8/3/2019 Stat 1 Report F 202

1/25

TERM PAPEROn

Dispersion:A statistical tool

Department of FinanceUniversity of Dhaka

8/3/2019 Stat 1 Report F 202

2/25

Date of Submission: July 24, 2010

Dr. M. Khairul HossainProfessorDepartment of FinanceUniversity of Dhaka

GROUP-2Apel Mahmood Rifat 15-007Sumaiya Amena 15-051Shakira Mahzabeen 15-085Khairul Bashar 15-153

Submitted To

Submitted By

8/3/2019 Stat 1 Report F 202

3/25

Letter of TransmittalJuly 24, 2010.Dr. M. Khairul Hossain

Professor

Department of Finance

University of Dhaka.

Subject: Submission of report named Dispersion: A statistical tool

Dear Sir,

We take the pleasure to inform you that, we are going to submit the report that you had

assigned us as a partial requirement for the course Business Statistics I (F-202)

The report is prepared on Dispersion: A statistical tool.

We sincerely hope that, you will enjoy going through this report, as we have felt great

pleasure to prepare it. If any other information is required for further clarification, we will bepleased to provide you with that.

We are thanking you heartily. We tried our best to make this report the best one. We think

this report can serve us all as a means of tool for solving business decision problems.

Finally, we would like to thank you for providing us the opportunity to work in such an

interesting and enthusiastic report as we have enjoyed as well as learned a lot in preparing

this report.

Sincerely,

Apel Mahmood Rifat

On behalf of group 2

15th

batch, Section-A

Department of Finance

University of Dhaka

8/3/2019 Stat 1 Report F 202

4/25

AcknowledgementFor the completion of this study we cant deserve all praise. There were a lot of people whohelped us by providing valuable information, advice and guidance for the completion of this

report in the scheduled time.

Course report is an essential part of BBA program as one can gather practical knowledge

within the short period of time by observing and doing the works of chosen topic. In this

regard our report has been arranged on Dispersion.

At first we like to pay our thanks to almighty Allah, for helping us to do all the works with

perfection.

We would like to pay our gratitude to our supervising course teacher Prof. Dr. M. KhairulHossain who instruct us in the right way and give us proper guidelines for preparing this

report.

At last we must mention the wonderful working environment and group commitment that has

enabled a lot deal to do and observe the process during our time.

8/3/2019 Stat 1 Report F 202

5/25

Table of Content

SerialNo.

Subject Page No.

01. Executive Summary 0702. Introduction 0803. Macroenvironmental Forces 1104. Pharmaceuticals Industry 1905. Square Pharmaceuticals Ltd. 2306. Impact of Macroenvironment in Square Pharma 3207. Beximco Pharmaceuticals Ltd. 3708. Macro environmental factors affecting Beximco Parma to

launch a new product

48

09. Findings 5810. Conclusion 5911. Reference 6012. Bibliography 61

8/3/2019 Stat 1 Report F 202

6/25

Introduction

Origin of the report

This report is generated under the academic supervision of our course teacher Prof. Dr. M.Khairul Hossain, Department of Finance, University of Dhaka. This report is prepared as the

requirement of Business statistics course. The topic is Dispersion: A statistical tool.

Methodology

The methodology of the report is inductive. The report is based on secondary information.

Secondary Information:The secondary sources of data are different reference books,

website etc.

Key Parts of the report

The main view of the report is to discuss Dispersion, as a statistical tool. Different measures

of dispersion and their use is discussed in this report.

Objectives of the report

Broad Objectives:The main objective of the study is to evaluate the impact of macro-

environment forces in decision making of launching new product in pharmaceutical industry.

Specific Objectives:

To be acquainted with the Pharmaceutical industry

To learn clear knowledge of macro-environment forces

To learn about new product launching process of Pharmaceutical industry

To have the practical knowledge of theoretical knowledge of Marketing theory

ScopeIn this report, at first we cover the preliminary concept of Dispersion. Then we go for the

classification of dispersion factors on launching a new product of a pharmaceuticals

company.

LimitationsThere were certain limitations of the problem we face in report preparing.

Unavoidable conditions:

Some of the unavoidable conditions also had a deterring effect on preparing thereport.

8/3/2019 Stat 1 Report F 202

7/25

Restrictions that we faced:

Lack of information, lack of technology etc. are the restrictions within the problem.

Absence of some information regarding data compilation:

While making the survey for data collection, we have faced problems. Some of the

information was really essential was hard to collect.

8/3/2019 Stat 1 Report F 202

8/25

Introduction

While measures of central tendency are used to estimate "normal" values of a dataset,

measures of dispersion are important for describing the spread of the data, or its variation

around a central value. Two distinct samples may have the same mean or median, but

completely different levels of variability, or vice versa. A proper description of a set of data

should include both of these characteristics. There are various methods that can be used to

measure the dispersion of a dataset, each with its own set of advantages and disadvantages.

In statistics, statistical dispersion (also called statistical variability or variation) is variability

or spread in a variable or a probability distribution. Common examples of measures of

statistical dispersion are the variance, standard deviation and inter quartile range.

Measures of dispersion express quantitatively the degree of variation or dispersion of values

in a population or in a sample. Along with measures ofcentral tendency, measures of

dispersion are widely used in practice as descriptive statistics. Some measures of dispersionare the standard deviation, the average deviation, the range, the interquartile range.

For example, the dispersion in the sample of 5 values (98,99,100,101,102) is smaller than the

dispersion in the sample (80,90,100,110,120), although both samples have the same central

location - "100", as measured by, say, the mean or the median . Most measures of dispersion

would be 10 times greater for the second sample than for the first one (although the values

themselves may be different for different measures of dispersion).

Dispersion is contrasted with location or central tendency, and together they are the most

used properties of distributions.

A measure of statistical dispersion is a real number that is zero if all the data are identical,

and increases as the data becomes more diverse. It cannot be less than zero.

Most measures of dispersion have the same scale as the quantity being measured. In other

words, if the measurements have units, such as meters or seconds, the measure of dispersion

has the same units.

http://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Variable_%28mathematics%29http://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Interquartile_rangehttp://www.statistics.com/resources/glossary/p/population.phphttp://www.statistics.com/resources/glossary/s/sample.phphttp://www.statistics.com/resources/glossary/c/centralt.phphttp://www.statistics.com/resources/glossary/d/descstats.phphttp://www.statistics.com/resources/glossary/s/standev.phphttp://www.statistics.com/resources/glossary/a/avgdev.phphttp://www.statistics.com/resources/glossary/r/range.phphttp://www.statistics.com/resources/glossary/i/intrqrtrng.phphttp://www.statistics.com/resources/glossary/m/mean.phphttp://www.statistics.com/resources/glossary/m/median.phphttp://en.wikipedia.org/wiki/Central_tendencyhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Units_of_measurementhttp://en.wikipedia.org/wiki/Units_of_measurementhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Central_tendencyhttp://www.statistics.com/resources/glossary/m/median.phphttp://www.statistics.com/resources/glossary/m/mean.phphttp://www.statistics.com/resources/glossary/i/intrqrtrng.phphttp://www.statistics.com/resources/glossary/r/range.phphttp://www.statistics.com/resources/glossary/a/avgdev.phphttp://www.statistics.com/resources/glossary/s/standev.phphttp://www.statistics.com/resources/glossary/d/descstats.phphttp://www.statistics.com/resources/glossary/c/centralt.phphttp://www.statistics.com/resources/glossary/s/sample.phphttp://www.statistics.com/resources/glossary/p/population.phphttp://en.wikipedia.org/wiki/Interquartile_rangehttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Variable_%28mathematics%29http://en.wikipedia.org/wiki/Statistics

8/3/2019 Stat 1 Report F 202

9/25

Importance

A study of dispersion enables us to get additional information about the composition of data.

Confining mean will not provide us this vital information.

Central tendency will only give information on the location of the data. Dispersion defines

the spread of the data. In addition, shape should also be part of the defining criteria of data.

So, dispersion describes location, spread & shape as best measures to define data.

Two different set of data can have different mean but same variability. On the other hand two

set of data can have same mean but different variability.

Shape A and B has the same mean but different variability

Curve ACurve B

Curve B

Curve A

Curve A and B have different mean but same variability.

8/3/2019 Stat 1 Report F 202

10/25

Variability or variation is something connected with human life and study is very important

for mankind. The total area of the earth may not be very important to a research minded

person but the area under different crops, area covered by forests, area covered by residential

and commercial buildings are figures of great importance because these figures keep on

changing form time to time and from place to place. Very large number of experts is engaged

in the study of changing phenomenon. Experts working in different countries of the worldkeep a watch on forces which are responsible for bringing changes in the fields of human

interest. The agricultural, industrial and mineral production and their transportation from one

part to the other parts of the world are the matters of great interest to the economists,

statisticians, and other experts. The changes in human population, the changes in standard

living, and changes in literacy rate and the changes in price attract the experts to make

detailed studies about them and then correlate these changes with the human life. Thus

variability or variation is something connected with human life and study is very important

for mankind.

The study of dispersion is very important in statistical data. Like-

Test the reliability of an average

Control the variability

Compare two or more sets of data with respect of their variability

Facilitate the use of other statistical techniques

If in a certain factory there is consistence in the wages of workers, the workers will besatisfied. But if some workers have high wages and some have low wages, there will be

unrest among the low paid workers and they might go on strikes and arrange demonstrations.

If in a certain country some people are very poor and some are very high rich, we say there is

economic disparity. It means that dispersion is large.

The idea of dispersion is important in the study of wages of workers, prices of commodities,

standard of living of different people, distribution of wealth, distribution of land among

framers and various other fields of life.

Measures of dispersion are known as averages of the second order because they indicate the

average deviation of individual observations from the mean.

Measures of dispersion can be described from two perspectives. They are:-

1. Absolute form

2. Relative form

A graphical representation is in the following:-

8/3/2019 Stat 1 Report F 202

11/25

Range: Considering the several measures of dispersion, the range is the first measure of the absolute

form. The range is based on the largest and the smallest values in the data set. It is known as the

simplest measure of dispersion. However, the range only provides information about the maximum

and minimum values and does not say anything about the values in between. It isthe difference

between the largest and the smallest values in a data set. In the form of an equation, after re-

arranging the data, it will be like this:

The range is widely used in statistical process control (SPC) applications because it is very easy to

calculate and understand.

Quartile Deviation: The quartile deviation is half the difference between the upper and lower

quartiles in a distribution. It is a measure of the spread through the middle half of a distribution. It

can be useful because it is not influenced by extremely high or extremely low scores. Quartile

Deviation is an ordinal statistic and is most often used in conjunction with the median. The formula

to calculate quartile deviation is:

Where, QD = Quartile Deviation

Q3 = Third Quartile

Q1 = First Quartile

Measures ofDispersion

AbsoulteForm

RangeQuartile

DeviationMean

DeviationStandardDeviation

RelativeForm

Coefficientof Range

Coefficientof QuartileDeviation

Coefficientof MeanDeviation

Coefficientof

Variation

Range = Largest value Smallest value

8/3/2019 Stat 1 Report F 202

12/25

Mean Deviation: A defect of the range is that it is based on only two values, the highest and the

lowest. It does not take into consideration all of the values. The mean deviation does. It measures

the mean amount by which the values in a population or sample vary from their mean. In terms of a

definition, mean deviation is the arithmetic mean of the absolute values of the deviations from the

arithmetic mean. The formula is:-

MD = l X X l

n

Where, X is the value of each observation

X is the arithmetic mean of the values

n is the number of observations in the sample

ll indicates the absolute value

Standard Deviation:The variance and the standard deviation are also based on the deviationsfrom the mean. However, instead of using the absolute value of the deviations, the variance and the

standard deviation square the deviations.

Features of standard deviation are as follows:

The standard deviation is the square root of the sample variance. Defined so that it can be used to make inferences about the population variance.

Calculated using the formula:

The values computed in the squared term, x i x bar, are anomalies, which is discussed in

another section

Not restricted to large sample data sets, compared to the root mean square anomaly

http://iridl.ldeo.columbia.edu/dochelp/StatTutorial/Climatologies/http://iridl.ldeo.columbia.edu/dochelp/StatTutorial/Climatologies/

8/3/2019 Stat 1 Report F 202

13/25

Variance: The arithmetic mean of the squared deviations from the mean is known as the variance.The variance is nonnegative and is zero only if all the observations are the same. The formula is:-

8/3/2019 Stat 1 Report F 202

14/25

Measures of Relative Dispersion

A measure of relative variation is the ratio of the measure of the absolute variation

to an average. It is sometimes called the co-efficient of variation because co-efficient

means a pure number that is independent of the unit of measurement. It should beremembered that while computing the relative variation the average used as base

should be the same one from which the absolute variations were measured.

The relative variations are:

Coefficient of range

The relative measure corresponding to a range called the coefficient of range, is

obtained by applying the following formula:

Coefficient of range =

In a frequency distribution, coefficient of range is calculated by taking the difference

between the lower limit of the lower class and the upper limit of the upper class.

Example:

The following are the prices of shares of a company from Monday to Saturday:

Day Price Day Price

Monday 200 Thursday 160

Tuesday 210 Friday 220

Coeffcient of range

Coeffcient of mean deviation

Coeffcient of quartile deviation

Coeffcient of variance

8/3/2019 Stat 1 Report F 202

15/25

Wednesday 208 Saturday 250

Solution:

Range= L S Here, Largest value = 160 and

=250 160 Smallest value = 250

Coefficient of range =

=

=0.219

Coefficient of quartile deviation

The relative measure corresponding to a quartile deviation called the coefficient of

quartile deviation is calculated as follows:

Coefficient of quartile deviation =

Coefficient of quartile deviation can be used to compare the degree of variation in

different distributions.

Coefficient of mean deviation

The relative measure corresponding to a mean deviation called the coefficient of

mean deviation is calculated as follows:

Coefficient of mean deviation =

If mean has been used while calculating the value of mean deviation in such a case

coefficient of mean deviation can be obtained by diving average deviation by the

mean.

Coefficient of variation

The relative measure corresponding to a variation is called the coefficient of

variation. This measure developed by Karl Pearson is the most commonly used

measure of relative variation. It is used in such problems where we want to compare

8/3/2019 Stat 1 Report F 202

16/25

the variability of two or more than two series. Coefficient of variation denoted by C.V

is obtained as follows:

C.V. =

Percentile: If the data are organized in ascending form and then which single data

divides the information into hundred, it is called percentile.

Percentile = () ; i= 1,2,3,..,99If the

() is in fraction, then

Percentile= value + *( ) + For frequency distribution,

Percentile=

Example: Find the percentile of 2, 4,6,8,10,12,14,16,18.Solution: Here, n= 9

Percentile= () th value=value=10.

Decile: In descriptive statistics, a decile is any of the nine values that divide the sorted data

into ten equal parts, so that each part represents 1/10 of the sample or population. Thus:

The 1st decile cuts off the lowest 10% of data, i. e., the 10th percentile.

The 5th decile cuts off lowest 50% of data, i. e., the 50th percentile, 2nd quartile, or

median.

The 9th decile cuts off lowest 90% of data, i. e., the 90th percentile.

http://en.wikipedia.org/wiki/Descriptive_statisticshttp://en.wikipedia.org/wiki/Descriptive_statistics

8/3/2019 Stat 1 Report F 202

17/25

Empirical Rule:

Provides significant information into the distribution of data around the mean,

approximating normality.1. The mean one standard deviation contains approximately 68.26% of the

measurements in the series.

2. The mean two standard deviations contain approximately 95.5% of the

measurements in the series.

3. The mean three standard deviations contain approximately 99.7% of themeasurements in the series.

Climatologists often use standard deviations to help classify abnormal climatic

conditions. The chart below describes the abnormality of a data value by how many

standard deviations it is located away from the mean. The probabilities in the third

column assume the data is normally distributed.

8/3/2019 Stat 1 Report F 202

18/25

Standard Deviations Away

From Mean

Abnormality Probability of Occurance

beyond -3 sd extremely subnormal 0.15%

-3 to -2 sd greatly subnormal 2.35%

-2 to -1 sd subnormal 13.5%-1 to +1 sd normal 68.0%

+1 to +2 sd above normal 13.5%

+2 to +3 sd greatly above normal 2.35%

beyond +3 sd extremely above normal 0.15%Oliver, John E. Climatology: Selected Applications. p 45.

Chebyshevs Theorem: A large standard deviation reveals that the observations are widely scatteredabout the mean. The Russian mathematician P. L. Chebyshev (1821-1894) developed a theorem that

allows us to determine the minimum proportion of the values that lie within specified number of

standard deviations of the mean. For example, according to Chebyshevs Theorem, at least three of

four values, or 75 percent, must lie between the mean plus two standard deviations and the mean

minus two standard deviations. This relationship applies regardless of the shape of the distribution.

Further, at least eight of nine values, o 88.9 percent will lie between plus three standard deviations

and minus three standard deviations of the mean. At least 24 of 25 values, or 96 percent, will lie

between plus and minus five standard deviations of the mean.

For any set of observations, the proportion of the values lie within k standard deviations of the mean

is at least 1 1/k2

, where k is any constant greater than 1.

8/3/2019 Stat 1 Report F 202

19/25

Which Measure of Variation to Use

The choice of a suitable measure of dispersion depends on the following three

factors:

1. The type of data available: If observations are few in numbers, avoid thestandard deviation. If they are generally skewed, avoid the mean deviation as

well. If they have gaps around the quartiles, the quartile deviation should be

avoided. If there are open-end classes, the quartile measure of variation

should be preferred.

2. The purpose of investigation: In an elementary treatment of statistical seriesin which a measure of variability is desired only for itself, any of the three

measures, namely, range, mean deviation, quartile deviation would be

acceptable. Probably the man deviation would be superior. In usual practice,

the measure of variability is employed in further statistical analysis. For such

a purpose, the standard deviation is by far the most popularly used. It is free

from those defects with which other measures suffer. It lends itself to the

analysis of variability in terms of normal curve of error. Practically, all

advanced statistical methods deal with variability and centre around the

standard deviation. Hence, unless the circumstances warrant for the use of

any other measure, we should make use of standard deviation for measuring

variability.

8/3/2019 Stat 1 Report F 202

20/25

A tabular format comparison among the measures of dispersion is drawn in the following:

Characteristics Range Quartile

Deviation

Mean Deviation Standard

DeviationClear Definition Yes Yes Yes Yes

Easily

Understandable

Yes Yes No Yes

Determination

Procedures

Easy Average Average Not that easy

For further

Algebraic Process

Not Eligible Not Eligible Not Eligible Eligible

Usage of all item

in a data set

No No Yes Yes

Effect of extreme

values

Yes Not much Yes Not much

Effect of sample

fluctuations

No Not much Not much Not much

From the above discussions, it is seen that standard deviation supports almost all the characteristics

of an ideal measures of dispersion. Therefore, we can say that, standard deviation is the ideal

measure of dispersion.

8/3/2019 Stat 1 Report F 202

21/25

Some practical applications of Measures of Dispersion:

The following data show the lifetime of laptops of two different brands.

Life time

(Years)

No. of laptop

Dell HP

2 4

4 6

6 8

8 10

10-12

20

15

25

30

35

15

20

20

25

15

i. Find which of the brands shows a greater lifetime?ii. Which of the brands you would prefer if the prices were same? Why?

Solution:

The brand, which has greater mean, has the greater lifetime. If the prices were same thebrand which has less variability has to be preferred. The brand which has less coefficient of

variance has less variability.

At first let us take Compaq,

Life time (Years) Dell (f) X fX 2 4 20 3 60 180

4 6 15 5 75 375

6 8 25 7 175 1225

8 10 30 9 270 2430

10-12 35 11 385 4235

8/3/2019 Stat 1 Report F 202

22/25

=

C.V.

8/3/2019 Stat 1 Report F 202

23/25

Life time (Years) HP(f) X fX 2 4 15 3 45 135

4 6 20 5 100 500

6 8 20 7140 980

8 10 25 9 225 2025

10-12 15 11165 1815

95 675 5455

=

C.V.

8/3/2019 Stat 1 Report F 202

24/25

Dell has the mean 7.72 and HP has the mean 7.11. As Dell has the greater mean than HP, so

Dell has the greater lifetime.

The covariance of Dell is 36.53% and the covariance of HP is 36.99%. If the prices were

same, Dell is more preferable as it has less variability and it indicates better quality and

higher consistency.

Application of Empirical Rule:

A sample of the rental rates at University Park Apartments approximates a symmetrical,

bell-shaped distribution. The sample mean is 500 taka and the standard deviation is 20 taka.

Using the empirical rule, we have to determine:

1. About 68 percent of the monthly food expenditures are between what two amounts?

2. About 95 percent of the monthly food expenditures are between what two amounts?

3. Almost all of the monthly expenditures are between what two amounts?

Solution:

1. About 68 percent of the monthly food expenditures are between

X 1s = 500 1(20)

That is 480 and 520 taka

2. About 95 percent of the monthly food expenditures are between

X 2s = 500 2(20)

That is 460 and 540 taka

8/3/2019 Stat 1 Report F 202

25/25

3. Almost all (99.7 percent) are between

X 3s = 500 3(20)

That is 440 and 560 taka.