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LESSON – 1

STATISTICS FOR MANAGEMENT

Session – 1 Duration: 1 hr

Meaning of Statistics

The term statistics mean that the numerical statement as well as statistical

methodology. When it is used in the sense of statistical data it refers to quantitative

aspects of things and is a numerical description.

Example: Income of family, production of automobile industry, sales of cars etc.

These quantities are numerical. But there are some quantities, which are not in

themselves numerical but can be made so by counting. The sex of a baby is not a

number, but by counting the number of boys, we can associate a numerical

description to sex of all newborn babies, for an example, when saying that 60% of all

live-born babies are boy. This information then, comes within the realm of statistics.

Definition

The word statistics can be used is two senses, viz, singular and plural. In

narrow sense and plural sense, statistics denotes some numerical data (statistical data).

In a wide and singular sense statistics refers to the statistical methods. Therefore,

these have been grouped under two heads – ‘Statistics as a data” and “Statistics as a

methods”.

Statistics as a Data

Some definitions of statistics as a data area) Statistics are numerical statement of facts in any department of enquiring placed

in relation to each other.

- Powley

b) By statistics we mean quantities data affected to a marked extent by multiple of

causes.

- Yule and Kendall

c) By statistics we mean aggregates of facts affected to a marked extent by

multiplicity of causes, numerically expressed, enumerated or estimated according

to reasonable standard of accuracy, collected in a systematic manner for pre-determined purpose and placed in relation to each other.

- H. Secrist

This definition is more comprehensive and exhaustive. It shows light on

characteristics of statistics and covers different aspects.

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Some characteristics the statistics should possess by H. Secrist can be listed as

follows.

Statistics are aggregate of facts

Statistics are affected to a marked extent by multiplicity of causes.

Statistics are numerically expressed

Statistics should be enumerated / estimated

Statistics should be collected with reasonable standard of accuracy

Statistics should be placed is relation to each other.

Statistics as a method

Definition

a) “Statistics may be called to science of counting”

- A.L. Bowley

b) “Statistics is the science of estimates and probabilities”.

- Boddington

c) Dr. Croxton and Cowden have given a clear and concise definition.

“Statistics may be defined as the collection, presentation, analysis and

interpretation of numerical data”.

According to Croxton and Cowden there are 4 stages.

a) Collection of Data

A structure of statistical investigation is based on a systematic collection of

data. The data is classified into two groups

i) Internal data and

ii) External data

Internal data are obtained from internal records related to operations of

business organisation such as production, source of income and expenditure,

inventory, purchases and accounts.

The external data are collected and purchased by external agencies. The

external data could be either primary data or secondary data. The primary data are

collected for first time and original, while secondary data are collected by published

by some agencies.

b) Organisations of data

The collected data is a large mass of figures that needs to be organised. The

collected data must be edited to rectify for any omissions, irrelevant answers, and

wrong computations. The edited data must be classified and tabulated to suit further

analysis.

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c) Presentation of data

The large data that are collected cannot be understand and analysis easily and

quickly. Therefore, collected data needs to be presented in tabular or graphic form.

This systematic order and graphical presentation helps for further analysis.d) Analysis of data

The analysis requires establishing the relationship between one or more

variables. Analysis of data includes condensation, abstracting, summarization,

conclusion etc. With the help of statistical tools and techniques like measures of

dispersion central tendency, correlation, variance analysis etc analysis can be done.

e) Interpretation of data

The interpretation requires deep insight of the subject. Interpretation involves

drawing the valid conclusions on the bases of the analysis of data. This work requires

good experience and skill. This process is very important as conclusions of results aredone based on interpretation.

We can define statistics as per Seligman as follows.

“Statistics is a science which deals with the method and of collecting,

classifying, presenting, comparing and interpreting the numerical data collected

to throw light on enquiry”.

Importance of statistics

In today’s context statistics is indispensable. As the use of statistics is

extended to various field of experiments to draw valid conclusions, it is foundincreased importance and usage. The number of research investigations in the field of

economics and commerce are largely statistical. Further, the importance and statistics

in various fields are listed as below.

a) State Affairs: In state affairs, statistics is useful in following ways

1. To collect the information and study the economic condition of people in the

states.

2. To asses the resources available in states.

3. To help state to take decision on accepting or rejecting its policy based on

statistics.

4. To provide information and analysis on various factors of state like wealth,

crimes, agriculture experts, education etc.

b) Economics: In economics, statistics is useful in following ways

1. Helps in formulation of economic laws and policies

2. Helps in studying economic problems

3. Helps in compiling the national income accounts.

4. Helps in economic planning.

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c) Business

1. Helps to take decisions on location and size

2. Helps to study demand and supply

3. Helps in forecasting and planning

4. Helps controlling the quality of the product or process

5. Helps in making marketing decisions

6. Helps for production, planning and inventory management.

7. Helps in business risk analysis

8. Helps in resource long-term requirements, in estimating consumer’s

preference and helps in business research.

d) Education: Statistics is necessary to formulate the polices regarding start of newcourses, consideration of facilities available for proposed courses.

e) Accounts and Audits:

1. Helps to study the correlation between profits and dividends enable to know

trend of future profits.

2. In auditing sampling techniques are followed.

Functions of statistics

Some important functions of statistics are as follows

1. To collect and present facts in a systematic manner.2. Helps in formulation and testing of hypothesis.

3. Helps in facilitating the comparison of data.

4. Helps in predicting future trends.

5. Helps to find the relationship between variable.

6. Simplifies the mass of complex data.

7. Help to formulate polices.

8. Helps Government to take decisions.

Limitations of statistics

1. Does not study qualitative phenomenon.

2. Does not deal with individual items.

3. Statistical results are true only on an average.

4. Statistical data should be uniform and homogeneous.

5. A statistical result depends on the accuracy of data.

6. Statistical conclusions are not universally true.

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7. Statistical results can be interpreted only if person has sound knowledge of

statistics.

Distrust of Statistics

Distrust of statistics is due to lack of knowledge and limitations of its uses, but

not due to statistical sciences.

Distrust of statistics is due to following reasons.

a) Figures are manipulated or incomplete.

b) Quoting figures without their context.

c) Inconsistent definitions.

d) Selection of non-representative statistical units.

e) Inappropriate comparison

f) Wrong inference drawn.

g) Errors in data collection.

Statistical Data

Statistical investigation is a long and comprehensive process and requires

systematic collection of data in large size. The validity and accuracy of the

conclusion or results of the study depends upon how well the data were gathered.

The quality of data will greatly influence the conclusions of the study and hence

importance is to be given to the data collection process.

Statistical data may be classified as Primary Data and Secondary Data basedon the sources of data collection.

♦ Primary data

Primary data are those which are collected for the first time by the investigator

/ researchers and are thus original in character. Thus, data collected by investigator

may be for the specific purpose / study at hand. Primary data are usually in the shape

of raw materials to which statistical methods are applied for the purpose of analysis

and interpretation.

♦ Secondary data

Secondary have been already collected for the purpose other than the problem

at hand. These data are those which have already been collected by some other

persons and which have passed through the statistical analysis at least once.

Secondary data are usually in the shape of finished products since they have been

already treated statistically in one or the other form. After statistical treatment the

primary data lose their original shape and becomes secondary data. Secondary data of

one organisation become the primary data of other organisation who first collect and

publish them.

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Primary Vs Secondary Data

Researcher originates primary data for specific purpose / study at hand whilesecondary data have already been collected for purpose other than research

work at hand.

Primary data collection requires considerably more time, relatively expensive.

While the secondary data are easily accessible, inexpensive and quickly

obtained.

Table – A compression of Primary and Secondary Data

Primary data Secondary data

Collection purpose For the problem at hand For other problems

Collection process Very involved Rapid and easy

Collection cost High Relatively low

Collection time Long Short

Suitability Its suitability is positive It may or may not suit the

object of survey

Originality It is original It is not original

Precautions No extra precautions

required to use the data

It should be used with

extra case

Limitations of secondary data

a) Since secondary data is collected for ‘some other purpose, its usefulness to

current problem may be limited in several important ways, including

relevancies and accuracy.

b) The objectives, nature and methods used to collect secondary data may not be

appropriate to present situation.

c) The secondary data may not be accurate, or they may not be completely

current or dependable.

Criteria for evaluating secondary data

Before using the secondary data it is important to evaluate them on following

factors

a) Specification and methodology used to collect the data

b) Error and accuracy of data.

c) The currency

d) The objective – The purpose for which data were collectede) The nature – content of data

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f) The dependability

Sources of data

Primary source – The methods of collecting primary data.

When data is neither internally available nor exists as a secondary source, then

the primary sources of data would be approximate.

The various method of collection of primary data are as follows

a) Direct personal investigation

- Interview

- Observation

b) Indirect or oral investigation

c) Information from local agents and correspondentsd) Mailed questionnaires and schedules

e) Through enumerations

Secondary source – The methods of collecting secondary data

i) Published Statistics

a) Official publications of Central Government

Ex : Central Statistical Organisation (CSO) – Ministry of planning

- National Sample Survey Organisation (NSSO)

- Office of the Registrar General and Census Committee – GOI

- Director of Statistics and Economics – Ministry of Agriculture

- Labour Bureau – Ministry of Labour etc.

ii) Publications of Semi-government organisation

Ex :

- The institute of foreign trade, New Delhi

- The institute of economic growth, New Delhi.

iii) Publication of research institutes

Ex :

- Indian Statistical Institute

- Indian Agriculture Statistical Institute

- NCRET Publications

- Indian Standards Institute etc.

iv) Publication of Business and Financial Institutions

Ex :

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- Trade Association Publications like Sugar factory, Textile mill, Indian

chamber of Industry and Commerce.

- Stock exchange reports, Co-operative society reports etc.

v) News papers and periodicals

Ex :

- The Financial Express, Eastern Economics, Economic Times, Indian

Finance, etc.

vi) Reports of various committees and commissions

Ex :

- Kothari commission report on education

- Pay commission reports

- Land perform committee reports etc.

vii) Unpublished statistics

- Internal and administrative data like Periodical Loss, Profit, Sales,

Production Rate, Balance Sheet, Labour Turnover, Budges, etc.

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LESSON – 1



Classification and Tabulation

The data collected for the purpose of a statistical inquiry some times consists

of a few fairly simple figures, which can be easily understood without any special

treatment. But more often there is an overwhelming mass of raw data without any

structure. Thus, unwieldy, unorganised and shapeless mass of collected is not capable

of being rapidly or easily associated or interpreted. Unorganised data are not fit for

further analysis and interpretation. In order to make the data simple and easily

understandable the first task is not condense and simplify them in such a way that

irrelevant data are removed and their significant features are stand out prominently.

The procedure adopted for this purpose is known as method of classification and

tabulation. Classification helps proper tabulation.

“Classified and arranged facts speak themselves; unarranged, unorganised

they are dead as mutton”.

- Prof. J.R. Hicks

♦ Meaning of Classification

Classification is a process of arranging things or data in groups or classes

according to their resemblances and affinities and gives expressions to the unity of

attributes that may subsit among a diversity of individuals.

♦ Definition of Classification

Classification is the process of arranging data into sequences and groups

according to their common characteristics or separating them into different but related

parts.

- Secrist

The process of grouping large number of individual facts and observations on

the basis of similarity among the items is called classification.

- Stockton & Clark

Characteristics of classification

a) Classification performs homogeneous grouping of data

b) It brings out points of similarity and dissimilarities.

c) The classification may be either real or imaginaryd) Classification is flexible to accommodate adjustments

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Objectives / purposes of classifications

i) To simplify and condense the large data

ii) To present the facts to easily in understandable form

iii) To allow comparisons

iv) To help to draw valid inferences

v) To relate the variables among the data

vi) To help further analysis

vii)To eliminate unwanted data

viii) To prepare tabulation

Guiding principles (rules) of classifications

Following are the general guiding principles for good classifications

a) Exhaustive: Classification should be exhaustive. Each and every item

in data must belong to one of class. Introduction of residual class (i.e.

either, miscellaneous etc.) should be avoided.

b) Mutually exclusive: Each item should be placed at only one class

c) Suitability: The classification should confirm to object of inquiry.d) Stability: Only one principle must be maintained throughout the

classification and analysis.

e) Homogeneity: The items included in each class must be homogeneous.

f) Flexibility: A good classification should be flexible enough to

accommodate new situation or changed situations.

Modes / Types of Classification

Modes / Types of classification refers to the class categories into which thedata could be sorted out and tabulated. These categories depend on the nature of data

and purpose for which data is being sought.

Important types of classification

a) Geographical (i.e. on the basis of area or region wise)

b) Chronological (On the basis of Temporal / Historical, i.e. with respect to time)

c) Qualitative (on the basis of character / attributes)

d) Numerical, quantitative (on the basis of magnitude)

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a) Geographical Classification

In geographical classification, the classification is based on the geographical

regions.

Ex : Sales of the company (In Million Rupees) (region – wise)

Region Sales

North 285

South 300

East 185

West 235

b) Chronological Classification

If the statistical data are classified according to the time of its occurrence, the

type of classification is called chronological classification.

Sales reported by a departmental store

MonthSales

(Rs.) in lakhs

January 22

February 26

March 32

April 25

May 27

June 30

c) Qualitative Classification

In qualitative classifications, the data are classified according to the presenceor absence of attributes in given units. Thus, the classification is based on some

quality characteristics / attributes.

Ex: Sex, Literacy, Education, Class grade etc.

Further, it may be classified as

a) Simple classification b) Manifold classification

i) Simple classification: If the classification is done into only two classes then

classification is known as simple classification.

Ex: a) Population in to Male / Female

b) Population into Educated / Uneducated

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ii) Manifold classification: In this classification, the classification is based on

more than one attribute at a time.

Ex :

d) Quantitative Classification: In Quantitative classification, the classification is

based on quantitative measurements of some characteristics, such as age, marks,

income, production, sales etc. The quantitative phenomenon under study is

known as variable and hence this classification is also called as classification by

variable.

Ex :

For a 50 marks test, Marks obtained by students as classified as follows

Marks No. of students

0 – 10 5

10 – 20 7

20 – 30 10

30 – 40 25

40 – 50 3

Total Students = 50

In this classification marks obtained by students is variable and number of

students in each class represents the frequency.

Tabulation

Meaning and Definition of Tabulation

Tabulation may be defined, as systematic arrangement of data is column and

rows. It is designed to simplify presentation of data for the purpose of analysis andstatistical inferences.

Population

Smokers Non-smokers

Illiterate Literate

Male Female

Male Female

Literate Illiterate

Male Female

Male Female

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Major Objectives of Tabulation

1. To simplify the complex data

2. To facilitate comparison

3. To economise the space

4. To draw valid inference / conclusions

5. To help for further analysis

Differences between Classification and Tabulation

1. First data are classified and presented in tables; classification is the basis for

tabulation.

2. Tabulation is a mechanical function of classification because is tabulationclassified data are placed in row and columns.

3. Classification is a process of statistical analysis while tabulation is a process of

presenting data is suitable structure.

Classification of tables

Classification is done based on

1. Coverage (Simple and complex table)

2. Objective / purpose (General purpose / Reference table / Special table or summary table)

3. Nature of inquiry (primary and derived table).

Ex:

a) Simple table: Data are classified based on only one characteristic

Distribution of marks

Class Marks No. of students

30 – 40 20

40 – 50 20

50 – 60 10

Total 50

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b) Two-way table: Classification is based on two characteristics

Class MarksNo. of students

Boys Girls Total

30 – 40 10 10 20

40 – 50 15 5 20

50 – 60 3 7 10

Total 28 22 50

Frequency Distribution

Frequency distribution is a table used to organize the data. The left column

(called classes or groups) includes numerical intervals on a variable under study. Theright column contains the list of frequencies, or number of occurrences of each

class/group. Intervals are normally of equal size covering the sample observations

range.

It is simply a table in which the gathered data are grouped into classes and the

number of occurrences, which fall in each class, is recorded.

♦ Definition

A frequency distribution is a statistical table which shows the set of all distinct

values of the variable arranged in order of magnitude, either individually or in groups

with their corresponding frequencies.

- Croxton and Cowden

A frequency distribution can be classified as

a) Series of individual observation

b) Discrete frequency distribution

c) Continuous frequency distribution

a) Series of individual observation

Series of individual observation is a series where the items are listed one after

the each observation. For statistical calculations, these observation could be arranged

is either ascending or descending order. This is called as array.

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Ex :

Roll No.

Marks obtained

in statistics

paper

1 83

2 80

3 75

4 92

5 65

The above data list is a raw data. The presentation of data in above formdoesn’t reveal any information. If the data is arranged in ascending / descending in

the order of their magnitude, which gives better presentation then, it is called arraying

of data.

Discrete (ungrouped) Frequency Distribution

If the data series are presented in such away that indicating its exact

measurement of units, then it is called as discrete frequency distribution. Discrete

variable is one where the variants differ from each other by definite amounts.

Ex :Assume that a survey has been made to know number of post-graduates in 10

families at random; the resulted raw data could be as follows.

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

This data can be classified into an ungrouped frequency distribution. The

number of post-graduates becomes variable (x) for which we can list the frequency of

occurrence (f) in a tabular from as follows;

Number of post

graduates (x)

Frequency

(f)

0 2

1 2

2 4

3 1

4 1

The above example shows a discrete frequency distribution, where the

variable has discrete numerical values.

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Continuous frequency distribution (grouped frequency distribution)

Continuous data series is one where the measurements are only

approximations and are expressed in class intervals within certain limits. In

continuous frequency distribution the class interval theoretically continuous from thestarting of the frequency distribution till the end without break. According to

Boddington ‘the variable which can take very intermediate value between the smallest

and largest value in the distribution is a continuous frequency distribution.

Ex :

Marks obtained by 20 students in students’ exam for 50 marks are as given

below – convert the data into continuous frequency distribution form.

18 23 28 29 44 28 48 33 32 43

24 29 32 39 49 42 27 33 28 29

By grouping the marks into class interval of 10 following frequency

distribution tables can be formed.


0 - 5 0

5 – 10 0

10 – 15 0

15 – 20 1

20 – 25 2

25 – 30 7

30 – 35 4

35 – 40 1

40 – 45 3

45 – 50 2

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LESSON – 1



Technical terms used in formulation frequency distribution

a) Class limits:

The class limits are the smallest and largest values in the class.

Ex :

0 – 10, in this class, the lowest value is zero and highest value is 10. the two

boundaries of the class are called upper and lower limits of the class. Class limit is

also called as class boundaries.

b) Class intervals

The difference between upper and lower limit of class is known as class

interval.

Ex :

In the class 0 – 10, the class interval is (10 – 0) = 10.

The formula to find class interval is gives on below

R

SLi

−=

L = Largest value

S = Smallest value

R = the no. of classes

Ex :

If the mark of 60 students in a class varies between 40 and 100 and if we want

to form 6 classes, the class interval would be

I= (L-S ) / K =6

40100 −=

6

60= 10 L = 100

S = 40

K = 6

Therefore, class intervals would be 40 – 50, 50 – 60, 60 – 70, 70 – 80, 80 – 90

and 90 – 100.

♦ Methods of forming class-interval

a) Exclusive method (overlapping)

In this method, the upper limits of one class-interval are the lower limit of next

class. This method makes continuity of data.

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Ex :


20 – 30 5

30 – 40 15

40 – 50 25

A student whose mark is between 20 to 29.9 will be included in the 20 – 30

class.

Better way of expressing is


20 to les than 30

(More than 20 but les than 30)

5

30 to les than 40 15

40 to les than 50 25

Total Students 50

b) Inclusive method (non-overlaping)

Ex :


20 – 29 5

30 – 39 15

40 – 49 25

A student whose mark is 29 is included in 20 – 29 class interval and a student

whose mark in 39 is included in 30 – 39 class interval.

♦ Class Frequency

The number of observations falling within class-interval is called its class

frequency.

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Ex : The class frequency 90 – 100 is 5, represents that there are 5 students scored

between 90 and 100. If we add all the frequencies of individual classes, the total

frequency represents total number of items studied.

♦ Magnitude of class interval

The magnitude of class interval depends on range and number of classes. The

range is the difference between the highest and smallest values is the data series. A

class interval is generally in the multiples of 5, 10, 15 and 20.

Sturges formula to find number of classes is given below

K = 1 + 3.322 log N.

K = No. of class

log N = Logarithm of total no. of observations

Ex : If total number of observations are 100, then number of classes could be

K = 1 + 3.322 log 100

K = 1 + 3.322 x 2

K = 1 + 6.644

K = 7.644 = 8 (Rounded off)

NOTE: Under this formula number of class can’t be less than 4 and not greater than

20.

♦ Class mid point or class marks

The mid value or central value of the class interval is called mid point.

Mid point of a class =2

class)of limitupper classof limit(lower +

♦ Sturges formula to find size of class interval

Size of class interval (h) = Nlog322.31

Range

+

Ex : In a 5 group of worker, highest wage is Rs. 250 and lowest wage is 100 per day.

Find the size of interval.

h = Nlog322.31

Range

+=

50log322.31

100250

+

−

= 55.57 ≅ 56

Constructing a frequency distribution

The following guidelines may be considered for the construction of frequency

distribution.

a) The classes should be clearly defined and each observation must belong to one

and to only one class interval. Interval classes must be inclusive and non-overlapping.

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b) The number of classes should be neither too large nor too small.

Too small classes result greater interval width with loss of accuracy. Too

many class interval result is complexity.

c) All intervals should be of the same width. This is preferred for easy

computations.

The width of interval =classesof Number

Range

d) Open end classes should be avoided since creates difficulty in analysis and

interpretation.

e) Intervals would be continuous throughout the distribution. This is important

for continuous distribution.

f) The lower limits of the class intervals should be simple multiples of the

interval.

Ex : A simple of 30 persons weight of a particular class students are as follows.Construct a frequency distribution for the given data.

62 58 58 52 48 53 54 63 69 63

57 56 46 48 53 56 57 59 58 53

52 56 57 52 52 53 54 58 61 63

♦ Steps of construction

Step 1

Find the range of data (H) Highest value = 70(L) Lowest value = 46

Range = H – L = 69 – 46 = 23

Step 2

Find the number of class intervals.

Sturges formula

K = 1 + 3.322 log N.

K = 1 + 3.222 log 30

K = 5.90 Say K = 6

∴ No. of classes = 6

Step 3

Width of class interval

Width of class interval =classesof Number

Range= 4883.3

6

23≅=

Step 4

Conclusions all frequencies belong to each class interval and assign this total

frequency to corresponding class intervals as follows.

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Class interval Tally bars Frequency

46 – 50 | | | 3

50 – 54 | | | | | | | 8

54 – 58 | | | | | | | 8

58 – 62 | | | | | 6

62 – 66 | | | | 4

66 – 70 | 1

Cumulative frequency distribution

Cumulative frequency distribution indicating directly the number of units that

lie above or below the specified values of the class intervals. When the interest of the

investigator is on number of cases below the specified value, then the specified valuerepresents the upper limit of the class interval. It is known as ‘less than’ cumulative

frequency distribution. When the interest is lies in finding the number of cases above

specified value then this value is taken as lower limit of the specified class interval.

Then, it is known as ‘more than’ cumulative frequency distribution.

The cumulative frequency simply means that summing up the consecutive

frequency.

Ex :

Marks No. of students‘Less than’cumulative

frequency

0 – 10 5 5

10 – 20 3 8

20 – 30 10 18

30 – 40 20 38

40 – 50 12 50

In the above ‘less than’ cumulative frequency distribution, there are 5 students

less than 10, 3 less than 20 and 10 less than 30 and so on.

Similarly, following table shows ‘greater than’ cumulative frequency

distribution.

Ex :

Marks No. of students ‘Less than’cumulative

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frequency

0 – 10 5 50

10 – 20 3 45

20 – 30 10 42

30 – 40 20 32

40 – 50 12 12

In the above ‘greater than’ cumulative frequency distribution, 50 students are

scored more than 0, 45 more than 10, 42 more than 20 and so on.

Diagrammatic and Graphic Representation

The data collected can be presented graphically or pictorially to be easy

understanding and for quick interpretation. Diagrams and graphs give visual

indications of magnitudes, groupings, trends and patterns in the data. These

parameter can be more simply presented in the graphical manner. The diagrams and

graphs help for comparison of the variables.

Diagrammatic presentation

A diagram is a visual form for presentation of statistical data. The diagram

refers various types of devices such as bars, circles, maps, pictorials and cartogramsetc.

Importance of Diagrams

1. They are simple, attractive and easy understandable

2. They give quick information

3. It helps to compare the variables

4. Diagrams are more suitable to illustrate discrete data

5. It will have more stable effect in the reader’s mind.

Limitations of diagrams

1. Diagrams shows approximate value

2. Diagrams are not suitable for further analysis

3. Some diagrams are limited to experts (multidimensional)

4. Details cannot be provided fully

5. It is useful only for comparison

General Rules for drawing the diagrams

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i) Each diagram should have suitable title indicating the theme with which

diagram is intended at the top or bottom.

ii) The size of diagram should emphasize the important characteristics of data.

iii) Approximate proposition should be maintained for length and breadth of

diagram.iv) A proper / suitable scale to be adopted for diagram

v) Selection of approximate diagram is important and wrong selection may

mislead the reader.

vi) Source of data should be mentioned at bottom.

vii)Diagram should be simple and attractive

viii) Diagram should be effective than complex.

Some important types of diagrams

a) One dimensional diagrams (line and bar)

b) Two-dimensional diagram (rectangle, square, circle)

c) Three-dimensional diagram (cube, sphere, cylinder etc.)

d) Pictogram

e) Cartogram

a) One dimensional diagrams (line and bar)In one-dimensional diagrams, the length of the bars or lines is taken into

account. Widths of the bars are not considered. Bar diagrams are classified mainly as

follows.

i) Line diagram

ii) Bar diagram

- Vertical bar diagram

- Horizontal bar diagram

- Multiple (compound) bar diagram

- Sub-divided (component) bar diagram

- Percentage subdivided bar diagram

i) Line diagram

This is simplest type of one-dimensional diagram. On the basis of size of the

figures, heights of the bar / lines are drawn. The distances between bars are kept

uniform. The limitation of this diagram are it is not attractive cannot provide more

than one information.

Ex : Draw the line diagram for the following data

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Year 2001 2002 2003 2004 2005 2006

No. of students passed in first

class with distinction5 7 12 5 13 15

2001 2002 2003 2004 2005 20064

6

8

10

12

14

16(15)

(13)

(5)

(12)

(7)

(5)

N o . o f s

t u d e n t s p a s s e d i n F C D

Year

Indication of diagram: Highest FCD is at 2006 and lowest FCD are at 2001 and 2004.

b) Simple bars diagram

A simple bar diagram can be drawn using horizontal or vertical bar. In

business and economics, it is very a common diagram.

Vertical bar diagram

The annual expresses of maintaining the car of various types are given below.

Draw the vertical bar diagram. The annual expenses of maintaining includes (fuel +

maintenance + repair + assistance + insurance).

Type of the car Expense in Rs. / Year

Maruthi Udyog 47533

Hyundai 59230

Tata Motors 63270

Source: 2005 TNS TCS Study

Published at : Vijaya Karnataka, dated: 03.08.2006

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47533

59230

63270

30000

35000

40000

45000

50000

5500060000

65000

70000

Maruthi Udyog Hyundai Tata Motors

Source: 2005 TNS TCS Study

Published at : Vijaya Karnataka, dated: 03.08.2006

Indicating of diagram

a) Annual expenses of Maruthi Udyog brand car is comparatively less

with other brands depicted

b) High annual expenses of Tata motors brand can be seen from diagram.

♦ Horizontal bar diagram

World biggest top 10 steel makers are data are given below. Draw horizontal

bar diagram.

Steel

maker

Arcelo

r Mittal

Nippo

nPOSCO JFE

BAO

Steel

US

Stee

l

NUCOR

RIVA Thyssen-

krupp

Tangshan

Prodn.

in

million

tonnes

110 32 31 30 24 20 18 18 17 16

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110

32

31

30

24

20

18

18

17

16

0 20 40 60 80 100 120

Arcelor Mittal

Nippon

POSCO

JFE

BAO Steel

US Steel

NUCOR

RIVA

Thyssen-krupp

Tangshan

T o p - 1 0 S t e e l M a k e r s

Production of Steel (Million Tonnes)

Source: ISSB Published by India Today

♦ Compound bar diagram (Multiple bar diagram)

Multiple bar diagrams are used to provide more information than simple bar

diagram. Multiple bar diagram provides more than one phenomenon and highly

useful for direct comparison. The bars are drawn side-by-side and different columns,shades hatches can be used for indicating each variable used.

Ex: Draw the bar diagram for the following data. Resale value of the cars (Rs. 000) is

as follows.

Year (Model) Santro Zen Wagonr

2003 208 252 248

2004 240 278 274

2005 261 296 302

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208

252 248240

278 274261

296 302

0

50

100

150

200

250

300

350

1 2 3Model of Car

V a l u e i n

R s .

Santro Zen Wagnor

Source: True value used car purchase data

Published by: Vijaya Karnataka, dated: 03.08.2006

Ex : Represent following in suitable diagram

Class A B C

Male 1000 1500 1500

Female 500 800 1000

Total 1500 2300 2500

1000

500

1500

800

1500

1000

0

500

1000

1500

2000

2500

P o p u l a t i o n

( i n N o s . )

1 2 3

Class

Male Female

27

1500

23002500

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Ex : Draw the suitable diagram for following data

Mode of

investment

Investment in 2004 in Rs. Investment in 2005 in Rs.

Investment %age Investment %age

NSC 25000 43.10 30000 45.45

MIS 15000 25.86 10000 15.15

Mutual Fund 15000 25.86 25000 37.87

LIC 3000 5.17 1000 1.52

Total 58000 100 66000 100

2004 20050

10

20

30

40

50

60

70

80

90

100

110

45.45

15.15

37.87

1.525.17

25.86

25.86

43.10

% o

f I n v e s t m e n t

Year

Two-dimensional diagram

In two-dimensional diagram both breadth and length of the diagram (i.e. area

of the diagram) are considered as area of diagram represents the data. The important

two-dimensional diagrams are

a) Rectangular diagram

b) Square diagram

a) Rectangular diagram

Rectangular diagrams are used to depict two or more variables. This diagram

helps for direct comparison. The area of rectangular are kept in proportion to the

values. It may be of two types.

i) Percentage sub-divided rectangular diagram

ii) Sub-divided rectangular diagram

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In former case, width of the rectangular are proportional to the values, the various

components of the values are converted into percentages and rectangles are divided

according to them. While later case is used to show some related phenomenon like

cost per unit, quality of production etc.

Ex : Draw the rectangle diagram for following data

Item ExpenditureExpenditure in Rs.

Family A Family B

Provisional stores 1000 2000

Education 250 500

Electricity 300 700

House Rent 1500 2800

Vehicle Fuel 500 1000

Total 3500 7000

Total expenditure will be taken as 100 and the expenditure on individual items

are expressed in percentage. The widths of two rectangles are in proportion to the

total expenses of the two families i.e. 3500: 7000 or 1: 2. The heights of rectangles

are according to percentage of expenses.

Item

Expenditure

Monthly expenditure

Family A (Rs. 3500) Family B(Rs. 7000)

Rs. %age Rs. %age

Provisional stores 1000 28.57 2000 28.57

Education 250 7.14 500 7.14

Electricity 300 8.57 700 10

House Rent 1500 42.85 2800 40

Vehicle Fuel 500 12.85 1000 14.28

Total 3500 100 7000 100

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0

20

40

60

80

100

B A

% o

f E x p e n d i t u r e

Family

Provisonal Stores Education

Electricity House Rent Vehicle Fuel

b) Square diagram

To draw square diagrams, the square root is taken of the values of the various

items to be shown. A suitable scale may be used to depict the diagram. Ratios are to

be maintained to draw squares.

Ex : Draw the square diagram for following data4900 2500 1600

Solution: Square root for each item in found out as 70, 50 and 40 and is divided by 10;

thus we get 7, 5 and 4.

0

1000

2000

3000

4000

5000

6000

754

321

4900

2500

1600

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Pie diagram

Pie diagram helps us to show the portioning of a total into its component parts.

It is used to show classes or groups of data in proportion to whole data set. The entire

pie represents all the data, while each slice represents a different class or group within

the whole. Following illustration shows construction of pie diagram.

Draw the pie diagram for following data

Revenue collections for the year 2005-2006 by government in Rs. (crore)s for

petroleum products are as follows. Draw the pie diagram.

Customs 9600

Excise 49300

Corporate Tax and dividend 18900

States taking 48800

Total 126600

Solution:

Item / Source Value in

crores

Angle of circle %ge

Customs 9600o

30.27360x126600

9600= 7.58

Excise 49300 20.140360x126600

49300= 39.00

Corporate Tax and Dividend 18900o

70.53360x126600

18900= 14.92

State’s taking 48800 80.138360x

126600

48800= 38.50

Total 126600 360o 100

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7.58

39

14.92

38.5

Customs

Excise

Corporate Tax

and Dividend

State’s taking

Source: India Today 19 June, 2006

Choice or selection of diagram

There are many methods to depict statistical data through diagram. No angle

diagram is suited for all purposes. The choice / selection of diagram to suit given set

of data requires skill, knowledge and experience. Primarily, the choice depends upon

the nature of data and purpose of presentation, to which it is meant. The nature of

data will help in taking a decision as to one-dimensional or two-dimensional or three-dimensional diagram. It is also required to know the audience for whom the diagram

is depicted.

The following points are to be kept in mind for the choice of diagram.

1. To common man, who has less knowledge in statistics cartogram and

pictograms are suited.

2. To present the components apart from magnitude of values, sub-divided bar

diagram can be used.

3. When a large number of components are to be shows, pie diagram is suitable.

Graphic presentation

A graphic presentation is a visual form of presentation graphs are drawn on a

special type of paper known are graph paper.

Common graphic representations are

a) Histogram

b) Frequency polygon

c) Cumulative frequency curve (ogive)

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Advantages of graphic presentation

1. It provides attractive and impressive view

2. Simplifies complexity of data

3. Helps for direct comparison

4. It helps for further statistical analysis

5. It is simplest method of presentation of data

6. It shows trend and pattern of data

Difference between graph and diagram

Diagram Graph

1. Ordinary paper can be used 1. Graph paper is required

2. It is attractive and easily

understandable

2. Needs some effect to understand

3. It is appropriate and effective to

measure more variable

3. It creates problem

4. It can’t be used for further analysis 4. Can be used for further analysis

5. It gives comparison 5. It shows relationship between

variables

6. Data are represented by bars,

rectangles

6. Points and lines are used to represent

data

Frequency Histogram

In this type of representation the given data are plotted in the form of series of

rectangles. Class intervals are marked along the x-axis and the frequencies are along

the y-axis according to suitable scale. Unlike the bar chart, which is one-dimensional,

a histogram is two-dimensional in which the length and width are both important. A

histogram is constructed from a frequency distribution of grouped data, where the

height of rectangle is proportional to respective frequency and width represents the

class interval. Each rectangle is joined with other and the blank space between the

rectangles would mean that the category is empty and there are no values in that class

interval.

Ex : Construct a histogram for following data.

Marks obtained (x) No. of students (f) Mid point

15 – 25 5 20

25 – 35 3 30

35 – 45 7 40

45 – 55 5 50

55 – 65 3 60

65 – 75 7 70

Total 30

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For convenience sake, we will present the frequency distribution along with

mid-point of each class interval, where the mid-point is simply the average of value of

lower and upper boundary of each class interval.

0

1

2

3

4

5

6

7

75655545352515

F r e q u e

n c y ( N o . o f s t u d e n t s )

Class Interval (Marks)

Frequency polygon

A frequency polygon is a line chart of frequency distribution in which either the values of discrete variables or the mid-point of class intervals are plotted against

the frequency and those plotted points are joined together by straight lines. Since, the

frequencies do not start at zero or end at zero, this diagram as such would not touch

horizontal axis. However, since the area under entire curve is the same as that of a

histogram which is 100%. The curve must be ‘enclosed’, so that starting mid-point is

jointed with ‘fictitious’ preceding mid-point whose value is zero. So that the

beginning of curve touches the horizontal axis and the last mid-point is joined with a

‘fictitious’ succeeding mid-point, whose value is also zero, so that the curve will end

at horizontal axis. This enclosed diagram is known as ‘frequency polygon’.

Ex : For following data construct frequency polygon.Marks (CI) No. of frequencies (f) Mid-point

15 – 25 5 20

25 – 35 3 30

35 – 45 7 40

45 – 55 5 50

55 – 65 3 60

65 – 75 7 70

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0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

8

10

A Frequency polygon

F r e q u e n c y

Mid point (x)

Cumulative frequency curve (ogive)

ogives are the graphic representations of a cumulative frequency distribution.

These ogives are classified as ‘less than’ and ‘more than ogives’. In case of ‘less

than’, cumulative frequencies are plotted against upper boundaries of their respective

class intervals. In case of ‘grater than’ cumulative frequencies are plotted against

upper boundaries of their respective class intervals. These ogives are used for

comparison purposes. Several ogves can be compared on same grid with different

colour for easier visualisation and differentiation.

Ex :

Marks

(CI)

No. of

frequencies (f)Mid-point

Cum. Freq.

Less than

Cum. Freq.

More than

15 – 25 5 20 5 30

25 – 35 3 30 8 25

35 – 45 7 40 15 22

45 – 55 5 50 20 15

55 – 65 3 60 23 10

65 – 75 7 70 30 7

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Less than give diagram

20 30 40 50 60 70

5

10

15

20

25

30

'Less than' ogive

L e s s t h a n C u m u l a t i v e F r e q u e n c y

Upper Boundary (CI)

Less than give diagram

10 20 30 40 50 60 70

10

15

20

25

30

35

'More than' ogive

M o r e t h a n O g i v e

Lower Boundary (CI)

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