Introduction To Quantitative Techniques 1a296h

UNIT 1 QUANTITATIVE TECHNIQUES: INTRODUCTION, DATA COLLECTION AND CLASSIFICATION

Quantitative Techniques: Introduction, Data Collection and Classification

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Unit Objectives Introduction Use of Quantitative Techniques Meaning and Types of Quantitative Techniques Statistics in Business The Concept of Statistics Data Collection and Data Organization

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1.0 1.1 1.2 1.3 1.4 1.5 1.6

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Structure

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

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• Describe frequency distribution

1.1 INTRODUCTION

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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.

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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


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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 decisionmaking 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 manager’s sense of decisionmaking. 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.

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A research team undertook an investigation of the use of techniques by businesses and their effectiveness in of organizational performance, across Europe. They sent 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.

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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.

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.


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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 decisionmaker 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 you need 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.

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1.3 MEANING AND TYPES OF QUANTITATIVE TECHNIQUES

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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 provides methods for collecting, analysing and meaningfully interpreting data.

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“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 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

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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 different variables 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. Self-Instructional Material

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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

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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, indicates trend, 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.

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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 is likely to prove successful. Statistics is also useful in the evaluation of investment opportunities by financial consultants. ants, 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.

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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 in Practical Statistics for Field Biology). Webster’s third new international dictionary defines statistics as a science of dealing with the collection, organization, presentation, analysis and interpretation of data.

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.


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Used as singular, Statistics is a study of the methods, theories and techniques by means of which the collected descriptions are summarized and interpreted.

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.

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

(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 conclusions about 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.

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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.

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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

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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 statistically valid 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 Self-Instructional Material

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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

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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 by Microsoft,’ 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.

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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.

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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 company morale.

1.5.3 Parameters and Statistics Differentiation between the 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.


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1.6 DATA COLLECTION AND DATA ORGANIZATION

1.6.1 Introduction


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

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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 (µ), population variance (σ ), standard deviation (σs). Statistics are usually denoted by Roman letters. Examples of statistics are sample mean ( x ), sample variance (s2) and sample standard deviation (s).

Data is classified into two broad categories (i) Quantitative Data, for example per capita income of different states of India or production of a particular commodity of an organization and (ii) Qualitative Data, for example classification of workers as skilled, unskilled and semi-skilled workers or asg 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 decisionmaking and predictions depends basically on how well the data was gathered. The quality of data will greatly affect the decisions.

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1.6.2 Collection of Data

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Almost 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 ment, 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.

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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 Self-Instructional Material

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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 (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.

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(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. Respondents are 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.

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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.

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

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• 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 s, 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

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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

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.


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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.

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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. The important step towards processing the data is organization of data.

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 the data 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:

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0, 2, 3, 1, 1, 3, 4, 2, 0, 3, 4, 2, 2, 1, 0, 4, 1, 2, 2, 3

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This 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:

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Number of Children (X) 0 1 2 3 4

Frequency (f) 3 4 6 4 3 Total = 20

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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

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25 ’’ ’’

’’ 30

15

30 ’’ ’’

’’ 35

25

35 ’’ ’’

’’ 40

30

40 ’’ ’’

’’ 45

15

45 ’’ ’’

’’ 50

10

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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 midpoint is 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.

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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>/=n determines the number of classes.

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Assuming we have n=50 observations. 2c >/= 50. 26=64. This rule suggests that there should be 6 classes in the frequency table.

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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 is desirable 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 _____________________________ _____________________ = = Range The 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 = 8.7 6


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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.


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 engers 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.

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Case Study

71

77

83

79

74

57

67

69

60

70

66

76

84

59

75

94

72

85

79

71

84

74

82

97

73

78

93

95

81

79

90

83

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

68

84

91

101

86

92

102

80

69

72 50 70 65 83 77 78 80 93

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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.

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Table 1.2 Frequency Distribution for engers

Tally (Days)

50 to 59

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Class (engers)

Frequency

Midpoint (M)

Relative Frequency

3

54.5

6%

7

64.5

14%

1111 11

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%

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60 to 69

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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 engers flew on Kingfisher. In 76% of 50 days, 70-99 engers flew on Kingfisher. At no time did the daily enger list exceed 109. The airline rarely carried fewer than 60 engers. 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:

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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 1.3 More Than Cumulative Frequency Distribution for the Number of engers

Cumulative Frequency (Days)

Relative Frequency

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 1.4 Less Than Cumulative Frequency Distribution for Number of engers

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Class (engers)

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Class (engers)


Relative Frequency

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%


Relative Frequency

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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Class (engers)


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Table 1.5 Relative Frequency Distribution for engers


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