Basics of statistics

P R E S E N T E D B Y : S O L U T I O N S A C A D E M Y
R A M A N S A C H D E V A
B . C O M ( H ) , M B A , M . C O M , M . P H I L , A M T , N E T .
P H O N E : 9 8 1 1 9 5 7 2 5 5 , 0 1 1 4 0 2 0 4 6 5 5 , 2 4 3 7 7 2 5 5
BASICS OF STATISTICS
By: Prof. Raman(DU)

ECONOMICS
• “Economics is that branch of knowledge that studies
consumption, production, exchange and distribution of
wealth”.
• “Economics is a science that studies human behaviour
as a relationship between ends and scarce means which
have alternative uses”.
By: Prof. Raman(DU)

ECONOMICS ACTIVITY
•Non-Economic Activities
•Economic Activities
By: Prof. Raman(DU)

NON-ECONOMIC ACTIVITIES
• Social activities
• Religious activities
• Political activities
• Charitable activities
• Parental activities
By: Prof. Raman(DU)

ECONOMIC ACTIVITIES
• Production
• Consumption
• Investment
• Exchange
• Distribution
By: Prof. Raman(DU)

NATURE OF ECONOMICS
• Economics as Science
a. Positive Science
b. Normative science
• Economics as an Art.
By: Prof. Raman(DU)

ECONOMICS AS POSITIVE SCIENCE
• A positive science is one which makes a real description
of an activity. It only answers what is? What was? It has
nothing to suggest about facts, positive economic deals
with what is or how the economic problems facing a
society are actually solved. Prof. Robbins held that
economics was purely a positive science. According to
him, economics should be neutral or silent between ends
; i.e., there should be no desire to learn about ethics of
economics decisions. Thus, in positive economics we
study be human decisions as facts which can be verified
with actual data.
By: Prof. Raman(DU)

ECONOMICS AS NORMATIVE SCIENCE
• A normative science is that science which refer to what
ought to be? What ought to have happened? Normative
economics deals with what ought to be or how the
economics problems should be solved. Alfred Marshall
and Pigou have considered the normative aspect of
economics, as it prescribes that cause of action which is
desirable and necessary to achieve social goals.
By: Prof. Raman(DU)

ECONOMICS AS AN ART
• Art is practical application of knowledge for achieving
some definite aim. It helps in solution of practical
problems. Art is the practical application of scientific
principles. Science lays down principles while art puts
these principles into practice. Economics is an art as it
gives us practical guidance in solution to various
economic problems.
By: Prof. Raman(DU)

STATISTICS
It is necessary to have quantitative measurements even
for things which are not basically quantifiable. This is
necessary for preciseness of statement. The systematic
treatment of quantitative expression is known as
‘Statistics’.
• Statistics can be defined in two ways:
• In Plural sense
• In Singular sense
By: Prof. Raman(DU)

STATISTICS IN PLURAL SEVSE
• According to Horace Secrist :
“By statistics we mean aggregates of facts affected
to a marked extent by multiplicity of numerically
expressed, enumerated or estimated according to
reasonable standards of accuracy, collected in a
systematic manner for a predetermined purpose and
placed in relation to each other.”
By: Prof. Raman(DU)

STATISTICS IN PLURAL SENSE
• Statistics are aggregates of facts
• Statistics are affected to a marked extent by multiplicity
of causes
• Statistics are numerically expressed
• Statistics are enumerated or estimated according to
reasonable standard of accuracy
• Statistics are collected in for a systematic manner
• Statistics are collected for a pre-determined purpose
• Statistics are placed in relation to each other
By: Prof. Raman(DU)

STATISTICS IN SINGULAR SENSE
• According to Croxton and Cowden, “Statistics may be
defined as a science of collection, presentation, analysis
and interpretation of numerical data.”
(a) Collection of data
(b) Organisation of data
(c) Presentation of data
(d) Analysis of data
(e) Interpretation of data
By: Prof. Raman(DU)

FUNCTIONS OF STATISTICS
• Statistics simplifies complex data
• Statistics presents the facts in a definite form
• Statistics provides a technique of comparison
• Statistics studies relationship
• Statistics helps in formulating policies
• Statistics helps in forecasting
• Statistics helps to test and formulate theories
By: Prof. Raman(DU)

IMPORTANCE OF STATISTICS
• Statistics and Economics
•Statistics and Economic Planning
•Statistics and Business Planning
•Statistics and Government
By: Prof. Raman(DU)

STATISTICS AND ECONOMICS
• Economic lays are evolved with the help of
statistical tools and the importance of statistics is
growing day-to-day in economic analysis. There
is a vital relationship among economics,
mathematics and statistics due to the
importance of mathematics and statistics in
economics. As a result, a new science has
evolved which is called Economics.
By: Prof. Raman(DU)

STATISTICS AND ECONOMICS
• Statistics and the study of consumption
• Statistics and the study of production
• Statistics and the study of exchange
• Statistics and the study of distribution
By: Prof. Raman(DU)

STATISTICS AND ECONOMIC
PLANNING
• Statistics is the most importance tool in
economic planning is the best use of
national resources, both human and
natural. Planning without statistics is a
leap in the dark. Every phase in planning –
drawing a plan, execution and review-is
based on statistics. The success of plan is
dependent upon sufficient and accurate
statistical data available at all these
stages. By: Prof. Raman(DU)

STATISTICS AND BUSINESS PLANNING
• The producer or the manufacturer has to
estimate demand for his goods in the
immediate as well as distant future. This is
done by market research for which all the
step of statistical method have to be
followed. The cost accountant uses
statistical tools to help the producer to fix
the prices of various commodities.
By: Prof. Raman(DU)

STATISTICS AND GOVERNMENT
• For the smooth administration of country, the
government has to be well versed with its all problems.
Before formulating policies the government must
consider certain statistics, like that of crimes, taxes,
wealth, trade, etc., so that there is no obstacle for the
promotion of economic development. Such policies may
develop the economic status of the people and the
nation, depending upon the accuracy of the statistical
law. Statistics is indispensable for all important functions
of the ministries of the state.
By: Prof. Raman(DU)

LIMITATIONS OF STATISTICS
• It dose not study the qualitative aspect of a
problem
• It dose not study individuals
• Statistical laws are true only on an average
• Statistics can be misused
• Statistics simply is one of the method of studying
a phenomenon
• Statistical results lack mathematical accuracy
By: Prof. Raman(DU)

INTRODUCTION TO SAMPLING
METHODS
• Sampling is very often used in our daily
life. For example while purchasing food
grains from a shop we usually examine a
handful from the bag to assess the quality
of the commodity. A doctor examines a few
drops of blood as sample and draws
conclusion about the blood constitution of
the whole body. Thus most of our
investigations are based on samples.
By: Prof. Raman(DU)

POPULATION
• In a statistical enquiry, all the items, which fall
within the purview of enquiry, are known as
Population or Universe. In other words, the
population is a complete set of all possible
observations of the type which is to be
investigated. Total number of students studying
in a school or college, total number of books in a
library, total number of houses in a village or
town are some examples of population.
By: Prof. Raman(DU)

CENSUS METHOD
• Information on population can be collected in
two ways –census method and sample method.
In census method every element of the
population is included in the investigation. For
example, if we study the average annual income
of the families of a particular village or area, and
if there are 1000 families in that area, we must
study the income of all 1000 families. In this
method no family is left out, as each family is a
unit.
By: Prof. Raman(DU)

MERITS AND LIMITATIONS OF CENSUS
METHOD
Merits:
1. The data are collected from each and every
item of the population
2. The results are more accurate and reliable,
because every item of the universe is required.
3. Intensive study is possible.
4. The data collected may be used for various
surveys, analyses etc.
By: Prof. Raman(DU)

MERITS AND LIMITATIONS OF CENSUS
METHOD
• Limitations:
1. It requires a large number of
enumerators and it is a costly method
2. It requires more money, labour, time
energy etc.
3. It is not possible in some circumstances
where the universe is infinite.
By: Prof. Raman(DU)

SAMPLING
• In our everyday life we have been using
sampling theory as we have discussed in
introduction. In all those cases we believe
that the samples give a correct idea about
the population. Most of our decisions are
based on the examination of a few items
that is sample studies.
By: Prof. Raman(DU)

SAMPLE
•Statisticians use the word sample to
describe a portion chosen from the
population. A finite subset of statistical
individuals defined in a population is
called a sample. The number of units
in a sample is called the sample size.
By: Prof. Raman(DU)

SAMPLING UNIT
• The constituents of a population which are
individuals to be sampled from the population
and cannot be further subdivided for the purpose
of the sampling at a time are called sampling
units. For example, to know the average income
per family, the head of the family is a sampling
unit. To know the average yield of rice, each
farm owner’ s yield of rice is a sampling unit.
By: Prof. Raman(DU)

SAMPLING FRAME
• For adopting any sampling procedure it is
essential to have a list identifying each
sampling unit by a number. Such a list or
map is called sampling frame. A list of
voters, a list of house holders, a list of
villages in a district, a list of farmers etc.
are a few examples of sampling frame.
By: Prof. Raman(DU)

REASONS FOR SELECTING A SAMPLE
1. Complete enumerations are practically
impossible when the population is infinite.
2. When the results are required in a short
time.
3. When the area of survey is wide.
4. When resources for survey are limited
particularly in respect of money and
trained persons.
5. When the item or unit is destroyed under
investigation.
By: Prof. Raman(DU)

PARAMETERS AND STATISTICS
• We can describe samples and populations
by using measures such as the mean,
median, mode and standard deviation.
When these terms describe the
characteristics of a population, they are
called parameters. When they describe
the characteristics of a sample, they are
called statistics.
By: Prof. Raman(DU)

PRINCIPLES OF SAMPLING
•Principle of statistical regularity
•Principle of Inertia of large
numbers
•Principle of Validity
•Principle of Optimisation
By: Prof. Raman(DU)

SAMPLING ERRORS
• Although a sample is a part of population,
it cannot be expected generally to supply
full information about population. So there
may be in most cases difference between
statistics and parameters. The
discrepancy between a parameter and its
estimate due to sampling process is
known as sampling error.
By: Prof. Raman(DU)

NON-SAMPLING ERRORS
•In all surveys some errors may occur
during collection of actual information.
These errors are called Non-sampling
errors.
By: Prof. Raman(DU)

ADVANTAGES OF SAMPLING
1. Sampling saves time and labour.
2. It results in reduction of cost in terms of money and
man-hour.
3. Sampling ends up with greater accuracy of results.
4. It has greater scope.
5. It has greater adaptability.
6. If the population is too large, or hypothetical or
destroyable sampling is the only method to be used.
By: Prof. Raman(DU)

LIMITATION OF SAMPLING
1. Sampling is to be done by qualified and
experienced persons. Otherwise, the information
will be unbelievable.
2. Sample method may give the extreme values
sometimes instead of the mixed values.
3. There is the possibility of sampling errors.
Census survey is free from sampling error.
By: Prof. Raman(DU)

TYPES OF SAMPLING
•1. Probability sampling.
•2. Non-probability sampling.
•3. Mixed sampling.
By: Prof. Raman(DU)

PROBABILITY SAMPLING
(RANDOM SAMPLING)
•A probability sample is one where the
selection of units from the population
is made according to known
probabilities. (e.g.) Simple random
sample, probability proportional to
sample size etc.
By: Prof. Raman(DU)

NON-PROBABILITY SAMPLING
• It is the one where discretion is used to
select ‘ representative’ units from the
population (or) to infer that a sample is ‘
representative’ of the population. This
method is called judgment or purposive
sampling.
By: Prof. Raman(DU)

MIXED SAMPLING
• Here samples are selected partly
according to some probability and partly
according to a fixed sampling rule; they
are termed as mixed samples and the
technique of selecting such samples is
known as mixed sampling.
By: Prof. Raman(DU)

METHODS OF SELECTION OF SAMPLES
1. Simple random sampling.
2. Stratified random sampling.
3. Systematic random sampling.
By: Prof. Raman(DU)

SIMPLE RANDOM SAMPLING
• A simple random sample from finite
population is a sample selected such that
each possible sample combination has
equal probability of being chosen. It is also
called unrestricted random sampling.
By: Prof. Raman(DU)

SIMPLE RANDOM SAMPLING WITHOUT
REPLACEMENT
• In this method the population elements
can enter the sample only once (i.e.) the
units once selected is not returned to the
population before the next draw.
By: Prof. Raman(DU)

SIMPLE RANDOM SAMPLING WITH
REPLACEMENT
In this method the population units may
enter the sample more than once. Simple
random sampling may be with or without
replacement.
By: Prof. Raman(DU)

METHODS OF SELECTION OF A SIMPLE
RANDOM SAMPLING
a) Lottery Method.
b) Table of Random numbers.
c) Random number selections using
calculators or computers.
By: Prof. Raman(DU)

LOTTERY METHOD
• This is the most popular and simplest method. In this
method all the items of the population are numbered on
separate slips of paper of same size, shape and color.
They are folded and mixed up in a container. The
required numbers of slips are selected at random for the
desire sample size.
• For example, if we want to select 5 students, out of 50
students, then we must write their names or their roll
numbers of all the 50 students on slips and mix them.
Then we make a random selection of 5 students.
By: Prof. Raman(DU)

TABLE OF RANDOM NUMBERS
As the lottery method cannot be used, when the population is infinite,
the alternative method is that of using the table of random numbers.
There are several standard tables of random numbers.
1. Tippett’ s table
2. Fisher and Yates’ table
3. Kendall and Smith’ s table are the three tables among them.
A random number table is so constructed that all digits 0 to 9 appear
independent of each other with equal frequency. If we have to select a
sample from population of size N= 100, then the numbers can be
combined three by three to give the numbers from 001 to 100.
By: Prof. Raman(DU)

Example 1:
Inan area there are 500 families. Using the following
extract from a table of random numbers select a sample of
15 families to find out the standard of living of those
families in that area.
4652 3819 8431 2150 2352 2472 0043 3488
9031 7617 1220 4129 7148 1943 4890 1749
2030 2327 7353 6007 9410 9179 2722 8445
0641 1489 0828 0385 8488 0422 7209 4950
By: Prof. Raman(DU)

• Solution:
In the above random number table we can start from any
row or column and read three digit numbers continuously row-
wise or column wise. Now we start from the third row, the
numbers are:
203 023 277 353 600 794 109 179
272 284 450 641 148 908 280
Since some numbers are greater than 500, we subtract 500
from those numbers and we rewrite the selected numbers as
follows:
203 023 277 353 100 294 109 179
272 284 450 141 148 408 280
By: Prof. Raman(DU)

RANDOM NUMBER SELECTIONS USING
CALCULATORS OR
COMPUTERS
• Random number can be generated
through scientific calculator or computers.
For each press of the key get a new
random numbers. The ways of selection of
sample is similar to that of using random
number table.
By: Prof. Raman(DU)

MERITS OF USING RANDOM NUMBERS
1. Personal bias is eliminated as a selection depends
solely on chance .
2. A random sample is in general a representative sample
for a homogenous population.
3. There is no need for the thorough knowledge of the
units of the population.
4. The accuracy of a sample can be tested by examining
another sample from the same universe when the
universe is unknown.
5. This method is also used in other methods of sampling.
By: Prof. Raman(DU)

LIMITATIONS OF USING RANDOM
NUMBERS
1. Preparing lots or using random number tables is tedious
when the population is large.
2. When there is large difference between the units of
population, the simple random sampling may not be a
representative sample.
3. The size of the sample required under this method is
more than that required by stratified random sampling.
4. It is generally seen that the units of a simple random
sample lie apart geographically. The cost and time of
collection of data are more.
By: Prof. Raman(DU)

STRATIFIED RANDOM SAMPLING
Of all the methods of sampling the procedure commonly
used in surveys is stratified sampling. This technique is
mainly used to reduce the population heterogeneity and to
increase the efficiency of the estimates. Stratification
means division into groups. In this method the population
is divided into a number of subgroups or strata. The strata
should be so formed that each stratum is homogeneous
as far as possible. Then from each stratum a simple
random sample may be selected and these are combined
together to form the required sample from the population.
By: Prof. Raman(DU)

TYPES OF STRATIFIED SAMPLING
1.Proportional sampling.
2. Non-proportional sampling.
By: Prof. Raman(DU)

PROPORTIONAL SAMPLING
In the proportional sampling 20 equal and
proportionate representation is given to
subgroups or strata. If the number of items is
large, the sample will have a higher size and vice
versa. The population size is denoted by N and
the sample size is denoted by ‘ n’ the sample size
is allocated to each stratum in such a way that the
sample fractions is a constant for each stratum.
That is given by n/N = c. So in this method each
stratum is represented according to its size.
By: Prof. Raman(DU)

NON-PROPORTIONATE SAMPLE
In non-proportionate sample, equal
representation is given to all the sub-
strata regardless of their existence in
the population.
By: Prof. Raman(DU)

Example 2:
A sample of 50 students is to be drawn from a population
consisting of 500 students belonging to two institutions A
and B. The number of students in the institution A is 200
and the institution B is 300. How will you draw the sample
using proportional allocation?
By: Prof. Raman(DU)

Solution:
There are two strata in this case with sizes N1= 200 and
N2= 300
and the total population N = N1+ N2= 500
The sample size is 50. If n1and n2are the sample sizes
n1=n/NxN2=50/500x200=20
n2=n/NxN2=50/500x300=30
The sample sizes are 20 from A and 30 from B. Then the
units from each institution are to be selected by simple
random sampling.
By: Prof. Raman(DU)

MERITS OF STRATIFIED SAMPLING
1. It is more representative.
2. It ensures greater accuracy.
3. It is easy to administer as the universe is sub-divided.
4. Greater geographical concentration reduces time and
expenses.
5. When the original population is badly skewed, this
method is appropriate.
6. For non –homogeneous population, it may field good
results.
By: Prof. Raman(DU)

LIMITATIONS OF STRATIFIED
SAMPLING
• To divide the population into homogeneous
strata, it requires more money, time and
statistical experience which is a difficult one.
• Improper stratification leads to bias, if the
different strata overlap such a sample will not be
representative one.
By: Prof. Raman(DU)

SYSTEMATIC SAMPLING
This method is widely employed because of
its ease and convenience. A frequently used
method of sampling when a complete list of
the population is available is systematic
sampling. It is also called Quasi-random
sampling.
By: Prof. Raman(DU)

SELECTION PROCEDURE
The whole sample selection is based on just a
random start . The first unit is selected with the
help of random numbers and the rest get selected
automatically according to some pre designed
pattern is known as systematic sampling. With
systematic random sampling every Kth element in
the frame is selected for the sample, with the
starting point among the first K elements
determined at random.
By: Prof. Raman(DU)

SELECTION PROCEDURE
For example, if we want to select a sample of 50 students from 500
students under this method Kth item is picked up from the sampling
frame and K is called the sampling interval.
Sampling interval ,
K = N/n = Population size/Sample size
k = 500/50 = 10
K = 10 is the sampling interval. Systematic sample consists
in selecting a random number say i K and every Kth Unit 22
subsequently. Suppose the random number ‘ i’ is 5, then we select 5,
15, 25, 35, 45,………. The random number ‘ i’ is called random start.
The technique will generate K systematic samples with equal
probability.
By: Prof. Raman(DU)

LIMITATIONS OF SYSTEMATIC
SAMPLING
1.Systematic sampling may not represent the
whole population.
2.There is a chance of personal bias of the
investigators.
Systematic sampling is preferably used when the
information is to be collected from trees in a
forest, house in blocks, entries in a register which
are in a serial order etc.
By: Prof. Raman(DU)

MERITS OF SYSTEMATIC SAMPLING
1. This method is simple and convenient.
2. Time and work is reduced much.
3. If proper care is taken result will be
accurate.
4. It can be used in infinite population.
By: Prof. Raman(DU)

FREQUENCY DISTRIBUTION
Frequency distribution is a series when a number
of
observations with similar or closely related values
are put in separate bunches or groups, each
group being in order of magnitude in a series. It i s
simply a table in which the data are grouped into
classes and the number of cases which fall in
each class are recorded. It shows the frequency
of occurrence of different values of a single
Phenomenon.
By: Prof. Raman(DU)

A frequency distribution is constructed for three
main reasons:
1. To facilitate the analysis of data.
2. To estimate frequencies of the unknown
population distribution from the distribution of
sample data and
3. To facilitate the computation of various
statistical Measures.
By: Prof. Raman(DU)

RAW DATA
The statistical data collected are generally
raw data or ungrouped data. Let us
consider the daily wages (in Rs. ) of 30
laborers in a factory.
80 70 55 50 60 65 40 30 80 90
75 45 35 65 70 80 82 55 65 80
60 55 38 65 75 85 90 65 45 75
By: Prof. Raman(DU)

RAW DATA
A better way to express the figures in an
ascending or descending order of magnitude and
is commonly known as array. But this does not
reduce the bulk of the data. The above data when
formed into an array is in the following form:
30 35 38 40 45 45 50 55 55 55
60 60 65 65 65 65 65 65 70 70
75 75 75 80 80 80 80 85 90 90
By: Prof. Raman(DU)

DISCRETE (OR) UNGROUPED
In this form of distribution, the frequency refers to discrete
value. Here the data are presented in a way that exact
measurement of units are clearly indicated. There are
definite difference between the variables of different
groups of items. Each class is distinct and separate from
the other class. Non-continuity from one class to another
class exist. Data as such facts like the number of rooms in
a house, the number of companies registered in a country,
the number of children in a family, etc.
By: Prof. Raman(DU)

Example:
In a survey of 40 families in a village, the number of
children per family was recorded and the following data
obtained.
1 0 3 2 1 5 6 2
2 1 0 3 4 2 1 6
3 2 1 5 3 3 2 4
2 2 3 0 2 1 4 5
3 3 4 4 1 2 4 5
Represent the data in the form of a discrete frequency
distribution.
By: Prof. Raman(DU)

Solution:
Frequency distribution of the number of children
Number of
children
Tally Marks Frequency
0
1
2
3
4
5
6
lll
llll ll
llll llll
llll lll
llll l
llll
llll l
3
7
10
8
6
4
6
Total 40
By: Prof. Raman(DU)

CONTINUOUS FREQUENCY
DISTRIBUTION
In this form of distribution refers to groups
of values. This becomes necessary in the
case of some variables which can take any
fractional value and in which case an exact
measurement is not possible. Hence a
discrete variable can be presented in the
form of a continuous frequency distribution.
By: Prof. Raman(DU)

DISTRIBUTION
Wage distribution of 100 employees
Weekly wages
(Rs.)
Number of
employees
50-100
100-150
150-200
200-250
250-300
350-400
4
12
22
33
16
8
5
Total 100
By: Prof. Raman(DU)

DISTRIBUTION
Nature of class:
The following are some basic technical terms when a
continuous frequency distribution is formed or data are
classified according to class intervals.
1. Class limits
2. Class Interval
3. Width or size of the class interval
4. Range
5. Mid-value or mid-point
6. Frequency
7. Number of class intervals
By: Prof. Raman(DU)

DISTRIBUTION
Range:
The difference between largest and
smallest value of the observation is
called The Range and is denoted by ‘ R’
i.e.
R = Largest value –Smallest value
R = L -S
By: Prof. Raman(DU)

DISTRIBUTION
Mid-value or mid-point:
The central point of a class interval is called the mid value
or mid-point. It is found out by adding the upper and lower
limits
of a class and dividing the sum by 2.
(i.e.) Mid-value =L+U/2
For example, if the class interval is 20-30 then the mid-
value is
20+30/2=25
By: Prof. Raman(DU)

DISTRIBUTION
Types of class intervals:
There are three methods of classifying the
data according to class intervals namely
a) Exclusive method
b) Inclusive method
c) Open-end classes
By: Prof. Raman(DU)

DISTRIBUTION
Exclusive method:
When the class intervals are so fixed that the upper limit of
one class is the lower limit of the next class; it is known as
the exclusive method of classification. The following data
are classified on this basis.
Expenditure
(Rs.)
No. of families
0 -5000
5000-10000
10000-15000
15000-20000
20000-25000
60
95
122
83
40
Total 400
By: Prof. Raman(DU)

DISTRIBUTION
Inclusive method:
In this method, the overlapping of the class intervals is
avoided. Both the lower and upper limits are included in
the class interval.
Class interval Frequency
5-9
10-14
15-19
20-29
30-34
35-39
7
12
15
21
10
5
Total 70
By: Prof. Raman(DU)

DISTRIBUTION
Open end classes:
A class limit is missing either at the lower end of the first
class interval or at the upper end of the last class interval
or both are not specified.
Salary Range No of
Workers
Below 2000
2000 –4000
4000 –6000
6000 –8000
8000 and
Above
7
5
6
4
3
By: Prof. Raman(DU)

DIAGRAMATIC AND GRAPHICAL
REPRESENTATION
One-dimensional diagrams:
In such diagrams, only one-dimensional measurement, i.e.
height is used and the width is not considered. These
diagrams are in the form of bar or line charts and can be
classified as
1. Line Diagram
2. Simple Diagram
3. Multiple Bar Diagram
4. Sub-divided Bar Diagram
5. Percentage Bar Diagram
By: Prof. Raman(DU)

REPRESENTATION
Line Diagram:
Line diagram is used in case where there are many items
to be shown and there is not much of difference in their
values. Such diagram is prepared by drawing a vertical
line for each item according to the scale. The distance
between lines is kept uniform. Line diagram makes
comparison easy, but it is less attractive.
Example 1: Show the following data by a line chart:
By: Prof. Raman(DU)

REPRESENTATION
By: Prof. Raman(DU)

REPRESENTATION
Simple Bar Diagram:
Simple bar diagram can be drawn either on
horizontal or vertical base, but bars on horizontal
base more common. Bars must be uniform width
and intervening space between bars must be
equal. While constructing a simple bar diagram,
the scale is determined on the basis of the highest
value in the series.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Represent the following data by a bar
diagram
By: Prof. Raman(DU)

REPRESENTATION
• Solution:
By: Prof. Raman(DU)

REPRESENTATION
Multiple Bar Diagram:
Multiple bar diagram is used for comparing two or
more sets of statistical data. Bars are constructed
side by side to represent the set of values for
comparison. In order to distinguish bars, they may
be either differently coloured or there should be
different types of crossings or dotting, etc. An
index is also prepared to identify the meaning of
different colours or dotting's.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Draw a multiple bar diagram for the following data.
By: Prof. Raman(DU)

REPRESENTATION
Solution:
By: Prof. Raman(DU)

REPRESENTATION
• Sub-divided Bar Diagram:
In a sub-divided bar diagram, the bar is sub-
divided into various parts in proportion to the
values given in the data and the whole bar
represent the total. Such diagrams are also called
Component Bar diagrams. The sub divisions are
distinguished by different colours or crossings or
dotting’s.
By: Prof. Raman(DU)

REPRESENTATION
Example 4:
Represent the following data by a sub-divided bar
diagram.
By: Prof. Raman(DU)

REPRESENTATION
Percentage bar diagram:
This is another form of component bar diagram. Here the
components are not the actual values but percentages of
the whole. The main difference between the sub-divided
bar diagram and percentage bar diagram is that in the
former the bars are of different heights since their totals
may be different whereas in the latter the bars are of equal
height since each bar represents 100 percent. In the case
of data having sub-division, percentage bar diagram will
be more appealing than sub-divided bar diagram.
By: Prof. Raman(DU)

REPRESENTATION
• Example:
Represent the following data by a percentage bar diagram
By: Prof. Raman(DU)

REPRESENTATION
Solution:
Convert the given values into percentages as follows
By: Prof. Raman(DU)

REPRESENTATION
Two-dimensional Diagrams:
In one-dimensional diagrams, only length 9 is taken into
account. But in two-dimensional diagrams the area
represent the data and so the length and breadth have
both to be taken into account. Such diagrams are also
called area diagrams or surface diagrams. The important
types of area diagrams are:
1. Rectangles
2. Squares
3. Pie-diagrams
By: Prof. Raman(DU)

REPRESENTATION
Rectangles:
Rectangles are used to represent the relative magnitude
of two or more values. The area of the rectangles are kept
in proportion to the values. Rectangles are placed side by
side for comparison. When two sets of figures are to be
represented by rectangles, either of the two methods may
be adopted.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Represent the following data by sub-divided percentage
rectangular diagram.
By: Prof. Raman(DU)

REPRESENTATION
Solution :
The items of expenditure will be converted into percentage
as shown below:
By: Prof. Raman(DU)

REPRESENTATION
Squares:
The rectangular method of diagrammatic
presentation is difficult to use where the values of
items vary widely. The method of drawing a
square diagram is very simple. One has to take
the square root of the values of various item that
are to be shown in the diagrams and then select a
suitable scale to draw the squares.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Yield of rice in Kgs. per acre of five countries are
Represent the above data by Square diagram.
By: Prof. Raman(DU)

REPRESENTATION
Solution:
To draw the square diagram we calculate as follows:
By: Prof. Raman(DU)

REPRESENTATION
Pie Diagram or Circular Diagram:
Another way of preparing a two-dimensional
diagram is in the form of circles. In such
diagrams, both the total and the component parts
or sectors can be shown. The area of a circle is
proportional to the square of its radius.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Draw a Pie diagram for the following data of production of
sugar in quintals of various countries.
By: Prof. Raman(DU)

REPRESENTATION
Solution:
The values are expressed in terms of degree as follows.
By: Prof. Raman(DU)

REPRESENTATION
Pie Diagram
By: Prof. Raman(DU)

REPRESENTATION
Pictograms and Cartograms:
Pictograms are not abstract presentation such as
lines or bars but really depict the kind of data we
are dealing with. Pictures are attractive and easy
to comprehend and as such this method is
particularly useful in presenting statistics to the
layman. When Pictograms are used, data are
represented through a pictorial symbol that is
carefully selected.
By: Prof. Raman(DU)

REPRESENTATION
Graphs:
A graph is a visual form of presentation of statistical data.
A graph is more attractive than a table of figure. Even a
common man can understand the message of data from the
graph. Comparisons can be made between two or more
phenomena very easily with the help of a graph.
How ever here we shall discuss only some important types of
graphs which are more popular and they are
1.Histogram
2. Frequency Polygon
3.Frequency Curve
4. Ogive
5. Lorenz Curve
By: Prof. Raman(DU)

REPRESENTATION
Histogram:
A histogram is a bar chart or graph showing
the frequency of occurrence of each value
of the variable being analyzed. In
histogram, data are plotted as a series of
rectangles. Class intervals are shown on
the ‘ X-axis’ and the frequencies on the
‘Y-axis’ .
By: Prof. Raman(DU)

REPRESENTATION
Example:
Draw a histogram for the following data.
By: Prof. Raman(DU)

REPRESENTATION
Frequency Polygon:
If we mark the midpoints of the top horizontal sides of the
rectangles in a histogram and join them by a straight line,
the figure so formed is called a Frequency Polygon. This is
done under the assumption that the frequencies in a class
interval are evenly distributed throughout the class. The
area of the polygon is equal to the area of the histogram,
because the area left outside is just equal to the area
included in it.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Draw a frequency polygon for the following data.
By: Prof. Raman(DU)

REPRESENTATION
Frequency Curve:
If the middle point of the upper boundaries of the
rectangles of a histogram is corrected by a smooth
freehand curve, then that diagram is called frequency
curve. The curve should begin and end at the base line.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Draw a frequency curve for the following data.
By: Prof. Raman(DU)

REPRESENTATION
Solution: FREQUENCY CURVE
By: Prof. Raman(DU)

REPRESENTATION
Ogives:
For a set of observations, we know how to construct a
frequency distribution. In some cases we may require the
number of observations less than a given value or more
than a given value. This is obtained by a accumulating
(adding) the frequencies up to (or above) the give
value. This accumulated frequency is called cumulative
frequency.
By: Prof. Raman(DU)

REPRESENTATION
There are two methods of constructing ogive namely:
1. The ‘ less than ogive’ method
2. The ‘ more than ogive’ method.
In less than ogive method we start with the upper limits of
the classes and go adding the frequencies. When these
frequencies are plotted, we get a rising curve. In more
than ogive method, we start with the lower limits of the
classes and from the total frequencies we subtract the
frequency of each class. When these frequencies are
plotted we get a declining curve.
By: Prof. Raman(DU)

REPRESENTATION
Example:
Draw the Ogives for the following data.
By: Prof. Raman(DU)

REPRESENTATION
Solution
By: Prof. Raman(DU)

REPRESENTATION
Lorenz Curve:
Lorenz curve is a graphical method of studying dispersion.
It was introduced by Max.O.Lorenz, a great Economist and
a statistician, to study the distribution of wealth and
income. It is also used to study the variability in the
distribution of profits, wages, revenue, etc. It is specially
used to study the degree of inequality in the distribution of
income and wealth between countries or between different
periods. It is a percentage of cumulative values of one
variable in combined with the percentage of cumulative
values in other variable and then Lorenz curve is drawn.
By: Prof. Raman(DU)

REPRESENTATION
Example 16:
In the following table, profit earned is given from the
number of companies belonging to two areas A and B.
Draw in the same diagram their Lorenz curves and
interpret them.
By: Prof. Raman(DU)

MEASURES OF CENTRAL TENDENCY
In the study of a population with respect to one in which we are
interested we may get a large number of observations. It is not
possible to grasp any idea about the characteristic when we
look at all the observations. So it is better to get one number for
one group. That number must be a good representative one for
all the observations to give a clear picture of that characteristic.
Such representative number can be a central value for all these
observations. This central value is called a measure of central
tendency or an average or a measure of locations. There are
five averages. Among them mean, median and mode are called
simple averages and the other two averages geometric mean
and harmonic mean are called special averages
By: Prof. Raman(DU)

Arithmetic mean or mean :
Arithmetic mean or simply the mean of a variable is
defined as the sum of the observations divided by the
number of observations. If the variable x assumes n
values x
1, x2… xn then the mean, x, is given by
This formula is for the ungrouped or raw data.
By: Prof. Raman(DU)

Example 1 :
Calculate the mean for 2, 4, 6, 8, 10.
Solution :
By: Prof. Raman(DU)

Short-Cut method :
Under this method an assumed or an arbitrary average
(indicated by A) is used as the basis of calculation of deviations
from individual values. The formula is
where, A = the assumed mean or any value in x
d = the deviation of each value from the assumed mean
By: Prof. Raman(DU)

Example : A student’ s marks in 5 subjects are 75, 68, 80, 92, 56. Find
his average mark.
Solution:
By: Prof. Raman(DU)

Grouped Data :
The mean for grouped data is obtained from the following
formula:
where x = the mid-point of individual class
f= the frequency of individual class
N = the sum of the frequencies or total frequencies.
By: Prof. Raman(DU)

Short-cut method :
A = any value in x
N = total frequency
c = width of the class interval
By: Prof. Raman(DU)

Example 3:
Given the following frequency distribution, calculate the
arithmetic mean
By: Prof. Raman(DU)

By: Prof. Raman(DU)

Merits of H.M :
1. It is rigidly defined.
2. It is defined on all observations.
3. It is amenable to further algebraic treatment.
4. It is the most suitable average when it is desired to give
greater weight to smaller observations and less weight to
the larger ones.
By: Prof. Raman(DU)

Demerits of H.M :
1. It is not easily understood.
2. It is difficult to compute.
3. It is only a summary figure and may not be the actual
item in the series
4. It gives greater importance to small items and is
therefore, useful only when small items have to be given
greater weightage.
By: Prof. Raman(DU)

Weighted Arithmetic mean
For calculating simple mean, we suppose that all the
values or the sizes of items in the distribution have equal
importance. But, in practical life this may not be so. In
case some items are more 20,50,10,99
important than others, a simple average computed is not
representative of the distribution. Proper weightage has to
be given to the various items.
By: Prof. Raman(DU)

If x1, x2…xn be the values of a variable x with respective
weights of w1, w2… wn assigned to them, then
Uses of the weighted mean:
Weighted arithmetic mean is used in:
a. Construction of index numbers.
b. Comparison of results of two or more universities where
number of students differ.
c. Computation of standardized death and birth rates.
By: Prof. Raman(DU)

Example:
Calculate weighted average from the following data
By: Prof. Raman(DU)

Solution:
By: Prof. Raman(DU)

Combined mean :
If the arithmetic averages and the number of items in two or
more related groups are known, the combined or the composite
mean of the entire group can be obtained by
The advantage of combined arithmetic mean is that, we can
determine the over, all mean of the combined data without
going back to the original data.
By: Prof. Raman(DU)

Example:
Find the combined mean for the data given below
n1= 20 , x1= 4 , n2 = 30, x2= 3
Solution:
By: Prof. Raman(DU)

Positional Averages:
These averages are based on the position of the given
observation in a series, arranged in an ascending or
descending order. The magnitude or the size of the values
does matter as was in the case of arithmetic mean. It is
because of the basic difference that the median and mode
are called the positional measures of an average.
By: Prof. Raman(DU)

Median :
The median is that value of the variate which divides the
group into two equal parts, one part comprising all values
greater, and the other, all values less than median.
Ungrouped or Raw data :
Arrange the given values in the increasing or decreasing
order. If the number of values are odd, median is the middle
value. If the number of values are even, median is the mean of
middle two values.
By formula
By: Prof. Raman(DU)

Example:
When odd number of values are given. Find median for
the following data
25, 18, 27, 10, 8, 30, 42, 20, 53
Solution: Arranging the data in the increasing order 8, 10,
18, 20, 25, 27, 30, 42, 53
The middle value is the 5th item i.e., 25 is the median
Using formula
By: Prof. Raman(DU)

Grouped Data:
In a grouped distribution, values are associated with
frequencies. Grouping can be in the form of a discrete
frequency distribution or a continuous frequency distribution.
Whatever may be the type of distribution , cumulative
frequencies have to be calculated to know the total number of
items.
Cumulative frequency : (cf)
Cumulative frequency of each class is the sum of the frequency
of the class and the frequencies of the pervious classes, i.e.
adding the frequencies successively, so that the last cumulative
frequency gives the total number of items.
By: Prof. Raman(DU)

Discrete Series:
Step1: Find cumulative frequencies.
Step2: find
Step3: See in the cumulative frequencies the value just
greater than
Step4: Then the corresponding value of x is median.
By: Prof. Raman(DU)

Example:
The following data pertaining to the number of members in
a family. Find median size of the family.
By: Prof. Raman(DU)

Continuous Series:
The steps given below are followed for the calculation of
median in continuous series.
Step1: Find cumulative frequencies.
Step2: Find
Step3: See in the cumulative frequency the value first greater than
Then the corresponding class interval is called the Median
class. Then apply the formula
Median =
Where l= Lower limit of the median class
m = cumulative frequency preceding the median
c = width of the median class
f =frequency in the median class.
N=Total frequency.
Note : If the class intervals are given in inclusive type convert
them into exclusive type and call it as true class interval and
consider lower limit in this.
By: Prof. Raman(DU)

Basics of statistics

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