Introduction to quantitative techniques

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Quantitative Techniques:

Introduction, Data

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Classification

UNIT 1 QUANTITATIVE

TECHNIQUES:

INTRODUCTION, DATA

COLLECTION AND

CLASSIFICATIONStructure

1.0 Unit Objectives

1.1 Introduction

1.2 Use of Quantitative Techniques

1.3 Meaning and Types of Quantitative Techniques

1.4 Statistics in Business

1.5 The Concept of Statistics

1.6 Data Collection and Data Organization

1.0 UNIT OBJECTIVES

After going through this unit, you will be able to:

Understand what quantitative techniques are

Discuss the significance of quantitative techniques

Explain the types of quantitative techniques

Differentiate between descriptive statistics and inferential statistics

Explain the various methods of data collection

Describe frequency distribution

1.1 INTRODUCTION

Decision-making is an essential part of management. Managers, at all levels, make

many decisions every day. Some decisions are routine and require little thought. Many,

however, are more complex as a manager has to co-ordinate limited human and capital

resources in the most efficient manner to achieve the given objectives of the organization.

This applies to all types of business organizations (a factory, a farm, a small business

enterprise or an MNC). A manager has to determine the objectives, analyse the existing

situation, seek alternatives, take decisions and implement them. Therefore, to carry out

the key managerial functions of planning, organizing, directing and controlling, the

management is engaged in a continuous process of decision-making.

Traditionally, decision-making has been considered as an art and/or a talent based

on inheritance, individual judgements and experience. However, in an increasingly

competitive, complex and fast changing world, the decision-makers are expected to

take fast and hopefully appropriate decisions on the basis of logic and hard analysis, and

not merely on judgment and experience.

As the complexity increases, management becomes more of a science than an

art. There is a greater need for supplementing the art of decision-making by systematic


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and scientific methods. Quantitative Analysis is the scientific approach to managerial

decision-making and Quantitative Techniques are essentially helpful supplements to

judgment and intuition.

1.2 USE OF QUANTITATIVE TECHNIQUES

The quantitative approach to decision-making is assuming an increasing degree of

importance in the theory and practice of management. The formal study and application

of quantitative techniques to practical decision-making started in the 20th century. During

World War II many new scientific and quantitative techniques were developed to assist

the military. These new developments were so successful that after World War II

companies started using similar techniques in managerial decision-making and planning.

Modern management is applying Quantitative Techniques to aid the process of decision-

making in an ever increasing measure. Besides traditional business areas, such as

production, finance and marketing, it is now possible and desirable to apply the quantitative

approach to areas, such as social responsibility, health care, ecology, public policy,

international relations, individual and group behaviour , organizational structure and

behaviour. Today, many organizations employ staff comprising operations research or

management science personnel or consultants to apply the principles of scientific

management to problems and opportunities. This is due to the fact that an intelligent

application of the appropriate tools can reduce an otherwise unwieldy and complex

problem to one of manageable dimensions. However, Quantitative Techniques are

expected only to supplement, and not to supplant the managers sense of decision-

making. A large number of business problems, in the relatively recent past, have been

given a quantitative representation with considerable degree of success.

Numerous professional journals regularly provide details of successful application

of such techniques to specific business problems. A number of researchers have tried to

establish the use that companies make of quantitative methods.

A research team undertook an investigation of the use of techniques by businesses

and their effectiveness in terms of organizational performance, across Europe. Theysent questionnaires to the chief executives of a sample of businesses in a number of

countries. Firms were asked whether Quantitative Techniques were used by the business

(with a detailed list of over 30 common techniques provided).The respondents were, by

and large, managers in these organizations, not mathematical or statistical specialists.

Two thirds (66%) of firms responding indicated they used at least some of the techniques.

Firms were also asked to comment on the perceived usefulness of the techniques for

their respective businesses.

Around 90% of firms using these techniques felt that these techniques were

useful (essential (24%), very useful (24%), useful (42%)). Only 8.7% of firms using

these techniques felt these techniques to be of little use and 1.3% felt there was no use

of these techniques.

Investigation of individual techniques revealed that basic statistics had been the

most popular quantitative technique as 89% of the firms were using statistical techniques.

Firms not using these techniques were also asked the reasons why such techniques

were not used. 39% of the firms not using these techniques responded that one of the

factors contributing to non-use was insufficient training and education of staff in the use

of these techniques.


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There is extensive empirical evidence that the relevant application of such

techniques has resulted in significant improvements in efficiency, particularly at the micro

economic level, and has led to improvements in decision-making in profit and non-profit

organizations. This is a stark illustration of why managers need to develop skills and

awareness in the quantitative area. Not to do so may put you as a manager and your

organization at a severe competitive disadvantage.

1.3 MEANING AND TYPES OF QUANTITATIVE

TECHNIQUES

The quantitative analysis approach consists of defining a problem, developing a model,

acquiring input data, developing a solution and analysing the results. The approach uses

concepts and/tools of mathematics and statistics, which provide the decision-maker with

systematic and powerful means of analysis and help in the decision-making process. In

other words, all the mathematical and statistical techniques used by managers in all

types of business organizations to help them with decision-making are called Quantitative

Techniques. These techniques are based on quantitative data, which provide the decision-

maker with a systematic and powerful means of analysis and helps in exploring business

policies. If you are studying business or are interested in any aspect of business youneed to know about quantitative methods as business is all about quantities, quantities of

goods produced and services provided, quantities of inputs and costs, quantities of revenues

and profit, and so on. Quantitative techniques offer the manager, a method of analysing

a problem (using proven techniques), providing information about that problem and of

assessing the potential outcomes from different decisions. These techniques are

particularly relevant to problems of complex business enterprises. They can broadly be

put under two groups (a) Statistical Techniques and (b) Operations research/Programming

Techniques.

Statistics is the science concerned with the collection organization, presentation,

analysis, and interpretation of data. Statistical Techniques involve analysis of the collected

data for the purpose of summarizing the information. The field of statistics providesmethods for collecting, analysing and meaningfully interpreting data.

Statistics has, as of now, established itself as a generic and versatile subject

of study. The more one gets to know of it, the more one imbibes its subtle impact in

terms of the mental ability to draw fairly valid conclusions even from limited data.

It is precisely owing to this reason that the applications of statistical methods have

fast spread its tentacles to the various important areas of human interest.

R.P. Hooda

Operations Research (OR) techniques, also known as Programming Techniques,

are used for building an information model for the available data. Building of information

model refers to quantifying the factors affecting the business with the help of different

variables and parameters, and describing interrelationships between these differentvariables in the form of mathematical equations. Based upon these equations, an optimal

solution for the business-related problems is obtained to get maximum profits out of

minimum costs. These techniques include a wide variety of techniques, such as linear

programming, theory of games, simulation and decision theory.

The techniques we shall be introducing are statistical techniques used by managers

in a wide variety of business organizations.


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1.4 STATISTICS IN BUSINESS

The use of statistical methods coupled with sound organization practice can be a

key to good management and effective leadership in business settings.

Richard A. Johnson and Dean W. Wichern

The complexity of the business environment necessitates the business decisions

to be based not on judgment and experience but on systematic and scientific methods

based on data. Decisions have to be based upon data which shows relationships, indicatestrend, and shows rates of change in various relevant variables. The careful generation

and analysis of data reduces uncertainty and leads to rapid learning and opportunities for

improvement that are generally not apparent from intuition or reports. Managers, at all

levels, make many decisions every day. Some decisions are routine and require little

thought. Many, however, are more complex and depend on numbers to suggest and

justify subsequent course of action. Statistics deals with ways of organizing, summarizing

and describing quantifiable data and methods of drawing inferences and generalizing

them. We may state that Statistics is the science concerned with the collection,

organization, presentation, analysis and interpretation of data. Statistical techniques involve

analysing the collected data for the purpose of summarizing the information. The field of

statistics provides methods for collecting, analysing and meaningfully interpreting data.From a management perspective, the ultimate goal of the firm is to be able to

offer desirable products and services as economically as possible. To accomplish this

goal, all managers, may not have to be statisticians. However, they need to know the

techniques of statistics. The two main duties of a professional, to make decisions and to

solve problems, can be accomplished through the application of statistical procedures as

virtually every area of business uses statistics in decision-making. Reason being Business

is all about quantities: quantities of goods produced and services provided, quantities of

inputs and costs, quantities of revenues and profit, and so on. Efforts at quality control,

cost minimization, product and inventory mix, and a host of other business matters can

be effectively managed by using proven statistical procedures. For those in marketing

research, statistics is of invaluable assistance in determining whether a new product islikely to prove successful. Statistics is also useful in the evaluation of investment

opportunities by financial consultants. Accountants, personnel managers, and

manufacturers, all find unlimited opportunities to utilize statistical analysis. Even the

medical researcher, concerned about the effectiveness of a new drug, finds statistics a

helpful ally. Through the application of precise statistical procedures, it is possible to

predict the future with some degree of accuracy. Any business firm faced with competitive

pressures can benefit considerably from the ability to anticipate business conditions

before they occur. If future sales is estimated with a reliable degree of accuracy,

management can easily take important decisions regarding inventory levels, raw material,

orders, employment requirements, and other aspect of business operations.

1.5 THE CONCEPT OF STATISTICS

Statistics deals with ways of organizing, summarizing and describing quantifiable data

and methods of drawing inferences and generalizing upon them (Jim Fowler, Lou Cohen,

Phil Jarvis inPractical Statistics for Field Biology). Websters third new international

dictionary defines statistics as a science of dealing with the collection, organization,

presentation, analysis and interpretation of data.


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The word statistics may be defined in two different but interrelated ways: (i) As

a plural noun; (ii) As a singular noun.

Statistics as a plural noun refers to numerical data (the factual data itself) arising

in any sphere of human experience. It is in this sense that the term is used when our

daily newspapers give birth-death statistics, crime statistics or soccer statistics of Calcutta

or when the food minister quotes statistics of sugar exports.

Used as singular, Statistics is a study of the methods, theories and techniques by

means of which the collected descriptions are summarized and interpreted.

Thus, statistics, in totality, refers to statistical (numerical) data and the method of

obtaining and analysing data.

1.5.1 Subdivisions within Statistics/Functions of Statistics

Managers apply some statistical techniques to virtually every branch of public and private

enterprise. These techniques are so diverse that statisticians commonly separate them

into two broad categories: Descriptive statistics and Inferential statistics.

(i) Descriptive Statistics

Descriptive statistics involves describing some characteristics of the numerical data. If

a business analyst is using data gathered on a group to describe or reach conclusionsabout that same group, the statistics are called descriptive statistics. It deals with the

collection, tabulation and presentation of data and the calculation of measures which

describe the data in various ways. It summarizes information in such a manner as to

make it more usable. By computing measures, such as percentages, means, standard

deviations, correlation coefficient, it may be possible to reduce the data to a manageable

proportion. By reducing the quantity of information by statistical methods, we increase

our ability to analyse and interpret the findings.

Descriptive statistics can be either univariate or bivariate. The univariate analysis

includes: frequency distributions, measures of central tendency, measures of dispersion.

The study of correlation and regression is a part of the bivariate and multivariate

descriptive statistics. Descriptive statistics is especially useful in instances where

interrelationship among two or more variables is to be analysed.

Many of the statistical data generated by businesses is descriptive. It might include

a number of employees on vacation during March, average salary of personnel in an

organization, corporate sales for a particular year and average return on investment for

a period of a few years.

(ii) Inductive/Inferential Statistics

Inferential or inductive statistics refers to those methods which are helpful in drawing

inferences about the characteristics of the population on the basis of sample data.

Alternatively, the inductive statistics may be termed as the logic of drawing statisticallyvalid conclusions about the totality of cases or items termed as population, in any statistical

investigation on the basis of examining a part of the population termed as sample and

which is drawn from the population in a scientific manner. If a researcher gathers data

from a sample and uses the statistics generated to reach conclusions about the population

from which the sample was taken, the statistics are inferential statistics. One application

of inferential statistics is in pharamaceutical research. Some new drugs are expensive to

produce, and therefore, tests must be limited to small samples of patients. Utilizing


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inferential statistics, researchers can design experiments with small randomly selected

samples of patients and attempt to reach to conclusions and make inferences about the

population. Market researchers use inferential statistics to study the impact of advertising

on various market segments. The advantage of using inferential statistics is that they

enable the researcher to study effectively a wide range of phenomenon without having

to conduct a census.

To understand the difference between descriptive and inferential statistics

definitions of (a) Population and sample, and (b) Parameters and statistic are helpful.

1.5.2 Population and Sample

Population:The collection or the aggregate of objects or the results of an operation are

called population or universe. A population/universe is the complete group of (collection

of all ) items, objects, persons about which knowledge is sought.

In theory of sampling, the parent population or simply the population means the

larger group from which the samples are drawn. Webster defines population as a

collection of all persons, objects, or items of interest.

The population can be a widely defined category, such as all automobiles, or it

can be narrowly defined, such as all Tata Motor cars produced from 2000 to 2010. A

population can be a group of people, such as all workers presently employed byMicrosoft, or it can be a set of objects, such as all dishwashers produced on February 3,

2011 by the General Electric Company at one particular plant. The researcher defines

the population to be whatever he or she is studying.

The population may be finite or infinite. Finite universe is one which has a definite

and certain number of items. A population containing only a finite (certain) number of

objects or individuals is called a finite population, viz., the population of books in the

library or of Indian students in the US, etc. A population with a number of objects so

large as to appear practically infinite is called an infinite population. When the number of

items is uncertain and infinite, the population is said to be an infinite population.

Data gathered from the whole population for a given measurement of interest,

refers to census. Most people are familiar with the Indian Census. Every ten years, the

government attempts to measure all persons living in the country. If a researcher is

interested in ascertaining the SAT scores for all students at the University of Arizona,

one way to do so is to conduct a census of all students currently enrolled at the University.

A sample is a portion of the whole. A part of the population selected from it (i.e.,

population) with the object of investigating its properties is called a sample. Researchers

often prefer to work with a sample of the population instead of the entire population. For

example, in conducting quality control experiments to determine the average life of light

bulbs, a light bulb manufacturer might randomly sample only 75 light bulbs during a

production run. Because of time and money limitations, a human resource manager

might take a sample of 40 employees instead of using a census to measure companymorale.

1.5.3 Parameters and Statistics

Differentiation between the terms parameter and statistic is important in the use of

inferential statistics.

A parameter is any descriptive measure of a population. All statistical measures

like mean, standard deviation, median computed from population data, i.e., all the statistical


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measures based on all items of the universe, are termed as parameters. The population

characteristic (or the constant) which we are estimating is called a parameter.

A statistic is any descriptive measure of a sample. Statistical measures worked

out on the basis of sample studies are termed as sample statistics. All statistical measures,

say mean and standard deviation worked out on the basis of sample studies, i.e., computed

from sample data, are termed as sample statistics.

Parameters are functions of the population values, while statistics are functions

of the sample observations. Parameters are usually denoted by Greek letters, for example

population mean (), population variance (), standard deviation (s). Statistics are

usually denoted by Roman letters. Examples of statistics are sample mean ( x ), sample

variance (s2) and sample standard deviation (s).

1.6 DATA COLLECTION AND DATA ORGANIZATION

Quantitative Analysis is the scientific approach to managerial decision-making. This

approach starts with data.

1.6.1 Introduction

Data is classified into two broad categories (i) Quantitative Data, for examplepercapita income of different states of India or production of a particular commodity of an

organization and (ii) Qualitative Data,for exampleclassification of workers as skilled,

unskilled and semi-skilled workers or assigning of ranks according to performance in an

interview. Like raw material for a factory, statistical data are raw material for statistics

and statistical methods/techniques. The field of statistics provides methods for collecting,

analysing and meaningfully interpreting data. The validity and accuracy of final judgment,

i.e., analysis and interpretation of results of data processing and manipulation for decision-

making and predictions depends basically on how well the data was gathered. The

quality of data will greatly affect the decisions.

1.6.2 Collection of DataAlmost every statistical effort begins with the process of collecting the necessary data

to be used in the study. Data collection, required under a number of circumstances,

involves collecting the values for variables. A Variable may be a continuous or a

discrete variable. For example, a marketing manager who needs to assess the effectiveness

of a new television advertisement, an operations manager who wishes to determine

whether a new technique is more effective in utilizing excess capacity in production and

a farmer who wants to assess the effect of fertilizer use on production of cereals, all of

them may need to collect data for further analysis. Data collection almost always involves

collecting data from a sample as collecting data from every item or individual in the

population would be too difficult or too time-consuming.

1.6.3 Types of Data: Primary Data and Secondary Data

Meaning:Depending upon the sources utilized, statistical data may be classified under

following two categories:

(i) Primary Data

(ii) Secondary Data


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Primary data refers to the data which is collected by the person performing the statistical

analysis. In other words, when the data is not available from an existing data source and

has to be collected by the user (himself/herself) for processing and analysis, then the

data is termed as Primary Data. Secondary data refers to the data which has been

collected earlier for some purpose other than the analysis currently being undertaken. In

other words, the data which has already been collected by others and already exists in

some form (published or unpublished) is termed as Secondary Data.

The Methods of Data Collection

Data may be collected through two sources: (a) Primary and (b) Secondary.

(a) Primary Data Sources

Conducting a field survey(s) is the only source of primary data collection. The survey

may be a Sample Survey or a Population/Census Survey.

Survey may be conducted by:

(a) Interviews (face to face or over telephone).

(b) Questionnaire: A set of questions is prepared by the researcher for extracting

information from the target population. The Questionnaire may be given to

respondents personally, through mail or by post/courier services. Respondentsare requested to return the filled questionnaire to the researcher.

(b) Secondary Data Sources

These may contain; external secondary data sources and internal secondary data sources.

External Secondary Data Sources

These are as follows:

o Newspapers and Business Magazines.

o Government Publications:National Income Statistics (CSO). Money and Finance

Statistics (Reserve Bank of India Bulletins) and Economic Survey (Govt. of India),etc.

o Non-Government Publications: The Bombay Stock Exchange, Economic

Intelligence Service (CMIE), etc.

o International Organizations: U.N, F.A.O, IMF, OECD, ILO and WTO

Publications, etc.

Internal Secondary Data Sources

The data generated, within an organization, or an institution, (in the process of routine

business activities), to plan and control organizational operations is referred to as internal

sources of secondary data. Such data is usually published in the form of annual reports

of the organization concerned. Financial accounts, production and sales records are

examples of such data. The data presented in these reports is primarily meant for internal

use.

1.6.4 Classification/Organization of Data

The collection of raw data in itself reveals very little. Merely collecting data is of no use

unless it is organized in such a way that it is revenant to the purpose for which the data

is needed. Suppose you were the manager of an electronic goods store that needed


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funding from banks for expansion. How would you present that data? Presenting the

bankers with data of thousands of transaction would overwhelm them and not be very

useful. You need to transform the transaction data into information by summarizing the

details of each transaction in some useful way that would convince the bankers that

your store is indeed a thriving business and you are a good candidate for funding from

the bank for expansion.

We must organize the data into some form so that, at a mere glance, we can get

an idea of what the data can tell us. When the raw data has been collected and edited,

it should be put into an ordered form so that it can be looked at more objectively. Theimportant step towards processing the data is organization of data.

Some common statistical tools of organizing a large collection of data, thereby

making it easier to more fully comprehend the information it contains are frequency

tables and various pictorial displays that can provide a handy visual representation of the

data. Data array, contingency tables and, stem and leaf designs also help in the

presentation of a large data set in a concise and discernible form.

Frequency Distributions

One way we can compress data is to use a frequency table or a frequency distribution.

Frequency distribution can be defined as a list of all the values obtained in thedata and the corresponding frequency with which these values occur in the data.

The frequency distribution can either be ungrouped or grouped.

Ungrouped Frequency Distributions

When the number of values of the variable is small, then we can construct an ungrouped

frequency distribution which is simply listing the frequency of occurrence against the

value of the given variable. As an example, let us assume that 20 families were surveyed

to find out how many children each family had. The raw data obtained from the survey

is as follows:

0, 2, 3, 1, 1, 3, 4, 2, 0, 3, 4, 2, 2, 1, 0, 4, 1, 2, 2, 3This data can be classified into an ungrouped frequency distribution. The number

of children becomes our variable (X) for which we can list the frequency of occurrence

(f) in a tabular form as follows:

Number of Chil dren(X) Frequency(f)

0 3

1 4

2 6

3 4

4 3

Total = 20

The above table is also known as discrete frequency distribution where the variable

has discrete numerical values.

Grouped Frequency Distribution

However, when the data set is very large, it becomes necessary to condense the data

into a suitable number of groups or classes of the variable values and then assign the

combined frequencies of these values into their respective classes. As an example, let


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us assume that 100 employees in a factory were surveyed to find out their ages. The

youngest person was 20 years of age and the oldest was 50 years old. We can construct

a grouped frequency distribution for this data so that instead of listing frequency by

every year of age, we can list frequency according to an age group. Also, since age is a

continuous variable, a frequency distribution would be as follows:

Age Group (Years) Frequency (f)

20 to less than 25 5

25 30 15

30 35 25

35 40 30

40 45 15

45 50 10

Total = 100

In this example, all persons between 20 years (including 20 years old) and 25

years (but not including 25 years old) would be grouped into the first class, and so on.

The interval of 20 to less than 25 is known as Class Interval (CI). A single representation

of a class interval would be the midpoint (or average) of that class interval. The midpointis also known as the class mark.

Constructing a Frequency Distribution

A frequency distribution (or frequency table) simply divides the data into classes and

records the number of observations in each class by tally marks, as shown in Table 1.2.

The number of classes in a frequency table is somewhat arbitrary. In general,

your table should have between 6 and 15 classes. Too few classes would not reveal

any details about the data; too many would prove as confusing as the list of raw data

itself.

A simple rule can be followed to approximate the number of classes, c:Number of classes is the lowest power to which 2 is raised. 2c>/=ndetermines

the number of classes.

Assuming we have n=50 observations. 2c>/= 50. 26=64. This rule suggests that

there should be 6 classes in the frequency table.

The class midpoint,M, is calculated as the average of the upper and lower

boundaries of that class. The midpoint for the first class in Table 1.2 is (50 + 59)/2

= 54.5.

The class interval is the range of values found within a class. It is determined by

subtracting the lower (or upper) boundary of one class from the lower (or upper) boundary

of the next class. The interval for the first class in Table 1.2 is (60 50) = 10. It isdesirable to make class intervals of equal size, since this facilitates statistical interpretations

in subsequent uses.

The class interval can be determined as:

Class interval for a frequency table CI Largest value Smallest value_____________________________

=_____________________

= RangeThe number of desired classes The number of classes


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Since you decided on six classes for your frequency table, the class interval

becomes:

CI =102 50

6

= 8.7

Since 8.7 is an awkward number, the interval can be slightly adjusted up or down.

For convenience, the interval of 10 was selected in forming Table 1.2.

Case Study

The Statistical Analysis Division of Kingfisher Airlines is to decide the size of planes.

As the resident statistician for Kingfisher Airlines, you are asked by the director of the

Statistical Analysis Division to collect and organize data on the number of passengers

who have chosen to fly on Kingfisher. This data is displayed in Table 1.1 for the past

50 days. However, in this raw form, it is unlikely that the director could gain any

valuable information regarding flight operations. It is difficult to arrive at any meaningful

conclusion by merely examining a bunch of numbers that have been jotted down. The

data must be organized and presented in some concise and revealing manner so that

the information they offer can be readily discerned.

Table 1.1Raw Data on the Number of Passengers for Kingfisher Airlines

68 71 77 83 79

72 74 57 67 69

50 60 70 66 76

70 84 59 75 94

65 72 85 79 71

83 84 74 82 97

77 73 78 93 95

78 81 79 90 83

80 84 91 101 86

93 92 102 80 69

A frequency distribution (or frequency table) simply divides the data into classes

and records the number of observations in each class, as shown in Table 1.2.

Table 1.2Frequency Distribution for Passengers

Class Tal ly F requency M idpoint Relative

(Passengers) (Days) (M ) F requency

50 to 59 111 3 54.5 6%

60 to 69 1111 11 7 64.5 14%70 to 79 1111 1111 1111 111 18 74.5 36%

80 to 89 1111 1111 11 12 84.5 24%

90 to 99 1111 111 8 94.5 16%

100 to 109 11 2 104.5 4%

50


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It can now be seen, for example, that on 18 of the 50 days, i.e., in 36% of 50 days,

between 70-79 passengers flew on Kingfisher. In 76% of 50 days, 70-99 passengers

flew on Kingfisher. At no time did the daily passenger list exceed 109. The airline rarely

carried fewer than 60 passengers.

Business Decision Based on Organization of Data:As the resident statistician

for Kingfisher Airlines, you may suggest the director of the Statistical Analysis Division

to have a 100 seater plane for the Kingfisher Airlines flights.

If the data in a frequency table is continuous, it is necessary to allow for fractional

values. Our class boundaries would have to appear as:

50 and under 60

60 and under 70

70 and under 80

:

Cumulative Frequency Distributions

Cumulative frequencies can be Less than or More than type. More than cumulative

frequency distribution can be obtained by adding successive frequencies from bottom to

top. Less than cumulative frequency distribution can be obtained by adding successive

frequencies from top to bottom.

Table 1.3More Than Cumulative Frequency Distribution

for the Number of Passengers

Class Cumulative Relative

(Passenger s) F requency (Days) F requency

50 or more 50 100%

60 or more 47 94%

70 or more 40 80%

80 or more 22 44%

90 or more 10 20%

100 or more 2 4%

110 or more 0

Table 1.4Less Than Cumulative Frequency Distribution

for Number of Passengers



Less than 50 0 o%

Less than 60 3 6%

Less than 70 10 20%

Less than 80 28 56%

Less than 90 40 80%

Less than 100 48 96%

Less than 110 50 100%


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Table 1.5Relative Frequency Distribution for Passengers



50 59 3 3 50 = 6%

60 69 7 7 50 = 14%

70 79 18 18 50 = 36%

80 89 12 12 50 = 24%

90 99 8 8 50 = 16%

100 109 2 2 50 = 4%

50 100%

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