STATISTICS-I

Unit 1

Definition of statistics:

1. Yule and Kendall defines that, By statistics we mean quantitative data affected to a marked extend

multiplicity of causes.

2. Dr. A.N.Bowley defined statistics as , Statistics are numerical statements of facts in a department of enquiry,

places in relations to each other.

3. Corner defines statistics as, Statistics are measurements, enumerations or estimates of natural or

social phenomena, systematically arranged so as to exhibit their inter

relations.

Importance of Statistics:

The methods and techniques available in statistics helps people to solve

various problems presented in statistical data. Hence it has a good application

and scope in the field of commerce, economics, physics, chemistry, botany,

zoology, psychology etc.

Statistics in Business:

Statistics is the most commonly used in business. It helps to take decision

regarding whether the company start a new business. The existing companies

can also make comparative study about their performance with the performance

of other companies through statistical analysis. The existing companies can also

project their future with regression and correlation analysis.

Statistics in Economics:

The problems in economics cannot be studied without the use of

statistics. The laws of economics always refer to statistics, in order to prove

their accuracy. The wider use of application of economics is not possible

without the knowledge of statistics.

Statistics in Astronomy:

Astronomers were the first who made recordings of the movements of

heavenly bodies and studied the eclipse and astronomical issues on the basis of

statistics.

Statistics in Education:

Statics is widely used in education. Research has become a common

feature in all branches of activities. Statistics is necessary for the formulation of

policies to start new courses, consideration of facilities available for new

courses etc. there are crores of people engaged in research work to test the past

knowledge and evolve new knowledge, and these are possible only through

statistics.

Statistics in Mathematics:

Statistics can be considered to be an important member of mathematics

family. Statistics indicates a quantitative study of many facts such as Average,

interpolation, extrapolation, correlation, regression analysis of the time series,

index numbers etc. To make these studies, the application of mathematics is

unavailable. Thus, we find that there is a close relationship between

mathematics and statistics.

Statistics in Management:

The important managerial activities like planning, directing and

controlling are properly executed with the help of statistical data and statistical

analysis and statistical analysis. Statistical techniques can also be used for the

payment of wages to the employees of the company.

Statistics in Banking and Finance:

Banking and Financial activities use statistics most commonly. Banks

also applies statistical techniques in calculating interest. Stock exchange,

financial institutions like Industrial Development Bank of India, State financial

corporation of India also uses statistics in projecting the future and to solve

various statistical problems.

Limitations of Statistics:

The following are the important limitations of statistics:

1. Statistics studies only quantitative phenomenon: Statistics studies only the quantitative phenomenon. It wont deal with the

phenomenon which could not be expressed in quantitative value. Hence

qualitative phenomenon like intelligence, honesty, studies etc. Cannot be

dealt with by statistics unless they are expressed in quantitative values.

2. Statistics deals with aggregates and not with individual measurement: Statistics deals with aggregate facts. It requires a series of figures for

calculating averages and for analysis. The individual measurement has no

recognition. It could not be taken into consideration for any statistical

analysis.

3. The statistical results are not perfectly accurate: The statistical theories wont give accurate results. The results would be

only approximate values. First of all the data collected for analysis may

not be accurate also.

4. Data must be accurate in statistics : Data for statistical analysis must be uniform. For example the data related

with the income of people in a locality could be mixed with the data

related to the expenditure of people. These tow phenomena should be

studied separately.

5. Statistics can be misused : Only experienced and efficient persons can handle statistics in a proper

way. Untrained and inefficient persons may not produce accurate results

with the help of statistical techniques.

Uses of Statistical Methods:

1. Classification and tabulation of raw data for the purpose of easy interpretation and analysis.

2. Various measures of dispersions to study the spreading of the observations and deviations from the average.

3. Various measures of averages for simplifying and condensing the numerical data.

4. Coefficient of dispersion for comparing two different sets of data. 5. Coefficient of correlation to study the degree of relationship between two

variables.

6. Regression analysis gives the relationship between the variables which is also used for the prediction of the value of one variable when the other is

known.

7. Various measures of time series for forecasting and for studying seasonal trend.

8. Index numbers of price, quantity, cost of living which are used in business and commercial economics. In particular, cost of living index

number is a parameter for determining the rates of dearness allowance to

the employees of all categories.

9. Statistical quality control is a measure to decide whether a production process is under control or not.

10. Sampling techniques such as random sampling, stratified sampling etc. which are used in business and socio-economic surveys.

11. Hypothesis testing for examples is used for testing the significance of the difference between the population parameter and sample statistics.

Functions of statistics:

1. It simplifies complexity: The important function of statistics is to collect the facts and figures in a

systematic manner and to present them in such a form that they are easier

to understand. The huge masses of data can be converted into a picture, a

diagram etc. which are easier to understand.

2. It presents the facts in a definite form: Statistics presents facts in a precise and definite form and thus help

proper comprehension of what is stated. Because of this definiteness, the

application of statistical methods has been increased and has attained

popularity in various sciences.

3. It facilitates comparison: Statistics helps in comparing the data with respect to time and location. It

also helps to compare one phenomenon with other. Comparison is one of

the main functions of statistics.

4. It enlarges individual experiences:

Statistics enlarges human knowledge and experience.in statistical study,

vague and indefinite ideas become clear and definite. It is a master key to

solve the problems of human life.

5. It foresees future: Statistical methods are very helpful in forecasting future trends on the

basis of the analysis of the past data, as modified in the light of current

conditions statistics helps in analysing past and present conditions and

making projections for the future.

6. It studies relationship: The extent of relationships between different data can be measured;

coefficient of correlation, regression etc. are the measures through which

we measure the function relationship.

7. It helps the government: The importance of statistics in the administration of the country is greatly

increased in the present times. The use of statistical data and statistical

techniques is so wide that all most all ministries and departments have

separate statistical units.

8. It tests the laws of other sciences: Statistics helps in testing the laws of other physical sciences and social

sciences. Hypothesis becomes a law, because its truthfulness is tested and

made acceptable through statistical proofs.

9. It helps in formulating and testing hypothesis: Statistical methods are helpful to develop new theories. It provides

guidance in the formulation of new policies and theories at all stages and

the drawing of plans in all the fields.

Distrust of statistics:

1. Deliberate twisting of facts. 2. Inconsistent definitions. 3. Failure to represent complete data. 4. Inappropriate comparison. 5. Wrong inference drawn. 6. Using misleading basis. 7. Improperly classified data. 8. Data collected by improper persons. 9. Inaccurate measurement. 10. Arithmetical errors. 11. Lack of technical knowledge of statistics. 12. Being biased opinion if the investigator.

Unit 2: Statistical Enquiry

Definition:-

By statistical enquiry or survey we mean a search for the collection

of facts of a given problem. The collected numerical facts are the raw materials

through which the problem is to be studied. The relevant collection of

information by any agency to tackle a particular problem under study is termed

as statistical enquiry or survey. It consists of two stages

1. Planning and design of enquiry. 2. Executing the survey or design.

1. Planning and design of enquiry:

At this stage it seems that the collection of data is the first step for any

statistical enquiry. But in a well-planned enquired data collection. The

following are to be need in lawful consideration:

a. Object of an enquiry: Statistical investigation may be conducted to serve the general purpose or

special purpose. Its object should be clearly defined; the enquiry must be

properly conducted with minimum possible waste of time, energy and

money.

b. Scope: The scope of enquiry should be decided with reference to the space, time

and the number of items to be covered. If the scope is not determined

unnecessary data may be collected and necessary data may be neglected.

c. Statistical unit: The investigators must determine the units in terms of which the data has

to be collected. A unit of measurement must be defined so that uniformity

can be maintained throughout the process of enquiry. It can be classified

into.

i. Unit for collection of data. ii. Units for statistical analysis and interpretation.

d. Sources of data: The two sources of information are primary and secondary.

a. Primary data : The data collected for the first time is known as primary

data and it can be collected through

1. Director personal interview 2. Indirect interview. 3. Information through agencies.

b. Secondary data : These data are already collected by someone and

available for the present study. The various sources of

secondary data are.

1. Published sources. 2. Unpublished sources.

e. Method of data collection : The two methods of data collection are:

1. Census method or complete enumeration method. 2. Sample method or partial enumeration method

If all the units of the universe under study are considered it is called

census enquiry.

If only a representation part of the universe is considered it is called

sample enquiry.

f. Frame : A list of all units under study is known as frame, it is important to

identify the unit which constitute population.

g. Standard of accuracy: It should be maintained in the desired enquiry and it should be

planned accuracy desired depends upon the scope if the enquiry.

h. Type of enquiry: There are other considerations as to the type of enquiry to be

undertaken-confidential or non-confidential, direct or indirect, census

or sample etc.

2. Executing the survey:

To conduct the enumeration we have to do proper selection of

enumerators and give training to them. Their work is also closely watched by

the supervisors, the collected data are handed over to the office. In office the

data is converted and processed by analysing the data into the form of a report

which is ready for publication.

1. Setting up of administrative team: An administrative organisation is needed for an enquiry and

the size of the organisation depends on the nature and scope of

enquiry. If the area is wide and covers a large geographical area,

regional offices may be set up apart from central office. If the

enquiry is limited to a small area no sub offices may be needed.

2. Design and form of questionnaire : The drafting of a schedule or questionnaire is an art. It is the

medium of communication between the investigator and the

respondents. Therefore it should be designed with care and caution.

There is no hard and fast rule in designing forms.

3. Selection of and training of field investigators: The success of survey depends on the work of enumerators.

Therefore they are properly selected and thoroughly trained for the

field work. Constant supervision is essential to see that the result is

off higher order. Enumerators must first understand the purpose of

study and the conduct of work. The respondent should not be

annoyed by irrelevant talks but the success of survey depends on

the ability of the enumerators who has patience, knack, pleasing

habit at the field of work.

4. Supervision of field work: A team consisting of supervisor and a group of enumerators

work in the field. A constant check by the supervisor ensures

accuracy. Generally the field of checks are done on a random basis.

The system or the method of field check-up should be kept secret

to be more effective.

5. Follow-up of non-responses: There may be respondence who do not supply desired

information, there must be a proper scheme to deal with such non-

response or non- availability of respondence. The respondence may

again be conducted.

6. Processing and analysis of data: A thorough check-up is done here. The data generally coded

transferred to cards or tapes/cd with the help of punching machines

or computers. This operation is important and so great caution is

required. Most of the work is tabulated by computers, use of

computers save time and energy.

7. Preparation of report: After the data has been analysed generally a report is drafted

to show the findings of the survey. It is the final step in execution.

The findings or the results of the investigation are given authority

in the form of the paper or as the report.

SAMPLING DESIGN

Population:

The population is a possible observation of the time which is to be

investigated. The term population does not refer to people but in technical term

is used to describe the complete group of persons or objects for which the y

Example: if we want to study the average weight of the students of the college

where 800 students are studying the population is 800 students.

Types of population:

1. Finite and infinite population. 2. Hypothetical and existent population.

1. Finite population: When the no. of observation can be counted it is called as finite

population.

Example: If we study the economic background of students of a college

say X all the students belonging to that college will constitute the

population and the member will be finite.

Infinite population:

When the no. of observation cannot be measured and is infinite

then it is an infinite population.

Example: the no. of stars in the sky, the no. of people watching a

particular television in the whole universe.

2. Existent population: A universe containing persons or concrete objects is known as

existent or real population.

Example: the no. of students in the university, the number of population

of the city, the no. of population in the employee

Hypothetical population:

A hypothetical universe which is also known as theoretical

population. It is the one which does not consist of concrete objects.

Example: If we toss a coin infinite no. of times it is a hypothetical

population.

Information or population can be collected in two ways:

1. Census method. 2. Sample method.

1. Census method: In census or universal coverage every element of the population is

included in the investigation where we make a complete enumeration of

all items in the population it is known as census method.

Example: if study the average expenditure of a particular university say X

and if there are 50000 students studying in that university we must study

the expenditure of all 50000 students. This method is known as census

method.

Merits:

a. The data are collected from each and every item of the population. b. The results are more accurate and reliable. c. Intensive study is possible. d. The data collected may be used for various surveys, analyses etc. Demerits:

a. It requires a large number of enumerators. b. It is a costly method. c. It requires more money, labour, time, energy etc. d. It is not possible when the universe is infinite.

2. Sample method:

In our daily life we have been using sampling without knowing

about it.

Example: a homemaker tests a small quantity of rice to see whether it has

been well cooked. But will not inspect the all the rice therefore in this

method only a part of group of population will be studied in the case of

sample enquiry.

Merits:

a. It saves time when the results are urgently required. b. It reduces cost since few items are selected for sampling. c. It has administrative convenience and more scientific. d. The degree of accuracy in this method is higher than census h and

every unit method.

SAMPLING METHODS

Methods of sampling: 1. Random sampling method:

a. Sample or unrestricted method. b. Restricted or stratified method.

Stratified sampling

Systematic sampling

Cluster sampling 2. Non-random sampling:

a. Judgement or purposive sampling b. Quota sampling c. Convenience sampling.

1. Random sampling method: A random sample is one where each item in the universe has an equal

chance of known opportunity of being selected A random sample is a

sample selected in such a way that any item in the population has equal

chance of being included.

Sample- simple random sampling It is the technique in which sample is so drawn that each and every

unit in the population has an equal and independent chance of being

included in sample. The two methods adopted are:

Lottery method Table of random numbers

Merits:

1. More scientific. 2. More representation. 3. Sampling error can be measured.

Demerits:

1. When the distribution is large this method cannot be used. 2. If the sample size is small, then it does not represent population.

Restricted random sampling When the population is having difficult segments with respect to

the variable under study then it is stratified sampling. First the population

is divided into two sub-groups and a sample is drawn from it. There are

two types of stratified sampling:

Proportional sampling Non proportional sampling

Merits

1. It ensures greater accuracy. 2. It is easy to administer and sub-divide.

Demerits

1. It requires more money, time and statistical experience.

Systematic sampling: It is also known as quasi random sampling. A systematic sample is

selected at random. When a complete list of population is available this

method is used, we average the items in numerical, geographical or

alphabetical.

Merits

1. It is simple and convenient. 2. The items and work is much reduced.

Demerits

1. It may not represent the whole population. 2. There is the element of personal bias of investigators.

Cluster sampling: It is also known as multi stage sampling. It refers to sampling

procedure which is carried out in several stages, the whole population is

divided into sampling units and these units are again divided into

subunits.This process will continue when we reach the least number.

Merits:

1. It introduces flexibility in the sampling method. 2. It is helpful in large scale survey and time consuming or

expensive.

3. It is valuable in under developed countries. Demerits:

1. It is less accurate than other models.

2. Non random sampling:

Judgement sampling: The investigator has the power to select or reject any item

investigation; the choice of the sample items depends on the judgement of

the investigator. He has the role to play in collecting information.

Merits

1. It is simple method. 2. It is used to obtain a more representative sample. 3. It is helpful to make public policy decision.

Demerits

1. Due to individual sample bias it may not be a representative one. 2. It is difficult to get correct sampling errors. 3. The estimates are not accurate. 4. Its results cannot be compared with other sampling studies.

Quota sampling: This sampling is similar to stratified sampling. It is used in USA

for investigating public opinion and consumer research. To collect data

the universe is divided into quota according to some good characteristics.

Each enumeration is then told to interview a certain number of persons

who are in quota. The selection of sample item depends on the personal

judgement.

Merits:

1. It saves time and money. 2. It will give quite reliable results. Demerits:

1. Personal prejudice and individual bias are there. 2. Sampling error cannot be determined.

Convenience sampling: The other name is chunk sampling. It is a convenient slice of

population which is commonly referred to as a sample. It is obtained by

selecting convenient population unit.

Merits

1. It is suitable when the universe is not clearly defined. 2. Sample unit is not clear. 3. Complete source list is not available.

Demerits

1. The result cannot be representative. 2. They are unsatisfactory. 3. They are bias.

Questionnaire:

It is the media of communication between investigator and the responder.

The success of investigator depends on construction of the questionnaire.

Point to be followed while forming a questionnaire:

1. The questionnaire should be brief. 2. The questions be simple understand. 3. Questions should be logically. 4. There must be choice like simple alternative questions, multiple

choice and specific information questions.

5. Proper words be should be used in questionnaire. 6. Questions of a sensitive and personal nature should be avoided. 7. Necessary instruction should be given to informanent. 8. Questions related to mathematical calculations should not be asked. 9. Questions must be capable of an objective answer. 10. A questionnaire should look attractive. 11. Pre-testing the questionnaire must be done before posting it. 12. The accuracy of questionnaire must be judged.

Pre- cautions required in the use of a questionnaire:

1. The person conducting the survey must introduce himself. 2. The aim and objective of the enquiry should be known to informants. 3. The number of questions should be restricted to the minimum. A

reasonable questionnaire may be 20-25 questions.

4. Instruction for filling the questionnaire should be given. 5. The questions should be attractive and intresting through proper layout. 6. The questionnaire should be pre-tested to find out its short comings if

any.

7. Questions of personal matter should be asked carefully.

Unit 3: Classification

Classification is the process of arranging the available facts into a

homogeneous groups or classes according to the resemblance and similarity.

Objects of classification:

The chief objects of classification are:

1. The condense of mass data. 2. To present the facts in a simple form. 3. To bring out clearly the point of similarity and dissimilarity. 4. To facilitate comparison. 5. To prepare data for tabulation 6. To eliminate unnecessary data. 7. To facilitate easy.

Rules of classification:

1. Exactness. 2. Mutually exclusive. 3. Stability. 4. Flexibility. 5. Suitability. 6. Homogeneity. 7. Mathematical accuracy.

Types of classification:

There are four types of classification:

1. Geographical classification. 2. Chronological classification. 3. Qualitative classification. 4. Quantitative classification.

1. Geographical classification: In this method the data are classified like states, districts, cities, talukas,

regions, zones etc.

Example:

Name of the town No. of employees

1. Madras 15000 2. Trichy 13000 3. Madurai 11000 4. Coimbatore 8000 5. Kanyakumari 5000

2. Chronological classification: In this type of data is classified according to the time of its occurrence,

such as years, months, weeks, days, hours etc.

Example: census data are expressed in decade, national income is

expressed every year, and departmental sales are expressed every month.

Time series are also called chronological classification. They further are

classified into the period of time and point of time.

3. Qualitative classification: In this method, the data is classified according to some quality or

attributes, such as gender, honesty, intelligence, literacy etc.

It can be further classified into two types:

a. Simple classification: If the data are classified into only two classes such as literate and

illiterate or honest or dishonest or male or female then this

classification is termed as simple classification.

b. Manifold classification: In this classification the datas are classified on the basis of more than

one attribute at a

time.

4. Quantitative classification: In this method, the data are classified according to some characteristics

which are capable of quantitative measurements like age, height, weight,

price, production, sales, income etc. then it is called quantitative

classification.

TABULATION

Tabulation: Tabulation is a systematic presentation of numeric data in columns

and rows in accordance with salient features and characteristics.

Croxton and Cowden state that

Either for ones own use or for the use of others the data must be

presented in a suitable form.

Definition:

Tabulation is the process of systematic and scientific presentation

of data in a compact form for further analysis. Tabulation is orderly

arrangement of data in columns and rows.

The main objectives of tabulation are:

1. To clarify the object of investigation. 2. To simplify the complex data. 3. To clarify the characteristic data. 4. To present fact in the minimum of space. 5. To facilitate comparison. 6. To detect errors and omission of data. 7. To facilitate statistical processing. 8. To help reference.

Rules of tabulation:

A good statistical table is an art. The following parts must be

present in all the tables:

1. Table number. 2. Title. 3. Head note. 4. Caption. 5. Stubs. 6. Body of the table. 7. Foot note. 8. Source note.

1. Table number: Table must be arranged with number in order to identify the table.

2. Title: Table must have title. The title should be clear, brief and concise. It should convey the content and purpose of the table.

3. Head note: It is a statement, given below the title and enclosed in brackets. For example, the unit of measurement is written as

headnote, such as in millions or in crores.

4. Captions: These are the headings for the vertical columns. They must be brief and self- explanatory. The main heading and sub

heading must be written in small letters.

5. Stubs: These are the heading or designation for the horizontal rows. Stubs are wider than the columns.

6. Body of the table: It contains the numerical information. It is the most important part of the table. The arrangement of the body is

generally in the form left to right in rows and from top to bottom

columns.

7. Foot note: If any explanation or elaboration regarding any items is necessary, foot notes should be given.

8. Source note: It refers to the source from where the information has been taken. It is useful to the reader to check the figures and gather

additional information.

Requirement (requisites) for good tabulation: The following are the requirement for a good tabulation:

1. Table must be simple. It should clearly convey the purpose for which it is prepared.

2. Columns and rows should not be too narrow or too wide. 3. Brief description should be given for the particulars of columns and

rows. If more details are required, it should be given as footnotes.

4. Table should clearly show the units of measurement. 5. Table should be an optimum one. It should appeal the readers. 6. Figures which are closely related should be kept together in rows and

columns.

Types of tables: Tables may be classified into four types:

1. Simple table. 2. Complex table. 3. General purpose table. 4. Special purpose table.

1. Simple table: Simple table may be defined as a table showing only one

characteristic. It is also called one way table. The model of one

way table or simple table is shown below.

Marks obtained by students of a class

Marks in Statistic No. of Students

Below 20 5

20 40 8

40 60 13

60 80 12

80 100 7

Total 45

2. Complex table: Complex table may be defined as tables showing two or

more characteristics. If the table shows two characteristics, it is

called two way table or double tabulation.

If the table shows three characteristics it is called triple tabulation,

if the table shows four or more characteristics it is called manifold

tabulation.

Marks scored by students in two classes

Marks No. of Students Total

Class A Class B

Below 20 5 10 15

20 40 8 15 23

40 60 13 20 33

60 80 12 18 30

80 100 7 7 14

Total 45 70 114

3. General table: General purpose tables are otherwise called reference tables

or repository tables. It provides information for general us.

Example: for this type of tables are published by statistical

department of government like census, tables about industrial and

agricultural production in various periods etc.

4. Special purpose table: Special purpose table is otherwise called summary tables or

derivative tables. Special purpose tables are derived from the

information of the general purpose tables.

Unit 4: Measures of Dispersion

Significance of measuring dispersion:

1. To determine the reliability of an average. 2. To facilitate comparison. 3. To facilitate control. 4. To facilitate the use of other statistical measures.

Properties of a good measure of dispersion:

1. Simple to understand. 2. Easy to calculate. 3. Rigidly defined. 4. Based on all items. 5. Amenable to further algebraic treatment. 6. Sampling stability. 7. Not unduly affected by extreme items.

Range:

It is defined as the difference between the largest observation and the

smallest observation.

Range = Largest observation Smallest observation = L S

Coefficient of range = L S

L S

Merits of range:

1. It is simple to compute and understand. 2. It gives a rough but quick answer. Demerits of range:

1. Only two extreme values are taken into account. 2. It is affected by extreme values. 3. It cannot be applied to open end cases. 4. It is not suitable for mathematical treatment. Uses of range:

1. It facilitates statistical quality control. 2. It facilitates weather forecasting.

Quartile Deviation:

Quartiles may be defined as those values which divide the total frequencies

into four equal parts. They are termed as lower quartile 1Q , Median 2Q and

upper quartile 3Q .

Quartile deviation = 3 12

Q Q

Quartile Coefficient = 3 1

3 1

Q Q

Q Q

Merits of quartile deviation:

1. It is easy to understand and calculate. 2. It can be used to calculate open end frequency distribution. 3. It is least affected by extreme values. Demerits of quartile deviation:

1. It is not based on all the items of the series. 2. It is not suitable for algebraic treatment. 3. It is very much affected by sampling fluctuation.

Mean deviation:

Mean deviation is the arithmetic mean of the deviations of a series computed

from any measure of central tendency.

Merits of mean deviation:

1. It is simple to understand and compute. 2. It is based on all items. 3. It is less affected by the extreme values. Demerits of mean deviation:

1. It is not suitable for algebraic treatment. 2. It is not very much accurate measure of dispersion. Uses of mean deviation:

1. It will help to understand standard deviation. 2. It is useful in marketing problems. 3. It is useful in forecasting business cycles.

Standard Deviation:

Standard deviation is the square root of the arithmetic mean of the

squares of deviations of all items of the distribution from arithmetic mean.

Relative measure of standard deviation is called the coefficient of variation.

Merits of standard deviation:

1. It is based on all items of the distribution. 2. It is amenable to algebraic treatment. 3. It is least affected by fluctuation of sampling. Demerits of standard deviation:

1. It is difficult to compute. 2. It is very much affected by the extreme values.

Properties of standard deviation:

1. Independent of change of origin. 2. Not independent of change of scale. 3. Fixed relationship among the measures of dispersion. Skewness:

Skewness or asymmetry is the attribute of a frequency distribution that

extends further on one side of the class with the highest frequency that on the

other.

Measures of Skewness: Measures of skewness indicate not only the extent of skewness but also

the direction. (i.e) the manner in which the deviations are distributed.

Absolute measures of Skewness:

(i) Karl Pearsons Coefficient of Skewness: 3

kp

Mean MedianS

(ii) Bowleys Coefficient of Skewness: 23 1

3 1

Q Q Median

Q QkS

Difference between Mean Deviation and Standard Deviation:

S.No Mean Deviation Standard Deviation

1 Actual sings + or - are ignored and

all deviations are taken as positive.

Actual signs + or - are ignored.

2 M.D can be computed from

Arithmetic Mean, Median or Mode.

S.D is computed from Arithmetic

Mean.

3 It is simple to calculate.

It is difficult to calculate.

4 Algebraic signs are ignored.

Algebraic signs are taken into

account.

Difference between Dispersion and Skewness:

S.No Dispersion Skewness

1 It shows up the spread of individaul

values about the central value.

It shows us the departure from

symmetry.

2 It is useful to study the variability

in data.

It is useful to study the

concentration in lower and higer

variables.

3 It judges the truthfulness of the

central tendency.

It judges the differences between

the central tendencies.

4 It is the type of averages of

deviation-average of the second

order.

It is not an average, but is

measured by the use of mean,

median and mode.

5 It shows the degree of variability.

It shows whether the concentration

is in higher or lower values.

LORENZ CURVE

Definition:

Lorenz curve is a graphical method of studying dispersion. It is a percentage

cumulative curve in which the percentage of cumulative values of one variable

is combined with the percentage of cumulative values of the other variable.

Procedures for drawing the Lorenz curve: Suppose X and Y are two variables considered for drawing the Lorenz

curve. The following steps are adopted for drawing the Lorenz curve.

1. Write down the cumulative values of X. 2. Express in percentage these cumulative values. 3. Write down the cumulative values of Y. 4. Express in percentage these cumulative values. 5. In graph paper, mark the percentages 0, 10, 20 100. 6. Join the diagonal points (0, 0) and (100,100). 7. Plot points (x, y) where x and y are the pairs of cumulative percentage

values of x and y.

8. Draw a smooth curve joining the plotted points.

Uses of Lorenz curve:

1. It is used to show the dispersion 2. It is a device used to show the measurement of economic inequalities in

distribution of income and wealth.

Unit 5: Vital Statistics

The term vital statistics refers to the numerical data or the techniques

used in the analysis of the data pertaining to the vital events occurring in the

given section of the population.

By vital we mean such as events of human life as fertility, morbidity, and

mortality. (Births, illness and deaths) marriage, divorce, separation, adoptions,

legitimations etc.

Methods of obtaining vital statistics:

1. Civil registration method 2. Population census method.

1. Civil registration method: The civil registration method may be defined as a unified process of

continuous, permanent and compulsory recording of the vital events and

characteristics thereof, as per the legal requirements of the country. In

India, the civil registration method covers registration births and deaths

only. It provides the best source of information on the vital statistics at all

levels. Overview of civil registration method is given as follows:

a. Registration of birth is a right of the child and is the first step towards establishing their identity.

b. Births and deaths are registered only at the place of their occurrence. c. Births and deaths are to be reported within 21 days of their

occurrence.

d. Persons giving the information about births/deaths are required to give it under their signature.

2. Population census method: The population census method was originally seen in providing data on

population at risk only. That is the denominator needed to estimate the

birth and death rates and other basic demographic parameters. However,

the rates so obtained in a sizeable number of developing countries were

too low to be accepted as true values.

Therefore, other specific questions were devised to gather the information

on fertility and mortality in the population censuses.

The strength of the census method is the fact that the population figures

by sex, age, place of birth, usual residence and other social and economic

variables are readily available at all levels of geographical sub-divisions

of the country.

Finally, census method is a very costly operation and requires long

advance planning, it is take only periodically at about 10 year interval,

which is one of the general limitation of census method.

Mortality:

The word mortality is used in relation to the occurrence of the deaths.

Measurement of mortality: 1. Crude death rate( C.D.R) 2. Specific death rate( S.D.R) 3. Standardised death rate(STDR)

1. Crude death rate: The simplest way of measuring the mortality is to relate the total

number of deaths during the period to the total population at the middle

of the period.

Crude death rate = 1000

x

x

P

D

Where xD is the number of deaths of persons aged x years.

and xP is the number of persons aged x years.

Crude death rate gives the probability that a person is randomly

chosen from the given population will die during the period from any

case whatsoever.It is simple to calculate. But the drawback of CDR is that

it is really crude measure of mortality and does not take into account the

age, sex and composition of the population.

2. Specific death rate: Specific death rate is the mortality rate calculated for a specific

class or section of the population. Specific death rates are the best

measures of mortality. Usually death rates are made specific with respect

to age and sex, although specify with respect to race, religion, cause of

death, etc. is also possible.

Age specific death rate at age x, 1000x

xx

P

Dm

3. Standardised death rate: Though specific death rates may be used to compare the mortality

position of two communities or places at different ages, they cannot

provide a comparison of over all mortality rates for these two populations

because, the specific death rates for one community may be higher at one

certain age but lower at other ages than the second community. What we

need for the purpose is single figure, some sort of average a weighted

average of age for specific death rates for each community so that the

figures will be directly comparable.

Standardised death rate of a community is the weighted arithmetic

mean of its age specific death rates using the figures of a given standard

population as weights. Standard population is the population of a larger

community and is denoted as sxP .

a. Direct method:

Standardised death rate =

sx

sxx

P

Pm

b. Indirect Method:

Standardised death rate for community A = (C.D.R) A

C

where the adjustment factor,

ax

sx

ax

sx

sx

sx

Pm

P

P

PmC

Fertility:

The word fertility is used in relation to the actual production of children

or occurrence of births, especially live births.

Measurement of Fertility:

1. Crude Birth Rate (C.B.R)

2. General Fertility Rate (G.F.R)

3. Total Fertility Rate (T.F.R)

1. Crude Birth Rate: Crude birth rate shows the rate at which population increases through

births. It is simple to calculate and has the sane defects as crude death rate.

Further-more, the total population used in the denominator is only the female

population and that too within a certain age interval in the child bearing

period.

Crude Birth Rate = 1000

x

x

P

B

Where xB is the number of births of persons aged x years.

and xP is the number of persons aged x years.

2. General Fertility Rate: It shows the rate at which females in the child bearing ages add to the

total population through births. Although GFR shows slight improvement

over crude birth rate, it is not a satisfactory measure. Because, the

probability of female bearing a child varies from age to age.

General Fertility Rate = 1000

xf

x

P

B

3. Total Fertility Rate:

Age specific Fertility Rate at age x years, 1000x

xx

P

Bi

f

Total fertility rate shows the rate at which a new born female

would on an average add to the total population, if the remained alive and

experienced fertility rates throughout the child bearing period. TFR is

only a hypothetical measure because all new females cannot be expected

to remain alive till the end of child-bearing period.

Total fertility rate = xi5

Population Growth:

Population growth is analysed by demographers in terms of four factors;

fertility, mortality, immigration and emigration.

Measurement of Population Growth:

1. Vital index.

2. Gross Reproduction Rate (G.R.R)

3. Net Reproduction Rate (N.R.R)

1. Vital index: The way of measuring the growth population using birth-death

ratio is called vital index.

Vital index

2. Gross Reproduction Rate: Gross Reproduction Rate measures the rate at which a new born

female would, on an average, add to the total female population, if she

remained alive and experienced the age-specific fertility rates till the end

of the child-bearing period. Like Total fertility rate, GRR is also a

hypothetical figure because; it does not take into account the mortality

experiences during the period. In the measurement of population growth,

it is appropriate to consider only the female births.

G.R.R =

or

G.R.R = xf i5 , where 1000

x

xf

xf

P

Bi

f

3. Net Reproduction Rate:

Net Reproduction Rate is the best measure of population growth. It can

never exceed Gross Reproduction Rate. Gross Reproduction Rate does not take

into consideration the fact of survival [(i.e.) some of the new born females

would die before reaching the child bearing period.] for correct measure of

population growth, the factor of mortality experienced by females must be

taken into account.

Net Reproduction Rate = 5 )( xxf i

Where is female age specific fertility rate

And x is the survival factor.