Unit – 1: Introduction to Statistics
Q.1. Define the term Statistics. What
are its Characteristics? Mention its Functions and Limitations.
Ans: Statistics: The word Statistics seems to have been derived
from the Latin word “status” or the Italian word Statista. All word means a political
state. In early year “statistics” equipped a collection of facts about the
people in the state for administration or political purpose.
A comprehensive definition was given by Prof.
Horace Secrist, which is a follows: “By
Statistics we mean aggregates of facts affected to a marked extent by
multiplicity of causes, numerically expressed, enumerated or estimated
according to a reasonable standards of accuracy, collected in a systematic
manner for a predetermined purpose and placed in relation to each other.”
Characteristics
of Statistics:
(i) Statistics are aggregates of facts. Single
and isolated figures are not statistics because they cannot be compared.
(ii) Statistics must be numerically expressed.
Statistical methods are applicable only to those data which can be numerically
expressed.
(iii) Statistics should be capable of
comparison and connected to each other.
(iv) Statistics should be collected in a
systematic manner.
(v) Statistics should be collected for a
definite purpose: The purpose should be specific and well defined.
Functions
and Limitations of Statistics:
The
functions of statistics are as follows:
(i) It presents fact in a definite form. Numerical
expressions of data are convincing.
(ii) It simplifies mass of figures. The data
presented in the form of table, graph or diagram, average or coefficients are
simple to understand.
(iii) It facilitates comparison. Once the data
are simplified they can be compared with other similar data.
(iv) It helps in prediction. Plans and
policies of organisations are invariably formulated in advance at the time of
their implementation.
(v) It helps in the formulation of suitable
policies. Statistics provide the basic material for framing suitable policies.
Limitations
of statistics are as follows:
(i) Statistics deals only with quantitative
characteristics. Data Which cannot be expressed in numbers are incapable of
statistical analysis. Qualitative characteristics like honesty, efficiency,
intelligence etc. cannot be studied directly.
(ii) Statistics deals with aggregates not with
individuals.
(iii) Statistical laws are not perfectly
accurate.
(iv) Statistical results are only an average.
Statistical results reveal only the average behavior.
(v) Statistics is only one of the methods of
studying a problem. Statistical tools do not provide the best solution under
all circumstances.
(vi) Statistics can be misused. The data
placed to an inexperienced person may reveal wrong results. Only persons having
fundamental knowledge of statistical methods can handle the data properly.
Q.2. What are various types of
Statistical Data? Mention their merits and demerits.
Ans: Statistical data are of two types
(a) Primary data
(b) Secondary data.
Primary
Data: Data which are collected for the first time for a specific
purpose are known as Primary data. For example: Population census, National
income collected by government, Textile Bulletin (Monthly), Reserve bank of
India Bulletin (Monthly) etc.
Secondary
Data: Data which are collected by someone else, used in investigation
are knows as Secondary data. Data are primary to the collector, but secondary
to the user. For example: Statistical abstract of the Indian Union, Monthly
abstract of statistics, Monthly statistical digest, International Labour
Bulletin (Monthly).
Merits and
Demerits of Primary Data:
Merits:
(a) They are reliable and accurate.
(b) If during collection, the Data are wrong
they can be checked again by cross examination.
(c) It
is more suitable if the field of enquiry is small.
Demerits:
(a) It the field of enquiry is too wide, it is
not suitable.
(b) Collection of primary data is costly and
time consuming.
(c) Personal Bias, prejudice and whims may
affect the data.
Merits and
Demerits of Secondary Data:
Merits:
(a) While using secondary data, time and
labour are saved.
(b) It may also be collected from unpublished
form.
(c) If secondary Data are available, they are
much quicker to obtain than primary data.
Demerits:
(a) Degree of accuracy may not be acceptable.
(b) Secondary Data may or may not fit the need
of the project.
(c) Data may be influenced by personal bias of
investigator.
Q.3. Distinguish between Primary data
and Secondary data.
Ans: Difference between Primary Data and Secondary Data:
(a) Primary data are those which are collected
for the first time and thus original in character. While Secondary data are
those which are already collected by someone else.
(b) Primary data are in the form of raw-material,
whereas Secondary data are in the form of finished products.
(c) Primary data are collected directly from
the people related to enquiry while Secondary data are collected from published
materials.
(d) Data are primary in the hands of
institutions collecting it while they are secondary for all others.
Q.4. What are various sources of
Secondary Data? Mention the points which should be considered before using
secondary data.
Ans: Sources of Secondary Data:
(a) Official publication by the central and state
governments, district Boards. (b) Publication by research institutions,
Universities etc. (c) Economic Journals. (d) Commercial Journals. (e) Reports
of Committees, commissions. (f) Publications of trade associations, Chamber of
Commerce etc.
Precautions
in the use of Secondary Data:
The following aspects should be considered
before use of secondary data:
(i) Suitability: The investigator must check
before using secondary data that whether they are suitable for the present
purpose or not.
(ii) Adequacy: The investigator has to
determine whether they are adequate for the present purpose of investigators.
(iii) Dependability: Dependability of
secondary data is determined by the following factors:-
(a) The authority which collected the data.
(b) Procedure of Sampling followed.
(c) Status of Investigator.
(iv) Units in which data are available.
Q.5. What are various essential
qualities of Secondary data? Explain some effective methods of collecting
primary data.
Ans: Qualities of Secondary Data:
(a) Data should be reliable
(b) Data should be suitable for the purpose of
investigator.
(c) Data should be adequate
(d) Data should be collected by trained
investigator.
Methods of
collecting primary Data
(a)
Direct Personal Observation: Under this method, the investigator
collects the data personally from the persons concerned. The information
obtained under this method is original in nature. This method is suitable when
the field of enquiry is small.
(b) Indirect Oral Investigation: - Under this
method, the investigator collects the data from third parties capable of
supplying the necessary information. This method is suitable where the
information to be obtained is of a complex nature and informants cannot be
approached directly.
(c) Schedule and questionnaire: - A list of
question regarding the enquiry is prepared and printed. Data are collected in
any of the following ways:-
(i) By sending the questionnaire to the
persons concerned with a request to answer the question and return the
questionnaire.
(ii) By sending the questionnaire through
enumerators for helping the informants.
(d) Local reports: - This method gives only
approximate results at a low cost.
Q.6. What are various stages involved
in statistical investigation? Explain them briefly.
Ans: Various stages in statistical investigation: There are five
stages in a statistical investigation which are given below:
(i) Collection of Data: Utmost care must be
exercised in collecting data as they are the foundation of statistical
analysis. If the data are faulty, the conclusions drawn can never be reliable.
(ii) Organisation of Data: Data collected from
published sources are generally in organised form but data collected from a
survey frequently needs organisation. Organising of data involves three steps
which are:
(a) Editing of data
(b) Classification of data according to some
common characteristics
(c) Tabulation.
(iii) Presentation of Data: Organised data can
be further presented in the form of Diagrams and Graphs.
(iv) Analysis: After collection, organisation
and presentation, data are analysed by adopting various statistical methods
such as measure of central tendency, measure of variation, correlation,
regression etc. to dig out information useful for decision-making.
(v) Interpretation: The last stage is
interpretation which is a difficult task and requires a high degree of skill,
care and experience. If the data have been analysed and not properly
interpreted, the whole object of investigation may be defeated and wrong
conclusion be drawn.
Q.7. What is Questionnaire? What are
its essential characteristics?
Ans: Questionnaire: A Questionnaire is simply a list of questions
in a printed sheet relating to survey which the investigators asks to the
informants and the answers of the informants are noted down against the
respective questions on the sheet. Choice of questions is a very important
parts of the enquiry whatever its nature.
Characteristics of an ideal Questionnaire:
(i) The Schedule of question must not be
lengthy.
(ii) It should be clear and simple.
(iii) Questions should be arranged in a
logical sequence.
(iv) Each question should be brief and must
aim to some particular information necessary for the investigation.
(v) Questions of personal matter like income
of property should be avoided.
(vi) The Units of information should be Cleary
shown in the sheet.
Q.8. What is Tabulation? Mention its
Objectives. What are its importance and limitations?
Ans: Tabulation: Tabulation refers to the systematic arrangement
of the information in rows and columns. Rows are the horizontal arrangement. In
simple words, tabulation is a layout of figures in rectangular form with
appropriate headings to explain different rows and columns. The main purpose of
the table is to simplify the presentation and to facilitate comparisons.
According to Neiswanger, "A statistical table is a systematic
organisation of data in columns and rows."
The
principal objectives of tabulation are stated below:
(i) To make complex data simple: When data are
arranged systematically in a table, such data become more meaningful and can be
easily understood.
(ii) To facilitate comparison: When different
data sets are presented in tables it becomes possible to compare them.
(iii) To economize space: A statistical table
furnishes maximum information relating to the study in minimum space.
(iv) To make data fit for analysis and
interpretation.
(v) To provide reference: A statistical table
can be used as a source of reference for other studies of similar nature.
Importance
of Tabulation:
a) Tabulation makes the data brief. Therefore,
it can be easily presented in the form of graphs.
b) Tabulation presents the numerical figures
in an attractive form.
c) Tabulation makes complex data simple and as
a result of this, it becomes easy to understand the data.
d) This form of the presentation of data is
helpful in finding mistakes.
e) Tabulation is useful in condensing the
collected data.
f) Tabulation makes it easy to analyze the
data from tables.
g) Tabulation is a very cheap mode to present
the data. It saves time as well as space.
h) Tabulation is a device to summaries the
large scattered data. So, the maximum information may be collected from these
tables.
Limitations
of Tabulation
a) Tables contain only numerical data. They do
not contain details.
b) Qualitative expression is not possible
through tables.
c) Tables can be used by experts only to draw
conclusions. Common men do not understand them properly.
Q.9. What do you mean by
Classification of Data? Mention its essential features.
Ans: Classification of Data: The
process of arranging the data in groups or classes according to their common
characteristics is technically known as classification. Classification is the
grouping of related facts into classes. It is the first step in tabulation.
In the words of Secrist, "Classification
is the process of arranging data into sequences and groups according to their
common characteristics or separating them into different but related
parts."
Essentials of classification
a) The classification must be exhaustive so
that every unit of the distribution may find place in one group or another.
b) Classification must conform to the objects
of investigation.
c) All the items constituting a group must be
homogeneous.
d) Classification should be elastic so that
new facts and figures may easily be adjusted.
e) Classification should be stable. If it is
not so and is changed for every enquiry then the data would not fit for an
enquiry.
f) The data must not overlap. Each item of the
data must be found in one class.
Q.10. Define the term population and
sample. What is sample and census survey? Distinguish between them.
Ans: Population and Sample
Population: Statistics is taken in relation to
a large data. Single and unconnected data is not statistics. In the field of a
statistical enquiry there may be persons, items or any other similar units. The
aggregate of all such units under consideration is called “Universe or
Population”.
Sample: If a part is selected out of the
universe then the selected part or portion is known as sample. Sample is only a
part of the universe.
Sample
survey and Census Survey:
Sample survey: It is a survey under which only
a part taken out of the universe is investigated. It is not essential to
investigate every individual item of the Universe.
Census survey and complete enumeration: Under
Census survey detail information regarding every individual person or item of a
given universe is collected.
Difference
between Census and Sample survey:
The following are the differences between
Census and Sample method of investigation:
(a) Under Census method, each and every
individual item is investigated whereas under sample survey only a part of
universe is investigated.
(b) There is no chance of sampling error in
census survey whereas sampling error cannot be avoided under sample survey.
(c) Large number of enumerators is required in
census whereas less number of enumerators is required in sample survey.
(d) Census survey is more time consuming and
costly as compared to sample survey.
(e) Census survey is an old method and it less
systematic than the sample survey.
Q.11. Mention the Merits and Demerits
of Census and Sample Survey.
Ans: Merits and Demerits of Census:
Merits:
(a) Since all the individuals of the universe
are investigated, highest degree of accuracy is obtained.
(b) Since there is no possibility of personal
bias affecting investigation, this method is free from sampling error.
(c) It
is more suitable if the field of enquiry is small.
(d) Since all the items of the universe are
taken into consideration, all the characteristics of the universe
Demerits:
(a) It the field of enquiry is too wide, it is
not suitable.
(b) Collection of primary data is costly and
time consuming.
(c) Personal Bias, prejudice and whims may
affect the data.
Merits and
Demerits of sample survey:
Merits:
(a) While using secondary data, time and
labour are saved.
(b) It may also be collected from unpublished
form.
(c) If secondary Data are available, they are
much quicker to obtain than primary data.
Demerits:
(a) Degree of accuracy may not be acceptable.
(b) Secondary Data may or may not fit the need
of the project.
(c) Data may be influenced by personal bias of
investigator.
Q.12.
What are various types of diagrams and graphs? Distinguish between diagrams and
graphs.
Ans: Types of diagrams and Graphs:
One of the
most effective and interesting alternative way in which a statistical data may
be presented is through diagrams and graphs. There are several ways in which
statistical data may be displayed pictorially such as different types of graphs
and diagrams. The commonly used diagrams and graphs to be discussed in
subsequent paragraphs are given as under:
Types of Diagrams/Charts:
a) Simple Bar Chart
b) Multiple Bar Chart or Cluster Chart
c) Staked Bar Chart or Sub-Divided Bar Chart or
Component Bar Chart
d) Simple Component Bar Chart
e) Percentage Component Bar Chart
f) Sub-Divided Rectangular Bar Chart
g) Pie Chart
Types of Diagrams/Charts:
a) Histogram
b) Frequency Curve and Polygon
c) Lorenz Curve
d) Histogram
Difference Between Diagrams And Graphs
There is no clear-cut line of demarcation
between a diagram and a graph yet:
a) A graph needs a graph paper but a diagram can
be drawn on a plain paper.
b) As diagrams are
attractive to look at, they are used for. Graphs on the other hand are more
useful to statisticians and research workers
for the purpose of further analysis.
c) For representing frequency distribution,
diagrams are rarely used when compared with graphs. For example, for the time series graphs are more
appropriate than diagrams.
Q.13. Compare between Tabular and Diagrammatical Presentation of
Data? Mention the uses and Limitations of diagrams and graphs.
Ans: Comparison Between Tabular And Diagrammatic Presentation
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Serial
No:
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Diagrams and Graphs
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Tabulation
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1.
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Diagrams and Graphs are meant for a lay man.
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Tables are meant for statisticians for the purpose of
further analysis.
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2.
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Diagrams give only an approximate idea.
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Tables contain precise figures. Exact values can be read
from tables.
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|
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3.
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Diagrams can be more easily compared, and can be interpreted by
a layman.
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Comparison and interpretations of tables can only be done
by statisticians and it is a difficult task.
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|
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4.
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Diagrams and graphs cannot present much information.
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Tables can present more information.
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5.
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Diagrams are more attractive and have a visual appeal.
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Tables are dry for a layman (may be attractive to a
statistician.)
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Uses of Diagrams and Graphs:
Diagrams and graphs are extremely useful due to the following
reasons:
(i) Information presented though
diagrams and graphs can be understood easily just in a bird’s eye view.
(ii) These are appealing and
fascinating to the eyes; Scholars take greater interest in presenting data
through these devices.
(iii) Diagrams and graphs produce a
greater lasting impression on the mind of the readers than the figures
presented in a table.
(iv) They facilitate ready comparison
of data over time and space. Graphs study economic relationship between two
variables.
However, graphic and diagrammatic
presentations have some limitations:
(i) Unlike a
table a diagram or a graph does not show the exact value of a variable.
(ii) Further, a
limited set of facts can be presented through such devices like diagram and
graph.
Q.14. What
are the points that should be kept in mind while constructing a diagram or
graph?
Ans: General Rules for Drawing Graphs
and Diagrams
Following points must be kept in mind while
constructing a diagram or graph. Every diagram or graph must have a serial
number. It is necessary to distinguish one from the other.
1. Serial number: Every
diagram or graph must have a serial number. It is necessary to distinguish one
from the other.
2. Title: Title must
be given to every diagram or graph. From the title one can know the idea
contained in it. The title should be brief and self-explanatory. It is usually
placed at the top.
3. Proper size and scale: A diagram
or graph should be of normal size and drawn with proper scale. The scale in a
graph specifies the size of the unit.
4. Cleanliness: Diagrams
must be as simple as possible. Further they must be quite neat and clean. They
should also be descent to look at.
5. Index: Every
diagram or graph must be accompanied by an index. This illustrates different
types of lines, shades or colors used in the diagram.
6. Footnote: Foot notes
may be given at the bottom of a diagram if necessary. It clarifies certain
points in the diagram.
Unit - 1: Measure of Central Tendency and
Dispersion (Part – B)
Q.N.1.What is average? Why is it called a measure of central
tendency?
Ans: - In the words of Croxton and Cowden, “An average value is a
single value within the range of the data that is used to represent all the
values in the series.”
The value of average lies between the maximum and minimum values
of the series. That is why it is also called measure of central tendency.
Q.N.2.What is the desirable properties of a good average.
Ans: - The following are the important properties which a good
average should satisfy
1.
It should be easy to understand.
2.
It should be simple to compute.
3.
It should be based on all the items.
4.
It should not be affected by extreme values.
5.
It should be rigidly defined.
6.
It should be capable of further algebraic treatment.
Q.N.3. Define Arithmetic Mean (A.M). What are its properties?
Explain its merits and demerits.
Ans: - Arithmetic Mean: - It is a value obtained by adding
together all the items and by dividing the total by the number of items. It is
also called average. It is the most popular and widely used measure for
representing the entire data by one value.
Arithmetic
mean may be either:
(i)
Simple arithmetic mean, or
(ii)
Weighted arithmetic mean.
Properties of
arithmetic mean:
1.
The sum of deviations of the items from the arithmetic mean is
always zero i.e. ∑(X–X) =0.
2. The Sum of
the squared deviations of the items from A.M. is minimum, which is less than
the sum of the squared deviations of the items from any other values.
3. If each
item in the series is replaced by the mean, then the sum of these substitutions
will be equal to the sum of the individual items.
Merits of A.M.:-
(i)
It is simple to understand and easy to calculate.
(ii)
It is affected by the value of every item in the series.
(iii)
It is rigidly defined.
(iv)
It is capable of further algebraic treatment.
(v)
It is calculated value and not based on the position in the
series.
Demerits of
A.M.:-
(i)
It is affected by extreme items i.e., very small and very large
items.
(ii)
It can hardly be located by inspection.
(iii)
In some cases A.M. does not represent the actual item. For
example, average patients admitted in a hospital is 10.7 per day.
(iv)
A.M. is not suitable in extremely asymmetrical distributions.
Q.N.4. Define Geometric Mean (G.M). Mention its merits and
demerits. What are its Uses?
Ans: - G.M:- It is defined as nth root of the
product of n items or values. i.e., G.M. = n√ (x1. x2. x3 ……xn)
Merits of
G.M.:-
(i)
It is not affected by the extreme items in the series.
(ii)
It is rigidly defined and its value is a precise figure.
(iii)
It is capable of further algebraic treatment.
(iv)
It is Useful in calculating index number.
Demerits of
G.M.:-
(i)
It is difficult to understand and to compute.
(ii)
It cannot be computed when one of the values
is 0 or negative.
Uses of G.M.:-
(i)
It is used to find average of the rates of
changes.
(ii)
It is Useful in measuring growth of
population.
(iii)
It is considered to be the best average for
the construction of index numbers.
Q.N.5. Define Harmonic Mean (H.M). Mention its merits and
demerits. What are its Uses?
Ans: - H.M.:- It is defined as the reciprocal of
the arithmetic mean of the reciprocal of the individual observations.
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|
|
N
|
|
H.M.
|
=
|
(1/x1 + 1/x2 + 1/x3 + ........ +1/xn)
|
Merits of
H.M.:-
(i)
Like AM and GM, it is also based on all observations.
(ii)
It is most appropriate average under conditions of wide variations
among the items of a series since it gives larger weight to smaller items.
(iii) It is capable
of further algebraic treatment.
(iv) It is
extremely useful while averaging certain types of rates and ratios.
Demerits of
H.M.:-
(i)
It is difficult to understand and to compute.
(ii)
It cannot be computed when one of the values
is 0 or negative.
(iii)
It is necessary to know all the items of a
series before it can be calculated.
(iv) It is
usually a value which may not be a member of the given set of numbers.
Uses of H.M.: If there are two
measurements taken together to measure a variable, HM can be used. For example,
tonne mileage, speed per hour. In the above example tonne mileage, tonne is one
measurement and mileage is another measurement. HM is used to calculate average
speed.
Q.N.6. Define Median. Mention its merits and demerits.
Ans: - Median : Median may be defined as the size (actual or
estimated) to that item which falls in the middle of a series arranged either
in the ascending order or the descending order of their magnitude. It lies in
the centre of a series and divides the series into two equal parts. Median is also
known as an average of position.
Merits of
Median:-
(i)
It is simple to understand and easy to calculate, particularly is
individual and discrete series.
(ii)
It is not affected by the extreme items in the series.
(iii)
It can be determined graphically.
(iv)
For open-ended classes, median can be
calculated.
(v)
It can be located by inspection, after arranging the data in order
of magnitude.
Demerits of
Median:-
(i)
It does not consider all variables because it
is a positional average.
(ii)
The value of median is affected more by
sampling fluctuations
(iii)
It is not capable of further algebraic
treatment.
Like mean, combined median cannot be calculated.
(iv) It cannot be
computed precisely when it lies between two items.
Q.N.7. Define Mode. Mention its merits and demerits.
Ans: - Mode: Mode is that value a dataset, which is repeated most often in the
database. In other words, mode is the value, which is predominant in the series
or is at the position of greatest density. Mode may or may not exist in a
series, or if it exists, it may not be unique, or its position may be somewhat
uncertain.
Merits of
Mode:-
(i)
Mode is the most representative value of
distribution, it is useful to calculate model wage.
(ii)
It is not affected by the extreme items in the series.
(iii)
It can be determined graphically.
(iv)
For open-ended classes, Mode can be calculated.
(v)
It can be located by inspection.
Demerits of
Mode:-
(i)
It is not based on all observations.
(ii)
Mode cannot be calculated when frequency
distribution is ill-defined
(iii)
It is not capable of further algebraic
treatment.
Like mean, combined mode cannot be calculated.
(iv) It is not
rigidly defined measure because several formulae to calculate mode is used.
Q.N.8. What is the relationship between mean, median and mode? Give the
formula.
Ans: - In a normal distribution Mean = Median = Mode.
In an asymmetrical distribution median is always in the middle but mean and
mode will interchange their positions or values. Mode = 3 Median - 2 Mean.
Q.N.9. What is the relation between arithmetic mean, geometric
mean and harmonic mean?
Ans: - A.M. > G.M. > H.M.
GM = √ (ARITHMETIC MEAN * HARMONIC MEAN)
Q.N.10. What is Dispersion? What purpose does a measure of
dispersion serve? What are its various types? Distinguish between absolute and
relative measures of dispersion.
Ans: - Dispersion: Dispersion is the measure of variation
of items. It measures the extent to which the items vary from central value.
Dispersion is also known as average of the second order. In the words of Brooks and Dick,” Dispersion
is the degree of the scatter or variation of the variable about a central
value.”
In the words Simpson and Kafka,” The
measurement of the Scatterness of the mass of figures in a series about an
average is called measure of variation or dispersion.”
Dispersion includes range, mean
deviation, quartile deviation, and standard deviation. Mean, Median and Mode are the average of 1st order.
Purpose of Measure of Dispersion:
Measures of Dispersion are needed for the following purposes:
(i)
To determine the reliability of an average.
(ii)
To serve as a basis for the control of the
variability.
(iii)
To compare two or more series with regard to
their variability.
(iv) To
facilitate the use of other statistical measures.
Measure
of dispersion may be broadly classified into two types:-
a. Absolute
measures of dispersion: It is classified into
i.
Range
ii.
Mean Deviation
iii.
Standard Deviation
iv.
Quartile Deviation
b.Relative measures of dispersion: It is
classified into
i.
Coefficient of Range
ii.
Coefficient of Mean Deviation
iii.
Coefficient of Variation
iv.
Coefficient of Quartile Deviation.
Difference
between absolute and relative measure of dispersion:
1.
Absolute measures are dependent on the unit of
the variable under consideration whereas the relative measures of dispersion
are unit free.
2.
For comparing two or more distributions,
relative measures and not absolute measures of dispersion are considered.
3.
As compared to absolute measures of
dispersion, relative measures of dispersion are difficult to compute and
comprehend.
Q.N.11.What is the desirable properties of a good measure of
dispersion (Variation).
Ans: - The following are the important properties which a good
measure of dispersion should satisfy:
1.
It should be simple to understand and easy to compute.
2.
It should be simple to compute.
3.
It should be based on all the items.
4.
It should not be affected by extreme values.
5.
It should be rigidly defined.
6.
It should be capable of further algebraic treatment.
7.
It should have sampling stability.
Q.N.12. Define Range. Mention its merits and demerits.
Ans: - Range: Range is
defined as the difference between the value of the smallest item and the value
of the largest item included in the distribution. It is the simplest method of
measuring dispersion. Symbolically,
Range=
Largest value (L) – Smallest Value (S)
The relative
measure corresponding to range, called the coefficient of range, is obtained by
applying the following formula: Coefficient of Range= (L- S)/ (L + S)
Merits of
Range:-
(i)
It is simple to understand and easy to calculate.
(ii)
It is less time consuming.
Demerits of
Range:-
(i)
It is not based on each and every item of the distribution.
(ii)
It is very much affected by the extreme values.
(iii)
The value of Range is affected more by
sampling fluctuations
(iv)
Range cannot be computed in case of open-end distribution.
Q.N.13. Define Quartile Deviation (Q.D). Mention its merits and demerits.
Ans: - Q.D: - The QD is half of the difference between the upper
and lower quartiles. Symbolically, QD= ½ (Q3- Q1)
QD is an absolute measure of dispersion. The relative measure
corresponding to QD, called the coefficient of QD, is obtained by applying the
following formula: Coefficient of Range= (Q3 - Q1) / (Q3
+ Q1)
Coefficient of QD can be used to compare the degree of variation
in different distributions.
Merits of
QD:-
(i)
It is based on 50% of the observations.
(ii)
It is not affected by the presence of extreme values.
(iii)
In case of open-end distribution, it can be computed.
Demerits of
QD:-
(i)
It is not based on each and every item of the distribution.
(ii)
It is not capable of further algebraic treatments.
(iii)
The value of Range is affected more by
sampling fluctuations
Q.N.14. Define Mean
Deviation (M.D).Mention its merits and demerits.
Ans: - M.D: For a given set of observation, MD is defined as the
arithmetic mean of the absolute deviation of the observations from an
appropriate measure of central tendency. The formula for computing MD is
MD= ∑│D│/ N
MD is an
absolute measure of dispersion. The relative measure corresponding to MD,
called the coefficient of MD, is obtained by dividing mean deviation by the
particular average used in computing mean deviation. Thus, if MD has been
computed from median, the coefficient of mean deviation shall be obtained by
dividing MD by median. Coefficient of MD = MD/ (Mean or Median)
Merits of
MD:-
(i)
It is simple to understand and easy to compute.
(ii)
It is based on each and every item of the data.
(iii)
MD is less affected by the values of extreme items than the
Standard deviation.
Demerits of
MD:-
(i)
The greatest drawback of this method is that algebraic signs are
ignored while taking the deviations of the items.
(ii)
It is not capable of further algebraic treatments.
(iii)
It is much less popular as compared to standard deviation.
Q.N.15. Define Standard Deviation (S.D). Mention its merits and
demerits.
Ans: - S.D: The standard deviation commonly denoted by ‘σ’ (Sigma)
is the most widely used measure of dispersion. It is the square root of the
second moment of dispersion and is calculated from the arithmetic mean. In
short, it may be defined as the root-mean-square deviation from the mean.
Merits of
SD:-
(i)
It is based on each and every item of the data and it is rigidly
defined.
(ii)
It is capable of further algebraic treatment. Combined SD of two
or more groups can be calculated.
(iii)
It is less affected by fluctuations of sampling than most other
measures of dispersion.
(iv)
For comparing variability of two or more series, co-efficient of
variation is considered as most appropriate and this is based on SD and Mean.
(v)
SD is most prominently used in further statistical work.
Demerits of
SD:-
(i)
It is not easy to calculate and to understand.
(ii)
It gives more weight to extreme items and less to those which are
nearer to mean.
Q.N.16.Why Standard Deviation (S.D) is regarded as best measure of
dispersion? Distinguish between SD and MD.
Ans: - SD is regarded as best measure of dispersion because: (Same
as advantages of SD)
Distinguish between mean deviation and standard deviation
1. Algebraic
signs are ignored while calculating mean deviations whereas standard deviation
takes into account algebraic sign also.
2. Mean
deviation can be computed either from median or mean whereas standard deviation
is computed always from arithmetic mean.
Q.N.17.What is coefficient of
variation? What purpose does it serve? Distinguish between variance and
coefficient of variation.
Ans: - Coefficient of variation: It is
the ratio of Standard deviation to the mean expressed as percentage. Coefficient of variation can be defined as
the coefficient of standard deviation with respect to mean which is generally
expressed in terms of percentage. The coefficient of variation is also known as
coefficient of variability. Symbolically, Coefficient of Variation (C.V.) =
(S.D / Mean)*100.
Purpose of Coefficient of variation:
Coefficient of variation is used to
compare the variability of two or more series. A series having greater
coefficient of variation is said to have more variable, i.e., less uniform,
less stable or less consistent. Again a series, having coefficient of variation
lesser is said to be less variable, i.e., more uniform, more stable or more
consistent.
Difference between variance and
coefficient of variation: Coefficient of variation is the percentage variation
in the mean while Variation is the total variation in the mean.
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