Statistics / Quantitative Techniques Study Material

What Is Statistics?
The word 'Statistics' is derived from the Latin word 'Statis' which means a "political
state." Clearly, statistics is closely linked with the administrative affairs of a state such as
facts and figures regarding defense force, population, housing, food, financial resources
etc. What is true about a government is also true about industrial administration units, and
even one’s personal life.
The word statistics has several meanings. In the first place, it is a plural noun which
describes a collection of numerical data such as employment statistics, accident statistics,
population statistics, birth and death, income and expenditure, of exports and imports etc.
It is in this sense that the word 'statistics' is used by a layman or a newspaper.
Secondly the word statistics as a singular noun, is used to describe a branch of applied
mathematics, whose purpose is to provide methods of dealing with a collections of data
and extracting information from them in compact form by tabulating, summarizing and
analyzing the numerical data or a set of observations.
The various methods used are termed as statistical methods and the person using them is
known as a statistician. A statistician is concerned with the analysis and interpretation of
the data and drawing valid worthwhile conclusions from the same.
It is in the second sense that we are writing this guide on statistics.
Lastly the word statistics is used in a specialized sense. It describes various numerical
items which are produced by using statistics ( in the second sense ) to statistics ( in the
first sense ). Averages, standard deviation etc. are all statistics in this specialized third
sense.
The word ’statistics’ in the first sense is defined by Professor Secrit as follows:-
"By statistics we mean aggregate of facts affected to a marked extent by multiplicity of
causes, numerically expressed, enumerated or estimated according to reasonable standard
of accuracy, collected in a systematic manner for a predetermined purpose and placed in
relation to each other."
This definition gives all the characteristics of statistics which are (1) Aggregate of facts
(2) Affected by multiplicity of causes (3) Numerically expressed (4) Estimated according
to reasonable standards of accuracy (5) Collected in a systematic manner (6) Collected
for a predetermined purpose (7) Placed in relation to each other.
In addition to this, one more stage i.e. organization of data is suggested

What Do Statisticians Do?
The word 'statistics' in the second sense is defined by Croxton and Cowden as follows:-
"The collection, presentation, analysis and interpretation of the numerical data."
This definition clearly points out four stages in a statistical investigation, namely:
1) Collection of data 2) Presentation of data
3) Analysis of data 4) Interpretation of data
In addition to this, one more stage i.e. organization of data is suggested
Statistics is a field that studies data. A statistician is involved with collecting,
summarizing, and interpreting this data. Many problems in statistics are motivated by the
world around us. For these problems, there is often an inherent degree of variability
among the data points. Statistics helps us find solutions to these problems by using
techniques to deal with this uncertainty in the data.
Statistics is a discipline which is concerned with: designing experiments and other data
collection, summarizing information to aid understanding, drawing conclusions from
data, and estimating the present or predicting the future.
In making predictions, Statistics uses the companion subject of Probability, which
models chance mathematically and enables calculations of chance in complicated cases.
Today, statistics has become an important tool in the work of many academic disciplines
such as medicine, psychology, education, sociology, engineering and physics, just to
name a few. Statistics is also important in many aspects of society such as business,
industry and government. Because of the increasing use of statistics in so many areas of
our lives, it has become very desirable to understand and practice statistical thinking.
This is important even if you do not use statistical methods directly.
It presents exciting opportunities for those who work as professional statisticians.
Statistics is essential for the proper running of government, central to decision making in
industry, and a core component of modern educational curricula at all levels."
Defines statistics as:
"The mathematics of the collection, organization, and interpretation of numerical data,
especially the analysis of population characteristics by inference from sampling."
A branch of mathematics dealing with the collection, analysis, interpretation, and
presentation of masses of numerical data.

The steps of statistical analysis involve collecting information, evaluating it, and drawing
conclusions.
The information might be:
A test group's favorite amount of sweetness in a blend of fruit juices
The number of men and women hired by a city government
The velocity of a burning gas on the sun's surface
Statisticians provide crucial guidance in determining what information is reliable and
which predictions can be trusted.
They often help search for clues to the solution of a scientific mystery, and sometimes
keep investigators from being misled by false impressions.
Statisticians work in a variety of fields, including medicine, government, education,
agriculture, business, and law.
WHAT DO STATISTICIANS DO?
Statisticians help determine the sampling and data collection methods, monitor the
execution of the study and the processing of data, and advise on the strengths and
limitations of the results. They must understand the nature of uncertainties and be able to
draw conclusions in the context of particular statistical applications.
Surveys: Survey statisticians collect information from a carefully specified sample and
extend the results to an entire population.
Sample surveys might be used to:
1. Determine which political candidate is more popular
2. Discover what foods teenagers prefer for breakfast
3. Estimate the number of children living in a given school district
Government Operations: Government statisticians conduct experiments to aid in the
development of public policy and social programs. Such experiments include:
1. Consumer prices
2. Fluctuations in the economy
3. Employment patterns
Population trends
Scientific Research: Statistical sciences are used to enhance the validity of inferences in:
1. Radiocarbon dating to estimate the risk of earthquakes
2. Clinical trials to investigate the effectiveness of new treatments
3. Field experiments to evaluate irrigation methods
4. Measurements of water quality

5. Psychological tests to study how we reach the everyday decisions in our lives
Business And Industry: Statisticians quantify unknowns in order to optimize resources.
They:
1. Predict the demand for products and services
2. Check the quality of items manufactured in a facility
3. Manage investment portfolios
4. Forecast how much risk activities entail, and calculate fair and competitive
insurance rates
Uses
1. To present the data in a concise and definite form : Statistics helps in classifying
and tabulating raw data for processing and further tabulation for end users.
2. To make it easy to understand complex and large data : This is done by presenting
the data in the form of tables, graphs, diagrams etc., or by condensing the data
with the help of means, dispersion etc.
3. For comparison : Tables, measures of means and dispersion can help in
comparing different sets of data..
4. In forming policies : It helps in forming policies like a production schedule, based
on the relevant sales figures. It is used in forecasting future demands.
5. Enlarging individual experiences : Complex problems can be well understood by
statistics, as the conclusions drawn by an individual are more definite and precise
than mere statements on facts.
6. In measuring the magnitude of a phenomenon:- Statistics has made it possible to
count the population of a country, the industrial growth, the agricultural growth,
the educational level (of course in numbers).
Limitations
1. Statistics does not deal with individual measurements. Since statistics deals with
aggregates of facts, it can not be used to study the changes that have taken place
in individual cases. For example, the wages earned by a single industry worker at
any time, taken by itself is not a statistical datum. But the wages of workers of
that industry can be used statistically. Similarly the marks obtained by John of
your class or the height of Beena (also of your class) are not the subject matter of
statistical study. But the average marks or the average height of your class has
statistical relevance.
2. Statistics cannot be used to study qualitative phenomenon like morality,
intelligence, beauty etc. as these can not be quantified. However, it may be
possible to analyze such problems statistically by expressing them numerically.
For example we may study the intelligence of boys on the basis of the marks
obtained by them in an examination.
3. Statistical results are true only on an average:- The conclusions obtained
statistically are not universal truths. They are true only under certain conditions.

This is because statistics as a science is less exact as compared to the natural
science.
4. Statistical data, being approximations, are mathematically incorrect. Therefore,
they can be used only if mathematical accuracy is not needed.
5. Statistics, being dependent on figures, can be manipulated and therefore can be
used only when the authenticity of the figures has been proved beyond doubt..
Distrust Of Statistics
It is often said by people that, "statistics can prove anything." There are three types of lies
- lies, demand lies and statistics - wicked in the order of their naming. A Paris banker
said, "Statistics is like a miniskirt, it covers up essentials but gives you the ideas."
Thus by "distrust of statistics" we mean lack of confidence in statistical statements and
methods. The following reasons account for such views about statistics.
1. Figures are convincing and, therefore people easily believe them.
2. They can be manipulated in such a manner as to establish foregone conclusions.
3. The wrong representation of even correct figures can mislead a reader. For
example, John earned Rs. 4000 in 1990 - 1991 and Jem earned Rs. 5000. Reading
this one would form the opinion that Jem is decidedly a better worker than John.
However if we carefully examine the statement, we might reach a different
conclusion as Jem’s earning period is unknown to us. Thus while working with
statistics one should not only avoid outright falsehoods but be alert to detect
possible distortion of the truth
Statistics Can Be Misused
In one factory which I know, workers were accusing the management for not providing
them with proper working conditions. In support they quoted the number of accidents.
When I considered the matter more seriously, I found that most of the staff was
inexperienced and thus responsible for those accidents. Moreover many of the accidents
were either minor or fake. I compared the working conditions of this factory to other
factories and I found the conditions far better in this factory. Thus by merely noting the
number of accidents and complaints of the workers, I would not dare to say that the
working conditions were worse. On the other hand due to the proper statistical knowledge
and careful observations I came to conclusion that the management was right.
Thus the usefulness of the statistics depends to a great extent upon its user. If used
properly, by an efficient and unbiased statistician, it will prove to be an efficient tool.
Collection of facts and figures and deriving meaningful information from them is an i As
an example, suppose "Jerry Greval" has a shoe company. His company wants to establish
their business in India, particularly in Mumbai. Let us see a few ways in which statistics
will be useful to him.

1. He does not wish to manufacture equal quantities of shoes ranging from size 1 to
10. Jerry would like to know which sizes are more in demand and which are in
less demand. Knowing this they can devise the manufacturing strategy.
2. Now the company wants to advertise the ’Brand name’ and thus their product in
the market. To make the product popular the brand name must be attractive: Jerry
selects the name ’Strong foot ’. The ‘Strong foot’ and its qualities have to be
made to look appealing to the people in Mumbai and this requires publicity.
Nothing is more appealing than what has been said in one’s own mother-tongue.
So Jerry wants to print and distribute leaflets among people For this he needs to
know the mother-tongues of various groups of people in Mumbai. This
information is the most important factor of his business. In order to get this
information, his company will have to appoint personnel who will go from door to
door and find out the necessary information about the shoe market, people’s
choice, their mother tongue etc. This process is known as taking a survey. The
objects under study are known as Individuals or Units and the collection of
individuals is known as the population.
Often it is not possible or practical to record observations of all the individuals of
the groups from different areas, which comprise the population. In such a case
observations are recorded of only some of the individuals of the population,
selected at random. This selection of some individuals which will be a subset of
the individuals in the original group, is called a Sample; i.e. instead of an entire
population survey which would be time-consuming, the company will manage
with a ‘Sample survey’ which can be completed in a shorter time.
Note that if a sample is representative of the whole population, any conclusion
drawn from a statistical treatment of the sample would hold reasonably good for
the population. This will of course, depend on the proper selection of the sample.
One of the aims of statistics is to draw inferences about the population by a
statistical treatment of samples.

CLASSIFICATION AND TABULATION
2.1 Introduction
In any statistical investigation, the collection of the numerical data is the first and the
most important matter to be attended. Often a person investigating, will have to collect
the data from the actual field of inquiry. For this he may issue suitable questionnaires to
get necessary information or he may take actual interviews; personal interviews are more
effective than questionnaires, which may not evoke an adequate response. Another
method of collecting data may be available in publications of Government bodies or other
public or private organizations.
Sometimes the data may be available in publications of Government bodies or other
public or private organizations. Such data, however, is often so numerous that one’s mind
can hardly comprehend its significance in the form that it is shown. Therefore it becomes,
very necessary to tabulate and summarize the data to an easily manageable form. In doing
so we may overlook its details. But this is not a serious loss because Statistics is not
interested in an individual but in the properties of aggregates. For a layman, presentation
of the raw data in the form of tables or diagrams is always more effective.
Tabulation
It is the process of condensation of the data for convenience, in statistical processing,
presentation and interpretation of the information.
A good table is one which has the following requirements :
1. It should present the data clearly, highlighting important details.
2. It should save space but attractively designed.
3. The table number and title of the table should be given.+
4. Row and column headings must explain the figures therein.
5. Averages or percentages should be close to the data.
6. Units of the measurement should be clearly stated along the titles or headings.
7. Abbreviations and symbols should be avoided as far as possible.
8. Sources of the data should be given at the bottom of the data.
9. In case irregularities creep in table or any feature is not sufficiently explained,
references and foot notes must be given.
10. The rounding of figures should be unbiased.
Classification
"Classified and arranged facts speak of themselves, and narrated they are as dead as
mutton" This quote is given by J.R. Hicks.
The process of dividing the data into different groups ( viz. classes) which are
homogeneous within but heterogeneous between themselves, is called a classification.

It helps in understanding the salient features of the data and also the comparison with
similar data. For a final analysis it is the best friend of a statistician.
Methods Of Classification
The data is classified in the following ways :
1. According to attributes or qualities this is divided into two parts :
(A) Simple classification
(B) Multiple classification.
2. According to variable or quantity or classification according to class intervals. -
Qualitative Classification : When facts are grouped according to the qualities (attributes)
like religion, literacy, business etc., the classification is called as qualitative
classification.
(A) Simple Classification : It is also known as classification according to Dichotomy.
When data (facts) are divided into groups according to their qualities, the classification is
called as 'Simple Classification'. Qualities are denoted by capital letters (A, B, C, D ......)
while the absence of these qualities are denoted by lower case letters (a, b, c, d, .... etc.)
For example ,

(B) Manifold or multiple classification : In this method data is classified using one or
more qualities. First, the data is divided into two groups (classes) using one of the
qualities. Then using the remaining qualities, the data is divided into different subgroups.
For example, the population of a country is classified using three attributes: sex, literacy
and business as,
MEASURES OF CENTRAL TENDENCY
Introduction
In the previous chapter, we have studied how to collect raw data, its classification and
tabulation in a useful form, which contributes in solving many problems of statistical
concern. Yet, this is not sufficient, for in practical purposes, there is need for further
condensation, particularly when we want to compare two or more different distributions.
We may reduce the entire distribution to one number which represents the distribution.
A single value which can be considered as typical or representative of a set of
observations and around which the observations can be considered as Centered is called
an ’Average’ (or average value) or a Center of location. Since such typical values tends to

lie centrally within a set of observations when arranged according to magnitudes,
averages are called measures of central tendency.
In fact the distribution have a typical value (average) about which, the observations are
more or less symmetrically distributed. This is of great importance, both theoretically and
practically. Dr. A.L. Bowley correctly stated, "Statistics may rightly be called the science
of averages."
The word average is commonly used in day-to-day conversations. For example, we may
say that Abert is an average boy of my class; we may talk of an average American,
average income, etc. When it is said, "Abert is an average student," it means is that he is
neither very good nor very bad, but a mediocre student. However, in statistics the term
average has a different meaning.
The fundamental measures of tendencies are:
(1) Arithmetic mean
(2) Median
(3) Mode
(4) Geometric mean
(5) Harmonic mean
(6) Weighted averages
However the most common measures of central tendencies or Locations are Arithmetic
mean, median and mode. We therefore, consider the Arithmetic mean.
4.2 Arithmetic Mean
This is the most commonly used average which you have also studied and used in lower
grades. Here are two definitions given by two great masters of statistics.
Horace Sacrist : Arithmetic mean is the amount secured by dividing the sum of values
of the items in a series by their number.
W.I. King : The arithmetic average may be defined as the sum of aggregate of a series of
items divided by their number.
Thus, the students should add all observations (values of all items) together and divide
this sum by the number of observations (or items).

Ungrouped Data
Suppose, we have 'n' observations (or measures) x1 , x2 , x3, ......., xn then the Arithmetic
mean is obviously
We shall use the symbol x (pronounced as x bar) to denote the Arithmetic mean. Since
(pronounced as sigma) to denote the sum. The symbol xi will be used to denote, in
general the 'i' th observation. Then the sum, x1 + x2 + x3 + .......+ xn will be represented by
or simply
Therefore the Arithmetic mean of the set x1 + x2 + x3 + .......+ xn is given by,
This method is known as the ''Direct Method".
Example A variable takes the values as given below. Calculate the arithmetic mean of
110, 117, 129, 195, 95, 100, 100, 175, 250 and 750.
Solution: Arithmetic mean =
= 110 + 117 + 129 +195 + 95 +100 +100 +175 +250 + 750 = 2021
and n = 10
Indirect Method (Assumed Mean Method)
A = Assumed Mean =
Calculations:
Let A = 175 then
i = -65, -58, -46, +20, -80, -75,-75, +0, + 75, +575

= 670 - 399
= 271/10 = 27.1
= 175 + 27.1
= 202.1
Example M.N. Elhance’s earnings for the past week were:
Monday $ 450
Tuesday $ 375
Wednesday $ 500
Thursday $ 350
Friday $ 270
Find his average earning per day.
Solution:
n = 5
Arithmetic mean =
Therefore, Elhance’s average earning per day is $389.
Definition of dispersion : The arithmetic mean of the deviations of the values of the
individual items from the measure of a particular central tendency used. Thus the
’dispersion’ is also known as the "average of the second degree." Prof. Griffin and Dr.
Bowley said the same about the dispersion.
In measuring dispersion, it is imperative to know the amount of variation (absolute
measure) and the degree of variation (relative measure). In the former case we consider
the range, mean deviation, standard deviation etc. In the latter case we consider the
coefficient of range, the coefficient mean deviation, the coefficient of variation etc.
Methods Of Computing Dispersion

(I) Method of limits:
(1) The range (2) Inter-quatrile range (3) Percentile range
(II) Method of Averages:
(1) Quartile deviation (2) Mean deviation
(3) Standard Deviation and (4) Other measures.
Range
In any statistical series, the difference between the largest and the smallest values is
called as the range.
Thus Range (R) = L - S
Coefficient of Range : The relative measure of the range. It is used in the comparative
study
Variance
The term variance was used to describe the square of the standard deviation R.A. Fisher
in 1913. The concept of variance is of great importance in advanced work where it is
possible to split the total into several parts, each attributable to one of the factors causing
variations in their original series. Variance is defined as follows:
Variance =
Standard Deviation (s. d.)
It is the square root of the arithmetic mean of the square deviations of various values

=

Merits : (1) It is rigidly defined and based on all observations.
(2) It is amenable to further algebraic treatment.
(3) It is not affected by sampling fluctuations.
(4) It is less erratic.
Demerits : (1) It is difficult to understand and calculate.
(2) It gives greater weight to extreme values.
Note that variance V(x) =
and s. d. (  and
Then V ( x ) =
Co-efficient Of Variation ( C. V. )
To compare the variations ( dispersion ) of two different series, relative measures of
standard deviation must be calculated. This is known as co-efficient of variation or the
co-efficient of s. d. Its formula is
C. V. =
Thus it is defined as the ratio s. d. to its mean.
Remark: It is given as a percentage and is used to compare the consistency or variability
of two more series. The higher the C. V. , the higher the variability and lower the C. V.,
the higher is the consistency of the data.
Combined Standard deviation : If two sets containing n1 and n2 items having means x1
and x2 and standard deviations s1 and 2 respectively are taken together then,
(1) Mean of the combined data is

(2) s.d. of the combined set is
Percentile
The nth percentile is that value ( or size ) such that n% of values of the whole data lies
below it. For example, a score of 7% from the topmost score would be 93 the percentile
as it is above 93% of the other scores.
Percentile Range
it is used as one of the measure of dispersion. it is a set of data and is defined as = P90 -
P10 where P90 and P10 are the 90th and 10th percentile respectively. The semi -
percentile range,
i.e. can also be used but it is not common in use.
Quartiles And Interquartile Range
If we concentrate on two extreme values ( as in the case of range ), we don’t get any idea
about the scatter of the data within the range ( i.e. the two extreme values ). If we discard
these two values the limited range thus available might be more informative. For this
reason the concept of interquartile range is developed. It is the range which includes
middle 50% of the distribution. Here 1/4 ( one quarter of the lower end and 1/4 ( one
quarter ) of the upper end of the observations are excluded.

Now the lower quartile ( Q1 ) is the 25th percentile and the upper quartile ( Q3 ) is the
75th percentile. It is interesting to note that the 50th percentile is the middle quartile ( Q2
) which is in fact what you have studied under the title ’ Median ". Thus symbolically
Inter quartile range = Q3 - Q1
If we divide ( Q3 - Q1 ) by 2 we get what is known as Semi-Iinter quartile range.
i.e. . It is known as Quartile deviation ( Q. D or SI QR ).
Therefore Q. D. ( SI QR ) =
Skewness, Moments And Kurtosis
The voluminous raw data cannot be easily understood, Hence, we calculate the measures
of central tendencies and obtain a representative figure. From the measures of variability,
we can know that whether most of the items of the data are close to our away from these
central tendencies. But these statical means and measures of variation are not enough to
draw sufficient inferences about the data. Another aspect of the data is to know its
symmetry. in the chapter "Graphic display" we have seen that a frequency may be
symmetrical about mode or may not be. This symmetry is well studied by the knowledge
of the "skewness." Still one more aspect of the curve that we need to know is its flatness
or otherwise its top. This is understood by what is known as " Kurtosis."
Skewness
It may happen that two distributions have the same mean and standard deviations. For
example, see the following diagram.

Although the two distributions have the same means and standard deviations they are not
identical. Where do they differ ?
They differ in symmetry. The left-hand side distribution is symmetrical one where as the
distribution on the right-hand is asymmetrical or skewed. For a symmetrical distribution,
the values, of equal distances on either side of the mode, have equal frequencies. Thus,
the mode, median and mean - all coincide. Its curve rises slowly, reaches a maximum (
peak ) and falls equally slowly (Fig. 1). But for a skewed distribution, the mean, mode
and median do not coincide. Skewness is positive or negative as per the positions of the
mean and median on the right or the left of the mode.
A positively skewed distribution ( Fig.2 ) curve rises rapidly, reaches the maximum and
falls slowly. In other words, the tail as well as median on the right-hand side. A
negatively skewed distribution curve (Fig.3) rises slowly reaches its maximum and falls
rapidly. In other words, the tail as well as the median are on the left-hand side.
Tests Of Skewness
1. The values of mean, median and mode do not coincide. The more the difference
between them, the more is the skewness.
2. Quartiles are not equidistant from the median. i.e. ( Q3 -Me ) = ( Me - Q1 ).
3 The sum of positive deviations from the median is not equal to the sum of the negative
deviations.
4. Frequencies are not equally distributed at points of equal deviation from the mode.
5. When the data is plotted on a graph they do not give the normal bell-shaped form.
Measure Of Skewness
1. First measure of skewness It is given by Karl Pearson
Measure of skewness Co-efficient of skewness
Skp = Mean - Mode J =
i.e. Skp = - Mo

Pearson has suggested the use of this formula if it is not possible to determine the mode
(Mo) of any distribution,
( Mean - Mode ) = 3 ( mean - median )
Skp = 3 ( - Mo ) Thus J =
Note : i) Although the co-efficient of skewness is always within
co-efficient lies within ± 3.
ii) If J = 0, then there is no skewness
iii) If J is positive, the skewness is also positive.
iv) If J is negative, the skewness is also negative.
Unless and until no indication is given, you must use only Karl Pearson’s formula.
Kurtosis
It has its origin in the Greek word "Bulginess." In statistics it is the degree of flatness or
’peakedness’ in the region of mode of a frequency curve. It is measured relative to the
’peakedness’ of the normal curve. It tells us the extent to which a distribution is more
peaked or flat-topped than the normal curve. If the curve is more peaked than a normal
curve it is called ’Lepto Kurtic.’ In this case items are more clustered about the mode. If
the curve is more flat-toped than the more normal curve, it is Platy-Kurtic. The normal
curve itself is known as "Meso Kurtic."
Moments
Moment is a familiar mechanical term for the measure of a force with reference to its
tendency to produce rotation. In statistics moments are used to describe the various

characteristics of a frequency distribution like center tendency, variation, skewness and
kurtosis.
Moments are calculated using the arithmetic mean. According to Waugh, the arithmetic
mean of the various powers of these deviations in any distribution are called the moments
of the distribution. Let ’ x ’ be the deviation of any item in a distribution from the
arithmetic mean of that distribution. The arithmetic mean of the various powers of these
deviations is the moments of the distribution. if we take the mean of the 1st power of the
deviations, we get the 1st moment about the mean, the mean of the squares of the
deviations gives the second moment about the mean, the mean of the cubes gives the
third moment about the mean and so on. The moments about mean are called the "central
moment" and are denoted by
1st central moment = 0
Since sum of deviations of items from the arithmetic mean is always zero.
For frequency distribution,
In many cases it is very difficult to calculate moments about actual moment, particularly
when actual mean is in fractions. In such case we first compute moments about an

arbitrary origin ’A’ and then convert these moments into moments about actual mean.
These are called ’ raw moments ’ which are denoted by
Thus we have
and so on. For frequency distribution
Now to obtain the central moment as
ients,
based upon four moments about the mean.

These are pure numbers and they provide information about the shape of the curve
obtained from the frequency distribution.
For symmetrical distribution, the moments of odd order about the mean vanish and
therefore m3 = 0 rendering 1 1 gives the measure of departure from symmetry
2 = gives the measure of flatness of
the mode and also defines the measure of Kurtosis or convexity of the curve.
Note : 2 2 = 0 then the curve is normal which is neither flat nor peaked
i.e. Meso kurtic.
2 2 > 0 then the curve is more peaked than a normal curve and is called
Lepto kurtic.
2 2 <> 0 then curve is flatter than a normal curve and is called Platy
kurtic.
CORRELATION - REGRESSION
Introduction
So far we have considered only univariate distributions. By the averages, dispersion and
skewness of distribution, we get a complete idea about the structure of the distribution.
Many a time, we come across problems which involve two or more variables. If we
carefully study the figures of rain fall and production of paddy, figures of accidents and
motor cars in a city, of demand and supply of a commodity, of sales and profit, we may
find that there is some relationship between the two variables. On the other hand, if we
compare the figures of rainfall in America and the production of cars in Japan, we may
find that there is no relationship between the two variables. If there is any relation
between two variables i.e. when one variable changes the other also changes in the same
or in the opposite direction, we say that the two variables are correlated.
W. J. King : If it is proved that in a large number of instances two variables, tend always
to fluctuate in the same or in the opposite direction then it is established that a
relationship exists between the variables. This is called a "Correlation."
Correlation
It means the study of existence, magnitude and direction of the relation between two or
more variables. in technology and in statistics. Correlation is very important. The famous
astronomist Bravais, Prof. Sir Fanci’s Galton, Karl Pearson (who used this concept in
Biology and in Genetics). Prof. Neiswanger and so many others have contributed to this
great subject.

Types of Correlation
1. Positive and negative correlation
2. Linear and non-linear correlation
A) If two variables change in the same direction (i.e. if one increases the other also
increases, or if one decreases, the other also decreases), then this is called a positive
correlation. For example : Advertising and sales.
B) If two variables change in the opposite direction ( i.e. if one increases, the other
decreases and vice versa), then the correlation is called a negative correlation. For
example : T.V. registrations and cinema attendance.
1. The nature of the graph gives us the idea of the linear type of correlation between
two variables. If the graph is in a straight line, the correlation is called a "linear
correlation" and if the graph is not in a straight line, the correlation is non-linear
or curvi-linear.
For example, if variable x changes by a constant quantity, say 20 then y also changes by a
constant quantity, say 4. The ratio between the two always remains the same (1/5 in this
case). In case of a curvi-linear correlation this ratio does not remain constant.
Degrees of Correlation
Through the coefficient of correlation, we can measure the degree or extent of the
correlation between two variables. On the basis of the coefficient of correlation we can
also determine whether the correlation is positive or negative and also its degree or
extent.
1. Perfect correlation: If two variables changes in the same direction and in the
same proportion, the correlation between the two is perfect positive. According
to Karl Pearson the coefficient of correlation in this case is +1. On the other hand
if the variables change in the opposite direction and in the same proportion, the
correlation is perfect negative. its coefficient of correlation is -1. In practice we
rarely come across these types of correlations.
2. Absence of correlation: If two series of two variables exhibit no relations
between them or change in variable does not lead to a change in the other
variable, then we can firmly say that there is no correlation or absurd
correlation between the two variables. In such a case the coefficient of
correlation is 0.
3. Limited degrees of correlation: If two variables are not perfectly correlated or is
there a perfect absence of correlation, then we term the correlation as Limited
correlation. It may be positive, negative or zero but lies with the limits

High degree, moderate degree or low degree are the three categories of this kind of
correlation. The following table reveals the effect ( or degree ) of coefficient or
correlation.
Degrees Positive Negative
Zero 0
Perfect c + 1 -1
+ 0.75 to +
1
- 0.75 to -1
+ 0.25 to +
0.75
- 0.25 to -
0.75
0 to 0.25 0 to - 0.25
6.5 Methods Of Determining Correlation
We shall consider the following most commonly used methods.(1) Scatter Plot
(2) Kar Pearson’s coefficient of correlation (3) Spearman’s Rank-correlation
coefficient.
1) Scatter Plot ( Scatter diagram or dot diagram ): In this method the values
of the two variables are plotted on a graph paper. One is taken along the
horizontal ( (x-axis) and the other along the vertical (y-axis). By plotting the
data, we get points (dots) on the graph which are generally scattered and hence
the name ‘Scatter Plot’.
The manner in which these points are scattered, suggest the degree and the
direction of correlation. The degree of correlation is denoted by ‘ r ’ and its
direction is given by the signs positive and negative.
i) If all points lie on a rising straight line the correlation is perfectly positive and
r = +1 (see fig.1 )
ii) If all points lie on a falling straight line the correlation is perfectly negative
and r = -1 (see fig.2)
iii) If the points lie in narrow strip, rising upwards, the correlation is high
degree of positive (see fig.3)

iv) If the points lie in a narrow strip, falling downwards, the correlation is high degree of
negative (see fig.4)
v) If the points are spread widely over a broad strip, rising upwards, the correlation is low
degree positive (see fig.5)
vi) If the points are spread widely over a broad strip, falling downward, the correlation is
low degree negative (see fig.6)
vii) If the points are spread (scattered) without any specific pattern, the correlation is
absent. i.e. r = 0. (see fig.7)
Though this method is simple and is a rough idea about the existence and the degree of
correlation, it is not reliable. As it is not a mathematical method, it cannot measure the
degree of correlation.
2) Karl Pearson’s coefficient of correlation: It gives the numerical expression for the
measure of correlation. it is noted by ‘ r ’. The value of ‘ r ’ gives the magnitude of
correlation and sign denotes its direction. It is defined as
r =
where
N = Number of pairs of observation
Note : r is also known as product-moment coefficient of correlation.
OR r =

OR r =
Now covariance of x and y is defined as
Example Calculate the coefficient of correlation for the following data.
Age
(years)
of
Husband
Age (years) of wife
Total
10 -
20
20 -
30
30 -
40
40 -
50
50 -
60
10 - 25
25 - 35
35 - 45
45 - 55
55 - 65
5
3
3
15
11
11
14
7
7
12
3
3
6
8
29
32
22
9
Total 8 29 32 22 9 100
Click here to enlarge

- -
in
Probable Error
It is used to help in the determination of the Karl Pearson’s coefficient of correlation ‘ r ’.
Due to this ‘ r ’ is corrected to a great extent but note that ‘ r ’ depends on the random
sampling and its conditions. it is given by
P. E. = 0.6745
i. If the value of r is less than P. E., then there is no evidence of correlation i.e. r is
not significant.
ii. If r is more than 6 times the P. E. ‘ r ’ is practically certain .i.e. significant.
iii. By adding or subtracting P. E. to ‘ r ’ , we get the upper and Lower limits within
which ‘ r ’ of the population can be expected to lie.
Symbolically e = r
P = Correlation ( coefficient ) of the population.

Example If r = 0.6 and n = 64 find out the probable error of the coefficient of correlation.
Solution: P. E. = 0.6745
= 0.6745
=
= 0.57
Spearman’s Rank Correlation Coefficient
This method is based on the ranks of the items rather than on their actual values. The
advantage of this method over the others in that it can be used even when the actual
values of items are unknown. For example if you want to know the correlation between
honesty and wisdom of the boys of your class, you can use this method by giving ranks to
the boys. It can also be used to find the degree of agreements between the judgements of
two examiners or two judges. The formula is :
R =
where R = Rank correlation coefficient
D = Difference between the ranks of two items
N = The number of observations.
Note: -
i)
agreement in the same direction
ii) When R = -
agreement in the opposite direction.
iii)

Computation:
i. Give ranks to the values of items. Generally the item with the highest value is
ranked 1 and then the others are given ranks 2, 3, 4, .... according to their values
in the decreasing order.
ii. Find the difference D = R1 - R2
where R1 = Rank of x and R2 = Rank of y
iii. Calculate D2 D2
iv. Apply the formula.
Note :
In some cases, there is a tie between two or more items. in such a case each items have
ranks 4th and 5th respectively then they are given = 4.5th rank. If three items are of
equal rank say 4th then they are given = 5th rank each. If m be the number of
items of equal ranks, the factor is added to S D2. If there are more than
one of such cases then this factor added as many times as the number of such cases, then
Example Calculate ‘ R ’ from the following data.
Student
No.:
1 2 3 4 5 6 7 8 9 10
Rank
in
Maths :
1 3 7 5 4 6 2 10 9 8
Rank
in
Stats:
3 1 4 5 6 9 7 8 10 2
Solution :
Student
No.
Rank
in
Maths
Rank
in
Stats
R1 - R2
D
(R1 - R2 )2
D2

(R1) (R2)
1 1 3 -2 4
2 3 1 2 4
3 7 4 3 9
4 5 5 0 0
5 4 6 -2 4
6 6 9 -3 9
7 2 7 -5 25
8 10 8 2 4
9 9 10 -1 1
10 8 2 6 36
N = 10 S D = 0 S D2 = 96
Calculation of R :
Example Calculate ‘ R ’ of 6 students from the following data.
Marks
in Stats
:
40 42 45 35 36 39
Marks
in
English
:
46 43 44 39 40 43
Solution:

Marks
in
Stats
R1
Marks
in
English
R2 R1 - R2
(R1 -R2)2
=D2
40 3 46 1 2 4
42 2 43 3.5 -1.5 2.25
45 1 44 2 -1 1
35 6 39 6 0 0
36 5 40 5 0 0
39 4 43 3.5 0.5 0.25
N = 6 S D = 0 S D2 = 7.50
Here m = 2 since in series of marks in English of items of values 43 repeated twice.
PROBABILITIES
7.1 Introduction
The theory of probability was developed towards the end of the 18th century and its
history suggests that it developed with the study of games and chance, such as rolling a
dice, drawing a card, flipping a coin etc. Apart from these, uncertainty prevailed in every
sphere of life. For instance, one often predicts: "It will probably rain tonight." "It is quite
likely that there will be a good yield of cereals this year" and so on. This indicates that, in
layman’s terminology the word ‘probability’ thus connotes that there is an uncertainty
about the happening of events. To put ‘probability’ on a better footing we define it. But
before doing so, we have to explain a few terms."
Trial

A procedure or an experiment to collect any statistical data such as rolling a dice or
flipping a coin is called a trial.
Random Trial or Random Experiment
When the outcome of any experiment can not be predicted precisely then the experiment
is called a random trial or random experiment. In other words, if a random experiment is
repeated under identical conditions, the outcome will vary at random as it is impossible to
predict about the performance of the experiment. For example, if we toss a honest coin or
roll an unbiased dice, we may not get the same results as our expectations.
Sample space
The totality of all the outcomes or results of a random experiment is denoted by Greek
of this sample space is known as a sample print.
Event
Any subset of a sample space is called an event. A sample space S serves as the universal
set for all questions related to an experiment 'S' and an event A w.r.t it is a set of all
possible outcomes favorable to the even t A
For example,
A random experiment :- flipping a coin twice
Sample space :- TT)}
The question : "both the flipps show same face"
Therefore, the event A : { (HH), (TT) }
Equally Likely Events
All possible results of a random experiment are called equally likely outcomes and we
have no reason to expect any one rather than the other. For example, as the result of
drawing a card from a well shuffled pack, any card may appear in draw, so that the 52
cards become 52 different events which are equally likely.
Mutually Exclusive Events
Events are called mutually exclusive or disjoint or incompatible if the occurrence of one
of them precludes the occurrence of all the others. For example in tossing a coin, there
are two mutually exclusive events viz turning up a head and turning up of a tail. Since
both these events cannot happen simultaneously. But note that events are compatible if it

is possible for them to happen simultaneously. For instance in rolling of two dice, the
cases of the face marked 5 appearing on one dice and face 5 appearing on the other, are
compatible.
Exhaustive Events
Events are exhaustive when they include all the possibilities associated with the same
trial. In throwing a coin, the turning up of head and of a tail are exhaustive events
assuming of course that the coin cannot rest on its edge.
Independent Events
Two events are said to be independent if the occurrence of any event does not affect the
occurrence of the other event. For example in tossing of a coin, the events corresponding
to the two successive tosses of it are independent. The flip of one penny does not affect in
any way the flip of a nickel.
Dependent Events
If the occurrence or non-occurrence of any event affects the happening of the other, then
the events are said to be dependent events. For example, in drawing a card from a pack of
cards, let the event A be the occurrence of a king in the 1st draw and B be the occurrence
of a king in the 1st draw and B be the occurrence of a king in the second draw. If the card
drawn at the first trial is not replaced then events A and B are independent events.
Note
(1) If an event contains a single simple point i.e. it is a singleton set, then this
event is called an elementary or a simple event.
(2) An event corresponding to the empty set is an "impossible event."
(3) An event corresponding to the entire sample space is called a ‘certain event’.
Complementary Events
Let S be the sample space for an experiment and A be an event in S. Then A is a subset of
S. Hence , the complement of A in S is also an event in S which contains the outcomes
which are not favorable to the occurrence of A i.e. if A occurs, then the outcome of the
experiment belongs to A, but if A does not occur, then the outcomes of the experiment
belongs to
It is obvious that A and are mutually exclusive. A = S.
If S contains n equally likely, mutually exclusive and exhaustive points and A contains m
out of these n points then contains (n - m) sample points.

Definitions of Probability
We shall now consider two definitions of probability :
(1) Mathematical or a priori or classical.
(2) Statistical or empirical.
1. Mathematical (or A Priori or Classic) Definition
If there are ‘n’ exhaustive, mutually exclusive and equally likely cases and m of them are
favorable to an event A, the probability of A happening is defined as the ratio m/n
Expressed as a formula :-
This definition is due to ‘Laplace.’ Thus probability is a concept which measures
numerically the degree of certainty or uncertainty of the occurrence of an event.
For example, the probability of randomly drawing taking from a well-shuffled deck of
cards is 4/52. Since 4 is the number of favorable outcomes (i.e. 4 kings of diamond,
spade, club and heart) and 52 is the number of total outcomes (the number of cards in a
deck).
If A is any event of sample space having probability P, then clearly, P is a positive
number (expressed as a fraction or usually as a decimal) not greater than unity. 0
number of cases not favorable to A are (n - m), the probability q that event A will not
happen is, q = or q = 1 - m/n or q = 1 - p.
Now note that the probability q is nothing but the probability of the complementary event
A i.e.
Thus p ( ) = 1 - p or p ( ) = 1 - p ( )
so that p (A) + p ( ) = 1 i.e. p + q = 1
The Laws of Probability

So far we have discussed probabilities of single events. In many situations we come
across two or more events occurring together. If event A and event B are two events
and either A or B or both occurs, is denoted by A B or (A + B) and the event that
both A and B occurs is denoted by AB. We term these situations as
compound event or the joint occurrence of events. We may need probability that A
or B will happen.
It is denoted by B) or P (A + B). Also we may need the probability that A and
B (both) will happen simultaneously. It is denoted by B) or P (AB).
Consider a situation, you are asked to choose any 3 or any diamond or both from a well
shuffled pack of 52 cards. Now you are interested in the probability of this situation.
Now see the following diagram.
(both) will happen simultaneousl
Consider a situation, you are asked to choose any 3 or any diamond or both from a well
shuffled pack of 52 cards. Now you are interested in the probability of this situation.
Now see the following diagram.
Click here enlarge
Now count the dots in the area which fulfills the condition any 3 or any
diamond or both. They are 16.

Thus the required probability
In the language of set theory, the set any 3 or any diamond or both is the union of the sets
‘any 3 which contains 4 cards ’ and ‘any diamond’ which contains 15 cards. The number
of cards in their union is equal to the sum of these numbers minus the number of cards in
the space where they overlap. Any points in this space, called the intersection of the two
sets, is counted here twice (double counting), once in each set. Dividing by 52 we get the
required probability.
Thus P (any 3 or any diamond or both)
In general, if the letters A and B stands for any two events, then
Clearly, the outcomes of both A and B are non-mutually exclusive.
Multiplication Law of Probability
If there are two independent events; the respective probability of which are known, then
the probability that both will happen is the product of the probabilities of their happening
To compute the probability of two or even more independent event all occurring (joint
occurrence) extent the above law to required number.
For example, first flip a penny, then the nickle and finally flip the dime.
On landing, probability of heads is for a penny
probability of heads is for a nickle

probability of heads is for a dime
Thus the probability of landing three heads will be or 0.125. (Note that all
three events are independent)
Conditional Probability
In many situations you get more information than simply the total outcomes and
favorable outcomes you already have and, hence you are in position to make yourself
more informed to make judgements regarding the probabilities of such situations. For
example, suppose a card is drawn at random from a deck of 52 cards. Let B denotes the
event ‘the card is a diamond’ and A denotes the event ‘the card is red’. We may then
consider the following probabilities.
Since there are 26 red cards of which 13 are diamonds, the probability that the card is
diamond is . In other words the probability of event B knowing that A has occurred
is .
The probability of B under the condition that A has occurred is known as condition
probability and it is denoted by P (B/A) . Thus P (B/A) = . It should be observed that
the probability of the event B is increased due to the additional information that the event
A has occurred.

Conditional probability found using the formula P (B/A) =
Justification :- P (A/B) =
Similarly P(A/B) =
In both the cases if A and B are independent events then P (A/B) = P (A) and P(B/A) =
P(B)
Therefore P(A) =
or P(B) =
Propositions
(1) If A and B are independent events then A and B' are also independent where B' is the
complementary event B.
(2) If A and B are independent events then A' and B' are also independent events.
(3) Two independent events cannot be mutually exclusive.
B' )
Binomial Distribution
Bernoulli’s trials : A series of independent trials which can be resulted in one of the two
mutually exclusive possibilities 'successes' or 'failures' such that the probability of the
success (or failures) in each trials is constant, then such repeated independent trials are
called as "Bernoulli’s trials".
A discrete variable which can results in only one of the two outcomes (success or failure)
is called Binomial.
For example, a coin flip, the result of an examination success or failures, the result of a
game - win or loss etc. The Binomial distribution is also known as Bernoulli’s
distribution, which expresses probabilities of events of dichotomous nature in repeated
trials.
When do we get a Binomial distribution ?

The following are the conditions in which probabilities are given by binomial
distribution.
1. A trial is repeated 'n' times where n is finite and all 'n' trials are identical.
2. Each trial (or you can call it an event) results in only two mutually exclusive,
exhaustive but not necessarily equally likely possibilities, success or failure.
3. The probability of a "success" outcome is equal to some percentage which is
4. ls). It is
defined as the ratio of the number of successes to the number of trials.
5. The events (or trials) are independent.
6. - p or 1 -
this is denoted by q. Thus p + q = 1.
Sup
probability of tails, such that p + q = 1 (note that p = q = 1/2 if the coin is fair) Then there
are three possible outcomes which are given below.
The sum of all these probabilities is q2 + 2 pq + p2 = (q + p)2. The terms of (q + p)2 in its
expansion give the probabilities of getting 0, 1, 2 heads.
The result obtained above can be generalized to find the probability of getting 'r' heads in
flipping n coins simultaneously.
The probabilities of getting 0, 1, 2, 3, ....., r, .....n heads in a flip of 'n' coins are the terms
of the expansion (q + p)n. Since the expansion is given by the Binomial Theorem, the
distribution is called Binomial Distribution.
Thus the Binomial formula is,

where n ! = n (n - 1) (n -2) .............3 . 2. 1
Properties of the Binomial distribution : We get below some important properties of
the Binomial distribution without derivations.
1. If x denotes the Binomial variate, expression of x i.e. the mean of the distribution
is given by,
2. The standard deviation of the Binomial distribution is determined by,
3. If in experiment, each of n trials, is repeated N times then expression of r
successes i.e. the expected frequency of r successes in N experiment is given by,
Normal Distribution
The normal distribution developed by Gauss is a continuous distribution of maximum
utility.
Definition : If we know a curve such that the area under the curve from x = a to x = b is
equal to the probability that x will take a value between a and b and that the total area
under the curve is unity, then the curve is called the probability curve.
If the curv
density or simply probability function.
Among all the probability curves, the normal curve is the most important one. The
corresponding function is called the normal probability function and the probability
distribution is called the normal distribution. The normal distribution can be considered
as the limiting form of the Binomial Distribution, however n, the number of trials, is very
large and neither P nor q is very small.

The normal distribution is given by
where y = ordinate, x = abscissa of a point on the curve, u = the mean of x,
x = a constant = 3.1416 and e = a constant = 2.7183.
The Normal Curve : The shape of a normal curve is like a bell. It is symmetrical about
the maximum ordinate If P and Q are two points on the x-axis (see figure), the shaded are
PQRS, bounded by the portion of the curve RS, the ordinates at P and Q and the x-axis is
equal to the probability that the variate x lies between x = a and x = b at P and Q
respectively. We have already seen that the total area under a normal curve is unity. Any
probability distribution, defined this way is known as the normal distribution. The
).

Properties of the normal distribution (Normal curve)
1. The normal curve is bell-shaped and symmetrical about the maximum ordinate at
the mean. This ordinate divides the curve into two equal parts. The part on
area under th
for the normal distribution, the mean, mode and median coincide. i.e. mean =
2. We know that the area under the normal curve is equivalent to the probability of
randomly drawing a value in the given range. The area is the greatest in the
middle, where the "hump" (where mean, mode and median coincide) and then
thin out towards out on the either sides of the curve, i.e. tails, but never becomes
zero. In other words, the curve never intersects x-axis at any finite point. i.e. x-
axis is its Asymptote.
3. Since the curve is symmetrical about mean. The first quartile Q1 and the third
quartile Q3 lie at the same distance on the two sides of the mean
Hence, middle 50% observations lie between
1. Since the normal curve is symmetrical its skewness is zero and kurtosis is 3. The
curve is meso kurtic.
2. The mean deviation is approximately.
3. As discussed earlier, the probability for the variable to lie in any interval ( a, b ) in
the range of variable is given by the are under the normal curve, the two ordinates
x = a and x = b, and the x-axis.
The area under the normal curve is distributed as follows :

-
-
-
These areas are shown in the following figure.
-
-
-
7.
The Standard Normal Variate ( Z-Score ) : The problem of finding the probability
get different normal curves, which multiplies it into too many problems if we are to find
All such problems can be reduced to a single one by reducing all normal distributions to a
single normal distribution called 'Standardized Normal Distribution' or to what is known
as the z-score.
To convert a value to a z-score is to express it in terms of how many standard deviations
reduc
Thus,

standard deviation. obviously
is denoted by N (0,1 ).
The areas under the curve between x = 0 and various ordinate x = a are in a table of
standard normal probabilities. This area is equal to the probability that x will assume a
value between x = 0 and x = a.
SAMPLING TECHNIQUES
INRODUCTION
Regardless of the method used to obtain the primary data (experimentation,
observation, or survey), the researcher has to decide whether the data is to be obtained
from every unit of the population under study, or only a representative portion of he
population will be used. The first approach that is, collecting data about each and
every unit of the population is called census method. The second approach, where
only a few units of population under study are considered for analysis is called
sampling method.
It is difficult to collect information about each of the population units as is done under
complete census method. Owing to the difficult associated with the census method or
complete enumeration survey, we resort to the sampling approach. The sampling is a
common activity in our day-to-day work. For example, if a housewife has to see
whether the pot of rice she is cooking is ready, she picks out a few rice grains and
examines them. On the basis of these few rice grains she takes a decision whether the
whole pot of rice is cooked. In his case, the housewife does sampling. Likewise in
most of our daily activities, we unknowingly take help of sampling techniques for
their effective performance. Thus, sampling is an important and all pervasive
activity.
The census method, is having tow main advantages, viz, in formation can be obtained
for each and every nit of population, and secondly, there is greater accuracy in
research results. The sampling techniques have got their own range of advantages
such as: (i) reduced cost owing to a study of selected units from the population. (ii)
greater speed due to smaller number of units to be studied, (iii) greater accuracy in
results because more trained and experienced experts can be engaged in collecting
data (iv) greater depth of data occurs because more details about the unit under study
can be obtained and (v) preservation of units is possible for reuse in case of
destructive nature of experiments.
Major disadvantages of the census method are many, viz, it is very costly, time
consuming and requires a lot of efforts and energy.

METHODS OF SAMPLING
There are two main categories under which various sampling methods can be put.
These two categories are : (i) probability sampling and (ii) non –probability sampling.
PROBABILITY SAMPLING
A Probability sample is also called random sample. It is chosen in such a way that
each member of the universe has a known chance of being selected. It is the
condition known chance that enables statistical procedures to be used on the data to
estimate sampling errors. The most frequently used probability samples are: simple
random samples, systematic samples, stratified samples, and cluster samples.
PROBABLITY SAMPLE :
1. SIMPLE RANDOM SAMPLE
2. SYSTEMATIC RANDOM SAMPLE
3. STRATIFIED RANDOM SAMPLE
4. CLUSTER SAMPLE
NON PROBABLITY SAMPLE
1. JUDGEMENT SAPMLE
2. CONVENIENCE SAMPLE
3. QUOTA SAMPLE
(I) SIMPLE RANDOM SAMPLING
Under simple random sampling each member of the population has a known and
equal chance of being selected. A selection tool frequently used with this design is
the random numbers table. For details readers may consult random number tables
available in the market and statistics books.
Suppose Hindustan Lever wants to determine the attitudes of their salesmen toward
their existing remuneration policies. Assume that there are 25000 such salesmen in
the organization and a simple random sample of 250 is to be used, the random
sample selection procedure that might be followed would be to assign a number from
0 to 2499 to each salesman. Then a table of random numbers can be consulted using
only four-digit numbers. The researcher is free to use a variety of methods to choose
the desired quantum of numbers from this table.
Lottery method is another random method for selecting the sample members. It is
assigning each salesman a number. Placing all these 2500 numbers (chits) in a

container and then randomly drawing out 250 numbers. A major assumption of this
process is that the numbers (chits) have to be thoroughly mixed-up within the
container so that the sequence of numbers placed in the container may not affect the
probability of their being drawn. After a number is drawn out, it is again placed back
into the container so that the probability of any number being selected remains known
and equal. Now, computer programmes also exist which can be used to generate the
desired quantum of random numbers.
(II) SYSTEMATIC SAMPLING
In this case the sample numbers are chosen in a systematic manner from the entire
population. Each member has a known chance of being selected, but not necessarily
equal one.
We want to select a sample of 250 from a population of 2500 employees, or one out
of every 10 since ratio of sample size to population size is 10 as shown below.
We randomly select a digit between one and ten say seven Thus we would then select
from our list of item numbers 7th 17th 27th etc…; up to 2497th item. This way we
complete a sample of 250 salesmen from a list of 2500. A gap of 10 after every item
included in the sample is called sampling interval.
The systematic procedure is often used in selective names from city directories,
telephone directories or almost any type of list. A systematic sample needs much less
work and can be developed much faster than a simple random sample. In both cases,
it is necessary that an existing list of the units of the population be studied. If such a
list is non-existing or cannot be developed , neither systematic samples nor random
samples can be used.
The advantage of this method is that it is more convenient to adopt than the simple
random sampling. The time and work involved in this method are relatively less. If
the population is sufficiently large, systematic sampling can often be expected to
yield results that are similar to those obtained by any other efficient method.
The disadvantage of this method is that it is a lesser representative design than simple
random sampling. If we are dealing with a population having hidden periodicities.
The major weakness of this selection process is that the system used may create a bias
in the results. The every 10th item selected may come out to be a leader or captain.
Thus a bias may enter and study conducted may lack representative ness of the
population. Another problem along these same lines is that a monotonic trend may
exist in the order of the population list and thus in the sample.
(II) STRATIFIED RANDOM SAMPLING
A stratified random sample is used when the researcher is particularly interested in
certain specific categories within the total population. The population is divided into

strata on the basis of recognizable or measurable characteristics of its members, e.g.
age income education etc. The total sample then is composed of members from each
strata so that the stratified sample is really a combination of a number of smaller
samples.
In a study to determine salesmen’s attitudes towards travel allowances, it is felt that
attitudes on this subject are closely related to the amount of traveling done by each of
these persons. Thus a stratified sample could be used with kilometers traveled per
month as the characteristic determining the makeup of the various strata. Table shown
such a breakdown using proportional allocation from each strata.
The salesmen in each of these four strata would seemingly be more homogenous in
terms of their attitudes towards travel allowance than the 2500 salesmen in total.
Thus it is possible to increase the accuracy of the result by taking the sample from
each stratum rather than using a sample selected from the entire population. The
stratified sample will be a probability sample as long as the individual units are
chosen from each stratum in a random manner.
CLASSIFIED OF SALESMEN ACCORDING TO KILOMETRES
TRAVELLED MOTHLY
Kilometers
traveled per
moth
No. of Salesman % age of total sales
force
No.in sample
Less than 200
km
250 10 25
201 -500 Km 1250 50 125
501- 750 Km 825 33 82
More than 750
KM
175 7 18
Total 2500 100 250
It is important to realize that the use of stratified samples will lead to more
accurate results only if the strata selected are logically related to the data sought. For
instance, in the previous study placing salesmen in strata on the basis of their weight
or colour of eyes would add nothing to the findings. On the other hand, using strata
such as years of services with the firm or geographic area served could be really
meaningful. Stratified sampling can be classified into two categories such as (i)
proportionate and (ii) disproportionate. These two types are discussed as follows:
(i) Proportionate stratified sampling: The breakdown of members per stratum
can be done on either a proportionate or disproportionate basis. A
proportionate stratified sampling is the method where the number of items in
each stratum is proportionate to their number in the population. Since 10 per

cent of the previous universe is composed of salesmen driving less than 200
km. monthly, this group will comprise 10 per cent of the sample. The same
relationship holds true for the other three strata.
(ii) Disproportinate stratified sampling: In certain cases composition of various
strata is such that if a proportionate sample were used, very little data would
be obtained about some of the strata. Let us assume a study is to be conducted
concerning characteristics of car owners, with the type of car owned being the
basis for stratification. While there are a large number of Maruti Ambasador
and Fiat/Premier owners in most cities, there are relatively few Chevrolet,
Toyota and Escort, Honda, Ford owners. Thus, if a proportionate stratified
sample of 500 car owners is to be obtained in a typical city, there might be
only two Chevrolet and four Toyota owners in the sample. Assume that
Chevrolet owners comprise 0.5 per cent of car owners, while Toyota owners
comprising1 percent of the total. With such a small representation, a little
could be learnt about the characteristics of these two types of car owners
In this situation, a disproportionate stratified sample should be used. This
means that in some of the strata the number of units would differ greatly from
their real representation in the universe. A smaller number of Maruti, Ambassador
and Fiat owners would be included in the sample that their number in the universe
warrants. Conversely, a large number of Chevrolet and Toyota owners would be
included.
A disproportionate stratified sample should be used when there appears to
be major variances in the values within certain strata. With a fixed sample size,
those strata exhibiting greatest variability are sampled more heavily than strata
that are fairly homogenous. Thus, using a disproportionate stratified sample
necessitates that the researcher have some previous knowledge about the
population being studied.
Although stratified random sampling will almost always provide more
reliable estimates than simple random samples of the same size, this gain in
accuracy will often be rather small. This means the researcher has to weigh the
additional time and effort involved in the stratification against the additional
accuracy obtained. Statistical procedures exist for determining the amount of this
gain.
(iii) CLUSTER SAMPLING
In this method the various units comprising the population are grouped in clusters and
the sample selection is made in such a way that each cluster has a known chance of
being selected. This is also called area sampling (multi-stage sampling). Experts
interpret a cluster sample as the one where a selected geographical area (a state,
district, a tehsil or a block) is sampled in its entirety.

A cluster sample is useful in two situations: (i) when there is incomplete data on
the composition of the population, and (ii) when it is desirable to save time and costs
by limiting the study to specific geographical areas.
For examples, if a study of consumers is to be made among households in Shimla
city. Because people are constantly moving during different seasons, no up-to-date
list is available on the compositions of the Shimla city households. Yet, if a
researcher wants to carry out a probability study, every household must have known
chance of being selected in a sample. Cluster sampling meets this condition.
The total Shimla city can be divided into clusters on the basis of municipality
divisions or census tracts. Assume a total of 50 terms of residential characteristics.
Since in cluster sampling only a small portion of the total population is included, it is
necessary to select certain clusters from the total group for further study. Each of the
50 clusters is assigned a number from 0 to 49. A decision then has to be made as to
how many clusters will be included in the sample. If the researcher desires that total
of six clusters be involved a table of random numbers can be used to select these six.
If the six numbers selected from the table are 7, 18, 25, 29, 39 and 46, then these six
clusters represented by these numbers would be included in the sample.
If the total sample size desired for our consumer study is 250 households, the
selection of these specific sample members can be done in two ways:
(1) An equal number of households can be selected from each of the six clusters i.e.
about 42 or (2) Each cluster can be represented in the sample on a somewhat
proportionate basis.
Another cluster sampling method is to assign numbers to each block of the six
selected clusters. No attempt is made at this time to count the households in these
blocks. Certain number of blocks are then randomly selected from each area. The
final step is to include in the sample every household of these selected blocks. This
technique minimizes the amount of data needed for sample selection since the only
specific data required are the block breakdowns for the six clusters that comprise the
sample.
The major advantage of cluster sampling is that complete data about the population is
not needed at the outset of the study. By constantly narrowing down the components
of the clusters complete data on any one cluster can be postponed until the last stage.
Another major advantage of cluster sampling is that it saves time and money when
personal interviews are used.
The major limitation of this method is that it leads to a substantial loss in precision
since units within each cluster tend to be rather heterogeneous.

NON-PROBABILITY SAMPLING.
In n on-probability sampling the chance of any particular unit in the population being
selected is unknown. Since randomness is not involved in the selection process, an
estimate of the sampling error cannot be made. But this does not mean that the
findings obtained from non-probability sampling are of questionable value. If
properly conducted their findings can be as accurace as those obtained from
probability sampling. The three more frequently used non-probability designs are
judgement convenience, and quota sampling.
(i) JUDGEMENT SAMPLING
A person knowledgeable about the population under study chooses sample members
he feels would be the most appropriate for the particular study. Thus a sample is
selected on the basis of his judgement.
(ii) CONVENIENCE SAMPLING
In this method, the sample nits are chosen primarily on the basis of the convenience
to the investigator. If 150 persons are to be selected from Ludhiana city, the
investigator goes to the famous localities like Chaura Bazar, Field Gunj, Industrial
Area and picks up 50 persons from each of these representative localities. The units
selected may be each person who comes across the investigator every 10 minutes.
(iii) QUOTA SAMPLING
In quota sampling the method is similar to the one adopted in stratified sampling.
Here also the population is divided into strata on the basis of characteristics of
population. The sample units are chosen so that each strata is represented in
proportion to its importance in the population.
In 100 heads of households in Mumbai are to be interviewed for their attitudes
towards a proposed city tax, the researcher may want to structure the sample on the
basis of household income. Suppose that 30 per cent of the Mumbai households
have monthly income less than Rs. 10,000 60 per cent have income between Rs.
10001 and Rs. 50,000 and the remaining 10 per cent have the come more than Rs.
50000. The sample of 100 households would then be comprised of 30 units from the
less than Rs. 10,000 in come category 60 units from the Rs. 10000 to Rs. 50000
category and 10 units from households with incomes exceeding Rs. 50,000.
In the quota sampling the units are selected on non-random manner while in the
stratified sampling they are chosen on the random basis. The interviewer might go
haphazardly to any residential area and interview people until desires number is
interviewed. Thus in the selection process, each member of the universe does not
have a known chance of being chosen.

PROBLEMS IN SAMPLING
Sampling encompasses data about only a portion of the universe. When a precise data
on each unit of the population is needed, sampling becomes dysfunctional. For
example, a static electricity board cannot take reading of a sampled number of
households’ meters for computing electricity bills to the whole of population of its
customers. Similarly, a bank cannot check accounts of customers on the basis of
average withdrawals on a particular day.
Even in those situations where sampling techniques are applicable, certain problems
exist. These problems pertain to the fact that how well the sample represents the
population from which it is drawn. There remain some discrepancies in accuracy and
reliability of representation of population in the sample.
These discrepancies are of two types:
(1) Sampling errors, and (2) data collection errors. The influence of these two types of
errors can be better shown by the formula:
S = P ± es edc
Where S = sample value
es = sampling error
edc = data collection error
P = true but unknown characteristic of population.
Thus the difference between the actual value of the character for the total
population and the value estimated from the sample, measures the total error which
again split into ‘es’ and ‘edc’.
1. SAMPLING ERROR
Hardly, a sample is an exact miniature (representation) of the total population. The
difference between the unknown values of population (parameters) and the values
obtained from the sample (statistics) are sampling errors.
Let us take an example of a college students’ study regarding how much of their total
expenses are personally earned by the students. Assume that in a city we take a
sample of 400 students. We find after interviewing 400 sample students that 30 per
cent of expenses are earned themselves by the students. Now, let us interview each
and every unit of the population. In the case, we reach a conclusion that 40 per cent of
the total expenses are earned themselves by the college students. We see that there
remains a discrepancy between the results from the sample and from the total
population (difference = 40-30 = 10%). This difference is not due to inappropriate
sampling but due to the fact that the sample of 400 is not an exact miniature
(representative) of the total population of the college students.
The size of these errors can be estimated and corrected in the results, if the
probability sampling is used. There exists no method for estimation and correction of
sampling errors in non-probability samples.

2. DATA COLLECTION ERRORS
In the data collection, there are certain errors which occur due to some unavoidable
reasons. These errors might be possible for distorting sample values from the
population values. These errors cannot be estimated and do not average out to the
actual population values. These errors are of for types as discussed in the ensuing
text.
(i) Non-response Errors
In all studies there exist some respondents who refuse to respond or are
difficult to approach. These respondents who do not participate in a survey
may be distinct or unique and may affect the result of the study
(ii) Selection Errors
Sometimes, the procedures adopted for the selection of units of population
are improper. There may be wrongly selected sampling frame or entire list
from which sample units are drawn may be wrong. Such selection leads to
questionable representativeness.
(iii) Measurement Errors
These errors intrude into the study owing to the style questions are asked
by the investigator or interpreted by the respondent. These errors may also
be due to the wrong recording of data by the interviewer. Such errors may
also affect results due to wrong editing, coding or interpretation of data.
(iv) Prediction Errors
Certain errors enter in the study due to the estimated or substitute data
used to predict certain future activities. The researcher is compelled to
accept such data because actual data may not be available. All these errors
can be rectified by proper training to the investigators.

Statistics / Quantitative Techniques Study Material

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