microfilms or in any
other way, and storage in data banks. Duplication of this publication of parts
thereof is permitted only under the provision of the copyright law 1965, in its
current version, and permission of use must always be obtained from CSC
Publishers.



IJSSC Journal is a part of CSC Publishers
http://www.cscjournals.org

(c) IJSSC Journal
Published in Malaysia

Typesetting: Camera-ready by author, data conversation by CSC Publishing Services - CSC Journals,
Malaysia



CSC Publishers, 2011

---

EDITORIAL PREFACE

The International Journal of Scientific and Statistical Computing (IJSSC) is an effective medium
for interchange of high quality theoretical and applied research in Scientific and Statistical
Computing from theoretical research to application development. This is the first issue of volume
first of IJSSC. International Journal of Scientific and Statistical Computing (IJSSC) aims to publish
research articles on numerical methods and techniques for scientific and statistical computation.
IJSSC publish original and high-quality articles that recognize statistical modeling as the general
framework for the application of statistical ideas.

The initial efforts helped to shape the editorial policy and to sharpen the focus of the journal.
Starting with volume 2, 2011, IJSSC appears in more focused issues. Besides normal
publications, IJSSC intend to organized special issues on more focused topics. Each special
issue will have a designated editor (editors) - either member of the editorial board or another
recognized specialist in the respective field.

This journal publishes new dissertations and state of the art research to target its readership that
not only includes researchers, industrialists and scientist but also advanced students and
practitioners. The aim of IJSSC is to publish research which is not only technically proficient, but
contains innovation or information for our international readers. In order to position IJSSC as one
of the top International journal in computer science and security, a group of highly valuable and
senior International scholars are serving its Editorial Board who ensures that each issue must
publish qualitative research articles from International research communities relevant to
Computer science and security fields.

IJSSC editors understand that how much it is important for authors and researchers to have their
work published with a minimum delay after submission of their papers. They also strongly believe
that the direct communication between the editors and authors are important for the welfare,
quality and wellbeing of the Journal and its readers. Therefore, all activities from paper
submission to paper publication are controlled through electronic systems that include electronic
submission, editorial panel and review system that ensures rapid decision with least delays in the
publication processes.

To build international reputation of IJSSC, we are disseminating the publication information
through Google Books, Google Scholar, Directory of Open Access Journals (DOAJ), Open J
Gate, ScientificCommons, Docstoc, Scribd, CiteSeerX and many more. Our International Editors
are working on establishing ISI listing and a good impact factor for IJSSC. I would like to remind
you that the success of the journal depends directly on the number of quality articles submitted
for review. Accordingly, I would like to request your participation by submitting quality manuscripts
for review and encouraging your colleagues to submit quality manuscripts for review. One of the
great benefits that IJSSC editors provide to the prospective authors is the mentoring nature of
the review process. IJSSC provides authors with high quality, helpful reviews that are shaped to
assist authors in improving their manuscripts.


Editorial Board Members
International Journal of Scientific and Statistical Computing (IJSSC)

---

EDITORIAL BOARD


EDITORIAL BOARD MEMBERS (EBMs)


Dr. De Ting Wu
Morehouse College
United States of America

---

TABLE OF CONTENTS




Volume 2, Issue 1, May 2011



Pages
1 - 15
Optimum Algorithm for Computing the Standardized Moments Using MATLAB 7.10(R2010a)
Karam Fayed

16 - 27
Application of Statistical Tool for Optimisation of Specific Cutting Energy and Surface
Roughness on Surface Grinding of Al-SiC35p Composites.
Dayanand Pai, Shreekantha Rao, Raviraj Shetty













International Journal of Scientific and Statistical Computing (IJSSC), Volume (2), Issue (1) : 2011

---

Karam A. Fayed

Optimum Algorithm for Computing the Standardized Moments
Using MATLAB 7.10(R2010a)


K.A.Fayed




karamfayed_1@hotmail.com
Ph.D.From Dept. of applied Mathematics and
Computing, Cranfield University, UK.
Faculty of commerce/Dept. of applied Statistics and Computing,
Port Said University, Port Fouad, Egypt.

Abstract

A fundamental task in many statistical analyses is to characterize the location and variability
of a data set. A further characterization of the data includes skewness and kurtosis. This
paper emphasizes the real time computational problem for generally the rth standardized
moments and specially for both skewness and kurtosis. It has therefore been important to
derive an optimum computational technique for the standardized moments. A new algorithm
has been designed for the evaluation of the standardized moments. The evaluation of error
analysis has been discussed. The new algorithm saved computational energy by
approximately 99.95%than that of the previously published algorithms.

Keywords:Statistical Toolbox, Mathematics, MATLAB Programming


1. INTRODUCTION
The formula used for Z -score appears in two virtually identical forms, recognizing the fact
that we may be dealing with sample statistics or population parameters. These formulae are
as follow:
x - x
z
i
=
i
Sample statistics (1)
s
x -
Z = i
i
Population statistics (2)

Where:

xi a row score to be standardized

n sample size
n
1
x =
x

i Sample mean
n i=1

Population mean
s Sample standard deviation

Population standard deviation
z Sample z score

Z Populationz score.

Subtracting the mean centers the distribution and dividing by the standard normalizes the
distribution. The interesting properties of Z score are that they have a zero mean (effect of
centering) and a variance and standard of one (effect of normalizing). We can use Z score to
compare samples coming from different distributions [1].


The most common and useful measure of dispersion is the standard deviation. The formula
for sample standard deviation is as follow:
International Journal of Scientific and Statistical Computing (IJSSC), Volume (2) : Issue (1) : 2011 1

---

Karam A. Fayed

n
1
s =
(x x 2) Sample standard deviation (3)
i -
n - 1 i=1
The population standard deviation is as follow:
n
1
=
(x
2
)
Population standard deviation (4)
i -
n i=1

2. MOMENTS
In statistics, the moments are a method of estimation of population parameters such as mean,
variance, skewness, and kurtosis from the sample moments.

a) Central Moments
Central moment is called moment about the mean. The central moments provide quantitative
indices for deviations of empirical distributions. The rthcentral is given by:
1 n
m
x
x
r
=
( - )r
i
n i=1
or :
1 n
m
x

r
=
( i - )r (5)
n i=1


Where:
m rth Sample and population central moments
r

b) Standardized Moment
The rth standardized moment in statistics is the rth central moment divided by r (standard
deviation raised to power r) as follow:
m r

=
r
r







(6)

Where:
rthstandardized moment
r
From Eq.(4), Eq.(5), & Eq.(6), We have:
r
1 ( x - )
1
r

=
=
( Z )
r
r
n

n
r
(1 / n ) ( x - )

m r
=
=
( (1 / n ) ( x
2
- )
)r
( m 2 )r
Therefore:
1
m
r
r

=
( Z )
=
r
(7)
n
( m 2 )r
Where:
m Second central moments
2

c) Computing Population Standardized Moments From Sample z Score
In the real world, finding the standard deviation of an entire population is unrealistic except in
certain cases such as standardized testing, where every element of a population is sampled.
In most cases, the standard deviation is estimated by examining a random sample taken from
the population as defined by eq.(3).
From eq.(5) & eq.(7), We have:
International Journal of Scientific and Statistical Computing (IJSSC), Volume (2) : Issue (1) : 2011 2

---

Karam A. Fayed

1
m
r
r

=
( Z ) =
r

n
( m 2 )r
r
1
( / n ) ( x - x )
=

( 1( / n) (x - x 2) )r

1
( / n ) ( x - x ) r
=

r / 2
r / 2
r
1
( / n )
( n - 1)
(
2
( x - x ) /( n - 1)) /2
1
( / n ) ( x -
r
x )
=
r / 2
(( n -
r
1) / n
/ 2
)
(S 2 )
1
/ 2
n
r
=


( x -
r
r
x ) / S
n n - 1

1
/ 2
n
r
=


r
z
n n - 1
Therefore:
1
2
n
rl

z
r
=


r (8)
n n - 1

Equation(8) represents the general equation for computing the rth standardized moments of
sample z-score.

d) Simplified Standardized Moments
rl 2
1 n
From eq.(8), the term

can be simplified using binomial theorem, since it
n n -1
can obtain the binomial series which is valid for any real number
as follow:

rl 2
1 n
The term

can be rewritten in the following form:
n n -1
rl 2
- rl 2
-rl 2
1 n
1 n - 1
1
1


=


=
1 -





n n - 1
n n
n
n

(10)
By replacing
and
we have:




For large values of ,we get:
International Journal of Scientific and Statistical Computing (IJSSC), Volume (2) : Issue (1) : 2011 3

---

Karam A. Fayed


Substituting Eq.(12) in eq.(8), we get:


Where:
rth simplified standardized moments.

e) Mathematical Formulae of Standardized and Simplified Moments
Using Eq.(8) & Eq.(13), we can get the following formulae:

Name
rth
Standardized moments
Simplified moments
Mean
1


Variance
2


Skewness
3


Kurtosis
4



f) Ratio Between Population and Sample z-Score
From Eq.(7) & Eq.(8), we can get the exact and simplified ratio of population and sample z-
score as follow:
Since:

We get:


And from Eq.(7) & Eq.(13), we can get:




Eq.(14) and Eq.(15) appear to be very dependent on the sample size. Therefore the ratio
between population and sample z-score(required for computing therth standardized moments)
depends on the sample size as given in Table_1. This table shows the variation. Figure_1
shows that the sample z score gets closer to population Z score. Therefore, computing
standardized moments using simplified technique is recommended for small sample size.


International Journal of Scientific and Statistical Computing (IJSSC), Volume (2) : Issue (1) : 2011 4

---