Inferential Statistics - rahimullahbaryalai.weebly.comrahimullahbaryalai.weebly.com/.../9/6/23967890/inferent… · Web viewInferential Statistics - rahimullahbaryalai.weebly.com

2013

Inferential StatisticsMuhammad Din Khalil

ContentsCHAPTER 1 INTRODUCTION TO STATISTICS...........................................................................................3

STRUCTURE............................................................................................................................................3

LEARNING OBJECTIVES:.........................................................................................................................3

1.1 INTRODUCTION, MEANING OF STATISTICS................................................................................3

1.2 STATISTICS IN THE PLURAL SENSE.............................................................................................4

1.3 STATISTICS IN THE SINGULAR SENSE.........................................................................................5

1.4 BRANCHES OF STATISTICS..........................................................................................................5

1.5 SCOPE AND APPLICATIONS OF STATISTICS................................................................................6

1.6 FUNCTIONS OF STATISTICS........................................................................................................6

1.7 LIMITATIONS OF STATISTICS......................................................................................................7

QUESTIONS:...........................................................................................................................................7

STRUCTURE............................................................................................................................................8

LEARNING OBJECTIVES:.........................................................................................................................8

2.1 INTRODUCTION – BASIC NOTIONS..................................................................................................8

DISCRETE VARIABLE VS CONTINUOUS VARIABLE..................................................................................9

STAGES OF STATISTICAL INVESTIGATION............................................................................................10

2.2 EXECUTION OF THE SURVEY....................................................................................................11

(i) Collection of data.....................................................................................................................11

(ii) Scrutiny, editing and presentation of data...........................................................................11

(iii) Analysis of data....................................................................................................................11

2.3 PRIMARY DATA & SECONDARY DATA PRIMARY DATA...................................................................11

2.3.1 Primary data...........................................................................................................................11

2.3.2 Secondary Data.......................................................................................................................12

Important sources of secondary data are –.................................................................................12

2.4 METHODS OF COLLECTION OF PRIMARY DATA.............................................................................12

1. Direct personal Interview............................................................................................................13

2. Indirect personal interview......................................................................................................13

3. Information through correspondents......................................................................................13

4. Method of questionnaire (Mail enquiry).................................................................................13

5. Method of schedule (Collection through enumerators)..........................................................13

2.5 DRAFTING QUESTIONNAIRE (DRAFTING SCHEDULE).....................................................................14

2.6 PILOT SURVEY (PRE-TEST)..............................................................................................................14

Prepared by Muhammad Din Khalil Page 1

2.7 Frequency Distribution..................................................................................................................15

2.7.1 Class Limits.......................................................................................................................15

STRUCTURE..........................................................................................................................................16

LEARNING OBJECTIVES:.......................................................................................................................16

3.1 MEASURES OF CENTRAL TENDENCY..............................................................................................16

3.2.1 Arithmetic Mean (Mean)........................................................................................................17

Arithmetic Mean for RAW Data.......................................................................................................17

Arithmetic Mean for Ungrouped Frequency Distribution................................................................18

Arithmetic Mean for Grouped Frequency Distribution....................................................................19

GEOMETRIC MEAN..............................................................................................................................20


CHAPTER 1 INTRODUCTION TO STATISTICS

Introduction to Statistics

STRUCTURE 1.1 Introduction, Meaning of Statistics

1.2 Statistics in the Plural Sense

1.3 Statistics in the Singular Sense

1.4 Branches of Statistics

1.5 Scope and Applications of Statistics

1.6 Functions of Statistics

1.7 Limitations of Statistics

LEARNING OBJECTIVES: After a proper study of this chapter the student is expected to understand:

1) Meaning of Statistics.

2) Characteristics and functions of Statistics.

3) Its application in different fields of activity.

1.1 INTRODUCTION, MEANING OF STATISTICSThis chapter introduces the student to the science of statistics, its meaning, characteristics, functions, its application in diverse fields of activity like Business Commerce and Industry. The need for the use of statistical tools in any decision making scenario is emphasized in this chapter. What is Statistics? Consider the following recent items from the mass media:


a) According to a recent survey conducted by a Government agency 81% of business graduates say their job is related to their specialization in MBA. In contrast 95% of health professionals say that their job is related to their major whereas only 30% of history graduates are working in their area of expertise.

b) Recent study on batting scores of great players in cricket showed that performance of these players changed significantly from 1st innings to 2nd innings of a test match. 80% of the players seemed to do well in the first innings of a test match as compared to the 2nd innings. Every day we are inundated with bits of information data like those of the above two examples Moreover all of us are either data producers or data users.

Obviously for data users, extracting information from the data is the most essential aspect of usage of the data. What specialized tools will enable us to become effective data users? The answer is statistics.

A common misconception is that a statistician is simply a “number cruncher” or a person who calculates and summarises numbers. Statistics involves numbers but there is much more to it than that. In simple words, statistics is the “science of data” a science that will enable us to become proficient data producers and efficient data users.

The word Statistics is derived from the Latin word Status which means State– a political state. Thus, statistics is closely linked with the administrative affairs of a state.

No organization can remain ignorant of facts and figures regarding its overall resource capability. Initially, such facts and figures were called statistics. Subsequently, the crude methods of collecting facts underwent a change. As a result of this, the collection and analysis of facts became more scientific. Thus, the gradual improvement shown in the process of collecting and analysing facts resulted in the formation of an independent branch of science, called Statistics.

The word Statistics is used in two senses:

a) In plural form, it stands for numerical facts (facts expressed in numbers) pertaining to a collection of objects.

b) In singular form, it stands for the science of collection organization, analysis and interpretation of numerical facts.

1.2 STATISTICS IN THE PLURAL SENSE

Prof. Horace Secrets defines statistics as follows: „By Statistics we mean, aggregate of facts affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a pre-determined purpose and placed in relation to each other.‟ Thus, according to Prof. Horace Secrist, the following characteristics of Statistics can be noticed.

1. Statistics means an aggregate of facts: Facts can be analysed only when there are more than one fact. Single fact cannot be analysed. Thus, the fact „ Mr. John is 180 Cm. Tall‟ cannot be statistically analysed. On the


other hand, if we know the heights of 40 students of a class, we can comment upon the average height, variation, etc. Hence, only a collection of many facts can be called statistics.

2. Statistics are affected to a marked extent by multiplicity of causes: The facts are the results of action and interaction of a number of factors. Thus, the statistics of yield of paddy is the result of factors such as fertility of soil, amount of rainfall, quality of seed used, quality and quantity of fertilizer used, etc. These factors, in turn, are the results of many other factors.

3. Statistics are numerically expressed: Only numerical facts can be statistically analysed. Therefore, facts such as „price decreases with increasing production‟ cannot be called statistics.

4. Statistics are enumerated or estimated according to reasonable Standards of accuracy: The facts should be enumerated (collected from the field) or estimated (computed) with required degree of Accuracy: The degree of accuracy differs from purpose to purpose. In meaning the length of screws, accuracy up to a millimetre may be Required, whereas, while measuring the heights of students in a class, Accuracy up to a centimetre is enough.

5. Statistics are collected in a systematic manner: The facts should be collected according to planned and scientific methods. Otherwise, they are likely to be wrong and misleading.

6. Statistics are collected for a pre-determined purpose: There must be a definite purpose for collecting facts. Otherwise, the facts become Useless and hence, they cannot be called statistics.

7. Statistics are placed in relation to each other: The facts must be Placed in such a way that a comparative and analytical study becomes Possible. Thus, only related facts which are arranged in logical order can be called statistics.

1.3 STATISTICS IN THE SINGULAR SENSE A. L. Bowley defined statistics as „The science of counting‟. Afterwards, he re-defined it as

„The science of averages‟. Both these definitions fail to give a complete notion of the science. A. L. Boddington defined Statistics as „The science of estimates and probabilities‟. This is a short and precise definition. According to Croxton and Cowden, „Statistics is the science of collection, presentation, analysis and interpretation of numerical data.‟ Thus, Statistics contains the tools and techniques required for the collection, presentation, analysis and interpretation of data. This definition is precise and comprehensive.

1.4 BRANCHES OF STATISTICS Mainly, “Statistics” is said to have two branches, namely, Statistical methods and Applied Statistics. Statistical Methods is the branch which deals with the general procedure of statistical analysis of numerical data. It supplies devices and techniques required for analysis of numerical data. Applied statistics deals with the application of statistical devices and techniques in the analysis of data relating to other sciences. Main branches of applied statistics are Biometry, Demography, Econometrics, Statistical Quality Control (SQC), Psychometric etc.


1.5 SCOPE AND APPLICATIONS OF STATISTICS Statistics is considered to be a distinct branch of study applicable to investigations in many branches of science. Statistical methods are applied to specific problems in Biology, Medicine, Agriculture, Commerce, Business, Economics, Industry, Insurance, Sociology, Psychology etc.

Statistics in Biology, Medicine, Agriculture, etc.: Statistical methods are in much use in the study of problems associated with Biological sciences. They are applied in the study of growth of plant, movement of fish population in the ocean, migration of birds, effect of newly invented medicines, theories of heredity, estimation of yield of crop, effect of fertilizer on yield, birth rate, death rate, population growth, growth of bacteria etc. The branch of Statistics which deals with problems in Biology is Biometry. The branch which deals with problems relating to population growth is Demography. It is well known that insurance premiums are based on the age composition of the population and the mortality rates. Actuarial science deals with the calculation of insurance premiums and dividends.

Statistics in Economics, Commerce, and Business etc.: Statistics is part and parcel of Economics, Commerce and Business. Statistical analysis of variations of price, demand and production are helpful to businessmen and economists. Cost of living index numbers help in economic planning and fixation of wages. They are used to estimate the value of money. Analysis of demand, price, production costs, inventory costs, etc., help in decision making in business activities. Management of limited resources and labour in obtaining maximum profit is done by statistical analysis of data. Planned recruitments and distribution of staff, proper quality control methods, careful study of demand for goods in the market, capture of market by advertisement, balance investment, etc. help the producer to extract maximum profit out of minimum capital. In industries, statistical quality control techniques help in increasing and controlling the quality of products at minimum cost. A government’s administrative system is fully dependent on production statistics, income statistics, labour statistics, economic indices of cost, price, etc. Economic planning of any nature is entirely based on statistical facts. Statistics has become so important today that hardly any science exists independent of this, and hence the statement ‘Sciences without Statistics bear no fruit; Statistics without Sciences has no root’.

1.6 FUNCTIONS OF STATISTICS Important functions of statistics are:

1. It simplifies complexity of the data: Complex numerical data are simplified by the application of statistical methods. For instance, complex data regarding varying costs and prices of commodities of daily use can be reduced to the form of cost of living index number. This can be understood easily.

2. It reduces the bulk of the data: Voluminous data could be reduced to a few figures making them easily understandable.

3. It adds precision to thinking: Statistics sharpens one’s thinking.

4. It helps in comparing different sets of figures: The imports and exports of a country may be compared among themselves or they may be compared with those of another country.


5. It guides in the formulation of policies and helps in planning: Planning and policy making by the government is based on statistics of production, demand, etc.

6. It indicates trends and tendencies: Knowledge of trend and tendencies helps future planning.

7. It helps in studying relationship between different factors: Statistical methods may be used for studying the relation between production and price of commodities.

1.7 LIMITATIONS OF STATISTICS Major limitations of Statistics are:

1. Statistics does not deal with qualitative data. It deals only with quantitative data: Statistical methods can be applied only to numerically expressed data. Qualitative characteristics can be studied only if an alternative method of numerical measurement is introduced.

2. Statistics does not deal with individual fact: Statistical methods can be applied only to aggregate of facts. Single fact cannot be statistically studied.

3. Statistical inferences (conclusions) are not exact: Statistical inferences are true only on an average. They are probabilistic statements.

4. Statistics can be misused: Increasing misuse of Statistics has led to increasing distrust in statistics.

5. Common men cannot handle Statistics properly: Only statisticians can handle statistics properly. An illogical analysis of statistical data leads to statistical policies.

QUESTIONS: 1. What do you mean by the word „Statistics‟?

2. State and explain Prof. Horace Secret’s definition of Statistics. (Define „Statistics‟. What are its characteristics?)

3. What are the functions of Statistics?

4. What are the limitations of Statistics?

5. What is the role of Statistics in Business, Commerce and Economics?


2nd Chapter Collection, Classification and Presentation of Data

STRUCTURE2.1 Introduction – Basic Notions

2.2 Primary Data & Secondary Data

2.3 Classification and Tabulation – Frequency Distribution

2.4 Graphical presentation of data

LEARNING OBJECTIVES:After a proper study of this chapter, the student should be able to,

1) Distinguish between primary and secondary data.

2) Understand (and to use) different methods of collecting data (primary and secondary).

3) Identify different sources of primary and secondary data.

4) Construct an appropriate frequency distribution (discrete or continuous) to a given raw data.

5) Draw Histogram, frequency polygon or ogives depending on the requirements.

2.1 INTRODUCTION – BASIC NOTIONSThis chapter is designed to make the student understand different methods of collecting primary data. Further, classification of the collected data is clearly explained. Presenting the data through frequency distributions – their construction along with the graphical techniques including histogram, frequency polygon and gives form integral parts of this chapter.

Some Basic Notions

In a statistical study, the objects whose characteristics are studied are called Units or Individuals. Thus, associated with any statistical study, we have a set of units. The totality of (collection of all) units under consideration is called Population or Universe. For example, in the study of financial condition of families of a particular tribe, the set of all families belonging to the tribe is the population. The families are the units. In the study of financial condition of families in India, the set of all families in India is the population.

A population may be that of people trees, fields, heights, etc. A population which contains finite number of units is a Finite Population. A population which contains infinite number of units is an Infinite Population. For instance, population of students belonging to a college is finite, whereas, population of heights of plants is infinite.


The units (members of a population) have some characteristics. These characteristics may be numerically measurable or may not be numerically measurable. A characteristic which is numerically measurable is a Quantitative characteristic and a characteristic which is not numerically measurable is a Qualitative characteristic.

Examples:

Quantitative Characteristic:

1. Height of a person. 2. Weight of an apple.3. Marks of a student4. Profit of an organization5. Temperature of a city

Qualitative Characteristic:

1. Intelligence of a child2. Skin-Colour of a person3. Love of a person with his Allah4. Fear of someone5.

In a population some characteristics remain the same for all units and some others vary from unit to unit. A quantitative characteristic which varies from unit to unit is a Variable. A qualitative characteristic which varies from unit in unit is an Attribute. For example, height weight etc. is variables and sex. Skin-colour etc. are attributes.

DISCRETE VARIABLE VS CONTINUOUS VARIABLEA variable may be Discrete or Continuous a variable which assumes only some specified values in a given range is a Discrete Variable. A variable which assumes all the values in the range is a Continuous Variable.

Examples:

Discrete variable

1. Number of children per family2. Number of seeds per bean pod

Continuous variable

1. Height of persons2. Weight of apples

A statistician is concerned with the study of variables which show statistical (stochastic non-mathematical) variation. Such a study is called Statistical investigation (Statistical survey). The person who conducts the statistical survey is called Investigator. The investigator plans the survey collects


the required data analyses them and finally draw condition. The investigator collects information about the units of the population either directly from them or through others. The persons who supply information are Informants Respondents. The investigator may directly collect information from the informants or he may collect them through his agents. These agents who collect and hand over information to the investigator are Enumerators.

For a statistical survey of a population the investigator may collect information from each and every unit belonging to the population or he may collect information from some selected representative units only. In the latter case, the group of representative units from whom information are collected is called Sample. Thus Sample is a representative portion of the population. A statistical survey in which sample is made use of is Sample survey and a survey in which the whole population is made use of is Census enumeration. The number of units in a sample is the sample size. It is denoted by „n‟.

For a statistical study, numerical data are collected such numerically expressed data are quantitative data. Data which are not numerically expressed are qualitative data.

A study of characteristics of units of a population by using statistical devices and techniques is called statistical Investigation or statistical survey. It is an analysis of statistical structure of the population with regard to some characteristics of the units. Statistical investigation may concern any theory or activity in Economics, Business, Industry, Medicine, etc. The investigation may be –

a. To prove a newly formed theory concerning a population. The theory may be “Tobacco chewing leads to mouth cancer”. It may be required to verify this theory.

b. To know the existing nature of the population. The investigator may be required to find out the cost of living index for a community of people or he may be required to find out the birth rate in a city.

c. To formulate a new theory concerning the population. An Economist may be interested in establishing a new theory regarding related variation of price and demand for a commodity.

STAGES OF STATISTICAL INVESTIGATIONMainly a statistical investigation has two stages. They are –

1. Planning and preparation.

2. Executive of the survey.

1. PLANNING AND PREPARATIONAt the outset, the survey should be properly planned. If the object of the survey is clearly known and the survey is well-planned, it can be systematically and successfully conducted. Pre-decision regarding the method of data collection, degree of accuracy, type of information to be collected, etc., are necessary for obtaining valid and complete data. Selection and training of enumerators and appointment of supervisory staff should be done. The plan of tabulation and presentation should be prepared in advance. The characteristics


which are to be studied and the methods that are to be adopted for the analysis should be predetermined. Proper outline for interpretation of data should be formed.

A well-planned survey is likely to given results of highest precision.

2.2 EXECUTION OF THE SURVEYAfter planning and making all the preparations for the survey, the survey is executed. The execution has four steps, namely (i) Collection of data (ii) Scrutiny, editing and presentation of data (iii) Analysis of data (iv) Interpretation of analysed data.

(i) Collection of dataThe validity of statistical inferences depends mainly on the data on which they are based. Incomplete and defective data lead to invalid conclusions. Thus, data should be collected with utmost care and hence, the stage of collection carries much importance. During the time of collection, the supervisory staff should ensure that enumerators are prompt in collecting accurate data. If there is non-response from any unit, special efforts should be made to secure data from such units.

(ii) Scrutiny, editing and presentation of dataThe collected data should be scrutinised for reliability, consistency and completeness. The volume of data can be reduced by systematically arranging them in the form of tables. The tabulated data, if necessary, may be represented by diagrams and graphs.

(iii) Analysis of dataThe data are analysed by using statistical devices and techniques. Statistical constants such as averages, measures of variation, correlation, trend etc., are derived. With the help of the derived values, assumptions (hypotheses) regarding the population are tested.

2.3 PRIMARY DATA & SECONDARY DATA PRIMARY DATA

2.3.1 Primary data Primary Data are the fresh data collected directly from the field. They are first-hand data.

Since primary data are collected specially for a particular purpose, extra care is taken to collect all the required data with required degree of accuracy. Primary data are reliable, complete and fresh.

Primary data are costly. Their collection consumes more time and labour than the collection secondary data, especially when the population is vast. If an investigator collects information regarding births and deaths directly from the people, the data are primary data.

Important methods of collecting primary data are:

1. Direct personal interview.

2. Indirect personal interview.


3. Information through correspondents.

4. Method of Questionnaire.

5. Method of Schedule

6. Electronic Media.

2.3.2 Secondary DataSecondary data are the data which the investigator does not collect directly from the field. They are the data which he borrows from others who have collected them for some other purpose.

Since Secondary data are primarily collected for some other purpose, they may not contain all the required information. The degree of accuracy may be different. They, generally, are not as dependable as primary data.

Since Secondary data are primarily collected for some other purpose, they may not contain all the required information. The degree of accuracy may be different. They, generally, are not as dependable as primary data.

Secondary data are cheap. Specially, when the population is vast and funds available is meagre, secondary data are preferred. Collection of secondary data consumes less time and labour than collection of primary data.

If an investigator collects the required data of births and deaths from the city corporation office records, the data are secondary data.

At the time of collection of secondary data, reliability of the source and validity of the data should be verified. The data should contain all the necessary information with required degree of accuracy.

Important sources of secondary data are –

a) Published sources:1. Reports and publications of Central and State Government departments. 2. Reports and publications of International bodies such as U.N.O., I.M.F., etc. 3. Publications of Banks, Research institutions, Administrative Offices, etc. 4. Magazines and Newspapers

b) Unpublished sources:1. Records maintained at Government offices, Municipal offices, Hujra offices, etc.

2. Records maintained by Research Institutions, Research scholars, etc.

2.4 METHODS OF COLLECTION OF PRIMARY DATAImportant methods of collecting primary data are –

1. Direct personal interview.

2. Indirect personal interview.


3. Information through correspondents.

4. Method of questionnaire (Mail enquiry).

5. Method of schedule (Collection through enumerators).

1. Direct personal InterviewHere, the investigator personally comes in contact with the units. The investigator interviews (observes) the units and extracts the required data from them. Since the investigator personally collects the data, the data are reliable. The investigator, here, can use his skill in extracting accurate data from the units. He can also collect, if necessary, additional interesting information. This method is costly and time consuming. It can be used only when the number of units to be contacted is small and when they can be reached easily. In this method, there is scope for bias.

2. Indirect personal interviewIn this method, the investigator does not contact the units directly but, he contacts persons who are in close association with the units. These persons (informants) supply information to the investigator.

The method is especially useful in collecting data regarding drinking habits of workers of a factory by contacting the bars of the area.

This method is costly; it consumes much time and labour. In this method, there is scope for bias. This method can be adopted only when the number of units is small.

3. Information through correspondentsIn this method, the investigator appoints his agents called correspondents at different place. These correspondents collect required data in their area and hand them over to the investigator.

This method is useful when data are to be collected regularly for a long period of time. The Government collects yield statistics, birth and death statistics, etc., through the village panchayat offices by this method.

4. Method of questionnaire (Mail enquiry)Questionnaire is a list of questions, answers for which are filled in by the informants and these answers are the required information for the investigation.

The questionnaire is sent to the informants by mail or otherwise. The informants are instructed to answer the questions (fill up the questionnaire) and send them back to the investigator. Thus, the investigator obtains the required data.

This method is highly useful when the area under investigation is vast. It is cheap. It consumes less time and labour.

5. Method of schedule (Collection through enumerators)Here, the investigator collects data through trained enumerators. The enumerators contact the informants and with the help of a schedule collect the required information. Schedule is a list of items on which the enumerators have to collect and record information. It is filled in by the


enumerators. Provided the enumerators are well-trained, the data collected by this method are reliable and accurate. This method is especially useful when the information are illiterate and they are spread over a vast area. The method is cheap.

This method requires intense training of the enumerators. Reliability of the data depends mainly on the training given to them and their integrity. In this method, there is scope for bias. The decennial population census of India is conducted by this method.

2.5 DRAFTING QUESTIONNAIRE (DRAFTING SCHEDULE)Success of questionnaire method of collection of data depends mainly on proper drafting of the questionnaire. In drafting a questionnaire, following general principles should be considered:

1. The number of questions should be as less as possible. If a lengthy questionnaire is unavoidable, it should be divided into two or more parts.

2. Questions should be short and simple.

3. Questions should be such that the answers to them are short. For instance, the question – „Are you married?‟ can be answered by saying „Yes‟ or „No‟ by the informant.

4. As far as possible, questions regarding personal matters should not be asked. The questions should be so framed that they do not hurt the feelings of the informants.

5. Questions should not be ambiguous, they should be objective. Any scope for misinterpretation should not be there.

6. Questions should be logically arranged.

7. Any clarification, if necessary, regarding any of the questions, should be supplied.

8. Questions should be so framed that validity of the information supplied by the informant can be cross checked.

9. A covering letter introducing the investigator and indicating the purpose of the survey should be attached to the questionnaire. It should supply necessary instructions to the informants regarding the return of the questionnaire, degree of accuracy, etc.

2.6 PILOT SURVEY (PRE-TEST)Pilot survey is a small trial survey (survey of a few units) undertaken before the main survey is conducted.

In a statistical survey, especially when a questionnaire or schedule is used, the investigator should conduct a pilot survey and verify whether the procedure to be adopted by him is efficient or not. Based on the experience of the pilot survey, he may introduce necessary changes in his procedure. He may alter ill-worded questions and make necessary changes in the Questionnaire.


Pilot survey will reduce inconveniences and loss due to ill-worded questions, misinterpretation of questions, non-response to questions, etc.

2.7 Frequency DistributionThe organization of a set of data in a table showing the distribution of the data into classes or groups together with the number of observation is each class or group is called a frequency Distribution. The number of observation falling in a particular class is referred to as the class frequency or simple frequency and is denoted by f. Data presented in the form of a frequency distribution are also called groped data while the data in the original form are referred to as ungrouped data.

2.7.1 Class LimitsThe class limits are defined as the number of values or the values of the variables which describe the classes; the small number is the lower class limit and the larger number is the upper class limits.


3rd Chapter Measure of Central Tendency

STRUCTURE 3.1 Measures of Central Tendency

3.2 Arithmetic Mean (Mean)

3.3 Median

3.4 Mode

3.5 Geometric Mean (G.M.)

3.6 Harmonic Mean (H.M.)

3.8 Questions

LEARNING OBJECTIVES: After the proper study of this chapter the student should be able to:

1) Understand the concept of Central Tendency.

2) Compute Arithmetic Mean, Median, Mode, Geometric Mean and Mode for any given data set.

3) Understand merits and demerits of each measure and thereby

Understand which measure to be used when?

3.1 MEASURES OF CENTRAL TENDENCY This chapter is devoted to different measures used to summarise the data. Different measures discussed are Mean, Median and Mode. Along with these three fundamental and trivial measures, two other measures, Geometric Mean, Harmonic mean are clearly introduced. We will present you its definition, method of computation, Interpretation and uses form the structure of the explanation for each measure.

Generally, in a frequency distribution, the values cluster around a central value. This property of concentration of the values around a central value is called Central Tendency. The central value around which there is concentration is called measure of central tendency (measure of location, average). Generally, a simple comparison of frequency distribution is made by comparing their measures of central tendency.


The term central tendency refers to the "middle" value or perhaps a typical value of the data, and is measured using the mean, median, or mode. Each of these measures is calculated differently, and the one that is best to use depends upon the situation.

3.2.1 Arithmetic Mean (Mean)

The arithmetic mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers. The symbol "μ" is used for the mean of a population. The symbol "X" is used for the mean of a sample. The formula for overline {X} is shown below:

Arithmetic Mean for RAW Data

The formula for X is essentially identical:

µ= ΣX/N for population Data

X=Σ x /n For Sample Data

Where ΣX is the sum of all the numbers in the sample and“n” is the number of numbers in the sample.

As an example, the mean of the numbers 1, 2, 3, 6, 8 is 20/5 = 4 regardless of whether the numbers constitute the entire population or just a sample from the population.

Exercise 1: Heights of six students are 163, 173, 168, 156, 162 and 165 Cms. Find the arithmetic mean.

Solution: The arithmetic mean of the heights is–

X=∑ Xn

=163+173+168+156+162+1656

¿ 9876

=164.5Cm

Example 2:

In a one-day cricket match, a bowler bowls 8 overs, he gives away 3, 5, 12, 0, 4, 1, 3 and 7 runs in these overs. Find the mean run rate per over.

Solution: The mean run rate is;


x= ∑ of the valuesnumber of values

=∑xn

x=3+5+12+0+4+1+3+78

=358

=3.375 runs per¿

Example 3: Ali scored the given number in four subject.

45, 32, 36, 41

He has been failed in all subjects. Find out the arithmetic mean for these numbers.

Solution:

x=∑Xin

Data: 45, 32, 36, 41

x=45+32+36+414

=1544

=38.5 the average numbers∈eachsubject

Arithmetic Mean for Ungrouped Frequency DistributionIf the observations X1, X2, X3, …., Xn, have frequencies f1, f2, f3, … fn, the arithmetic mean is

Example 4: In an office there are 84 employees, their salaries are given below.

Salary (Rs) 2430 2590 2870 3390 4720 5160Employees 4 28 31 16 3 2

i. Find the mean salary of the employees.ii. What is the total salary of the employees?

Solution:

Salary Employees fx


(Rs.)(X)

(f)

243025902870339047205160

428311632

97207252088970542401416010320

Total 84 249930

i. The mean salary of the employees is –

x=∑fxn

=24993084

=Rs .2975.36

ii. Total salary of the employee is – ∑fx = Rs.249930

Example 5: The list of goals of Afghanistan Football team has scored in 36 matches are given below.

Number of Goals (Xi) 0 1 2 3 4Number of Matches (fi) 3 4 10 15 4

The formula is x=∑fixin

xi fi fixi0 3 01 4 42 10 203 15 454 4 16∑ 36 85

x=∑fixin

Putting the values

x=8536

=2.36 is the average goals scoredby AfghanistanFootballTeam


Arithmetic Mean for Grouped Frequency DistributionExample 6: The age limit of employees of an organization and their number is given below. Find out the arithmetic mean.

Age of the employees

Number of employees (fi)

30-39 340-49 450-50 1060-69 1

Solution:

The formula to find out the Arithematic mean for a group data is;

x=∑fixin

Class Limits (Age) Frequency fi Mid-Point (Xi) fixi

30-39 3 34.5 103.540-49 4 44.5 17850-50 10 54.5 54560-69 1 64.5 64.5∑ 18 891

x=∑fixin

x=89118

=49.50

Thus, the average age of each employee is 49.5 years.

GEOMETRIC MEANThe geometric mean of n values is the nth root of the product of the values. It is denoted by G. Thus, geometric mean of the values x1, x2, x3, ...... xn is –

G= n√X 1+X 2+X 3+…+Xn

If logaritham are used. Then the formula will be;

G=Antilog ⦋❑❑


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