Enterprise Statistics B.B.A.LLB 1st sem Notes

 

 

 

Created By Shubham Yadav

 

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SUBJECT: BUSINESS STATISTICS  

 

AN INTRODUCTION TO BUSINESS STATISTICS 

OBJECTIVE: The
intention of the current lesson is to allow the scholars to grasp  the
that means, definition, nature, significance and limitations of
statistics.  

Kya karoge padke kiska bhala hua hai, Lol

“A data of statistics is sort of a data of
international 

language of algebra; it could show of use at any time
underneath  

any circumstance”………………………………………Bowley.  

STRUCTURE:  

1.1 Introduction  

1.2 That means and Definitions of Statistics  

1.3 Varieties of Information and Information Sources  

1.4 Varieties of Statistics  

1.5 Scope of Statistics  

1.6 Significance of Statistics in Enterprise  

1.7 Limitations of statistics  

1.8 Abstract  

1.9 Self-Take a look at Questions  

1.10 Shock

1.1 INTRODUCTION  

For a layman, ‘Statistics’ means numerical
info expressed in quantitative  phrases. This info could relate
to things, topics, actions, phenomena, or  areas of house. As a
matter of reality, information don’t have any limits as to their reference,  protection, and
scope. On the macro degree, these are information on gross nationwide product and 
shares of agriculture, manufacturing, and providers in GDP (Gross Home
Product).  

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On the micro degree, particular person corporations,
howsoever small or massive, produce intensive  statistics on their
operations. The annual experiences of corporations comprise number of information  on
gross sales, manufacturing, expenditure, inventories, capital employed, and different
actions.  These information are sometimes subject information, collected by using
scientific survey methods.  Except commonly up to date, such information are the
product of a one-time effort and have  restricted use past the scenario
which will have referred to as for his or her assortment. A scholar  is aware of statistics
extra intimately as a topic of examine like economics, arithmetic, 
chemistry, physics, and others. It’s a self-discipline, which scientifically offers
with information,  and is usually described because the science of information. In coping with
statistics as information,  statistics has developed acceptable strategies of
gathering, presenting, summarizing,  and analysing information, and thus
consists of a physique of those strategies.  

          1.2 MEANING AND DEFINITIONS OF
STATISTICS  

            Within the
starting, it could be famous that the phrase ‘statistics’ is used somewhat curiously
in  two senses plural and singular. Within the plural sense, it refers to a
set of figures or information.  Within the singular sense, statistics refers back to the
complete physique of instruments which are used to  acquire information, organise and interpret
them and, lastly, to attract conclusions from them.  It needs to be famous that
each the features of statistics are essential if the quantitative  information are
to serve their goal. If statistics, as a topic, is insufficient and consists
of  poor methodology, we couldn’t know the best process to extract
from the info the  info they comprise. Equally, if our information are
faulty or that they’re  insufficient or inaccurate, we couldn’t attain
the best conclusions despite the fact that our  topic is effectively
developed.  

A.L. Bowley has outlined statistics as: (i) statistics is
the science of counting, (ii)  Statistics could rightly be referred to as the
science of averages, and (iii) statistics is the  science of measurement
of social organism thought to be a complete in all its mani-

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festations. Boddington outlined as:
Statistics is the science of estimates and  chances. Additional, W.I.
King has outlined Statistics in a wider context, the science  of
Statistics is the tactic of judging collective, pure or social phenomena
from the  outcomes obtained by the evaluation or enumeration or assortment of
estimates.  

Seligman explored that statistics is a science that offers with the
strategies of gathering,  classifying, presenting, evaluating and
deciphering numerical information collected to throw  some gentle on any sphere
of enquiry. Spiegal defines statistics highlighting its position in 
decision-making notably underneath uncertainty, as follows: statistics is
involved  with scientific methodology for gathering, organising, summa
rising, presenting and  analyzing information in addition to drawing legitimate
conclusions and making cheap selections  on the premise of such
evaluation. In line with Prof. Horace Secrist, Statistics is the 
combination of info, affected to a marked extent by multiplicity of causes,
numerically  expressed, enumerated or estimated in line with cheap
requirements of accuracy,  collected in a scientific method for a
pre-determined goal, and positioned in relation  to one another.  

From the above definitions, we will spotlight
the foremost traits of statistics as  follows:  

(i) Statistics are the aggregates of info. It means a single determine just isn’t
statistics.  For instance, nationwide revenue of a rustic for a single yr
just isn’t statistics however  the identical for 2 or extra years is
statistics.  

(ii) Statistics are affected by a variety of elements. For instance, sale of a product  relies upon
on a variety of elements reminiscent of its worth, high quality, competitors, the 
revenue of the customers, and so forth. 

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(iii) Statistics have to be moderately correct. Fallacious figures, if analysed, will lead
to  misguided conclusions. Therefore, it’s crucial that conclusions have to be
primarily based  on correct figures.  

(iv) Statistics have to be collected in a scientific method. If information are collected in a  haphazard
method, they won’t be dependable and can result in deceptive 
conclusions.  

(v) Collected in a scientific method for a pre-determined
goal  (vi) Lastly, Statistics needs to be positioned in
relation to one another. If one collects information  unrelated to one another,
then such information will probably be complicated and won’t result in  any logical
conclusions. Information needs to be comparable over time and over house.  1.3 TYPES OF DATA AND DATA
SOURCES  

Statistical information are the fundamental uncooked materials
of statistics. Information could relate to an exercise of  our curiosity, a
phenomenon, or an issue scenario underneath examine. They derive as a  consequence
of the method of measuring, counting and/or observing. Statistical information, 
subsequently, discuss with these features of an issue scenario that may be
measured,  quantified, counted, or categorized. Any object topic
phenomenon, or exercise that  generates information by means of this course of is
termed as a variable. In different phrases, a variable  is one which exhibits a
diploma of variability when successive measurements are recorded.  In
statistics, information are categorized into two broad classes: quantitative information
and  qualitative information. This classification relies on the form of
traits which are  measured.  

Quantitative information are these that may be quantified in particular
models of measurement.  These discuss with traits whose successive
measurements yield quantifiable  observations. Relying on the character of
the variable noticed for measurement,  quantitative information will be additional
categorized as steady and discrete information. 

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Clearly, a variable could also be a steady
variable or a discrete variable.  (i) Steady information symbolize the
numerical values of a steady variable. A  steady variable is the
one that may assume any worth between any two  factors on a line phase,
thus representing an interval of values. The values  are fairly exact and
shut to one another, but distinguishably totally different. All  traits
reminiscent of weight, size, peak, thickness, velocity, temperature,  tensile
power, and so forth., symbolize steady variables. Thus, the info recorded 
on these and related different traits are referred to as steady information. It could
be  famous {that a} steady variable assumes the best unit of
measurement.  Most interesting within the sense that it permits measurements to the
most diploma of  precision.  

(ii) Discrete information are the values assumed by a discrete
variable. A discrete  variable is the one whose outcomes are measured in
fastened numbers. Such information  are basically depend information. These are derived
from a strategy of counting, such  because the variety of objects possessing or not
possessing a sure attribute.  The variety of clients visiting a
departmental retailer on a regular basis, the incoming  flights at an airport, and the
faulty objects in a consignment obtained on the market,  are all examples of
discrete information.  

Qualitative information discuss with qualitative traits of a
topic or an object. A  attribute is qualitative in nature when its
observations are outlined and famous in  phrases of the presence or absence of
a sure attribute in discrete numbers. These information  are additional categorized
as nominal and rank information.  

(i) Nominal information are the end result of classification into two or
extra classes of  objects or models comprising a pattern or a inhabitants
in line with some high quality  attribute. Classification of scholars
in line with intercourse (as males and 

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females), of staff in line with ability (as
expert, semi-skilled, and unskilled),  and of staff in line with the
degree of schooling (as matriculates,  undergraduates, and post-graduates),
all consequence into nominal information. Given any  such foundation of classification, it
is at all times potential to assign every merchandise to a  explicit class and make a
summation of things belonging to every class. The  depend information so obtained
are referred to as nominal information.  

(ii) Rank information, alternatively, are the results of assigning
ranks to specify order  by way of the integers 1,2,3, …, n. Ranks could
be assigned in line with the  degree of efficiency in a take a look at. a contest, a
competitors, an interview, or a  present. The candidates showing in an
interview, for instance, could also be assigned  ranks in integers starting from I
to n, relying on their efficiency within the  interview. Ranks so assigned
will be considered as the continual values of a  variable involving
efficiency as the standard attribute.  

Information sources could possibly be seen as of two sorts,
viz., secondary and first. The 2 can  be outlined as underneath:  

(i) Secondary information: They exist already in some type: printed or
unpublished –  in an identifiable secondary supply. They’re, typically,
out there from  printed supply(s), although not essentially within the type
really required.  

(ii) Major information: These information which don’t exist already in any
type, and thus  must be collected for the primary time from the first
supply(s). By their very  nature, these information require contemporary and first-time
assortment protecting the entire  inhabitants or a pattern drawn from
it.  

1.4 TYPES OF STATISTICS  

There are two main divisions of statistics
reminiscent of descriptive statistics and inferential  statistics. The time period descriptive
statistics offers with gathering, summarizing, and 

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simplifying information, that are in any other case fairly
unwieldy and voluminous. It seeks to  obtain this in a way that
significant conclusions will be readily drawn from the  information. Descriptive
statistics could thus be seen as comprising strategies of bringing out  and
highlighting the latent traits current in a set of numerical information. It
not  solely facilitates an understanding of the info and systematic
reporting thereof in a  method; and likewise makes them amenable to additional
dialogue, evaluation, and  interpretations.  

Step one in any scientific inquiry is
to gather information related to the issue in  hand. When the inquiry relates
to bodily and/or organic sciences, information assortment  is generally an
integral a part of the experiment itself. The truth is, the very method wherein 
an experiment is designed, determines the form of information it could require
and/or  generate. The issue of figuring out the character and the form of
the related information is  thus mechanically resolved as quickly because the design of
experiment is finalized. It’s  potential within the case of bodily sciences.
Within the case of social sciences, the place the  required information are sometimes
collected by means of a questionnaire from a variety of fastidiously  chosen respondents,
the issue just isn’t that merely resolved. For one factor,  designing the
questionnaire itself is a essential preliminary drawback. For an additional, the
quantity  of respondents to be accessed for information assortment and the standards
for choosing them  has their very own implications and significance for the
high quality of outcomes obtained. Additional,  the info have been collected, these
are assembled, organized, and introduced within the  type of acceptable tables
to make them readable. Wherever wanted, figures,  diagrams, charts, and
graphs are additionally used for higher presentation of the info. A helpful 
tabular and graphic presentation of information would require that the uncooked information be
correctly  categorized in accordance with the targets of investigation
and the relational evaluation  to be carried out. . 

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A effectively thought-out and sharp information
classification facilitates simple description of the  hidden information
traits via a wide range of abstract measures. These embody 
measures of central tendency, dispersion, skewness, and kurtosis, which represent
the  important scope of descriptive statistics. These type a big a part of
the subject material  of any fundamental textbook on the topic, and thus they
are being mentioned in that order  right here as effectively.  

Inferential statistics, often known as inductive statistics, goes
past describing a  given drawback scenario via gathering,
summarizing, and meaningfully  presenting the associated information. As an alternative, it
consists of strategies which are used for drawing  inferences, or making broad
generalizations, a few totality of observations on the  foundation of
data about part of that totality. The totality of observations
about  which an inference could also be drawn, or a generalization made, is
referred to as a inhabitants or  a universe. The a part of totality, which is
noticed for information assortment and evaluation to  acquire data in regards to the
inhabitants, known as a pattern.  

The specified details about a given
inhabitants of our curiosity; may additionally be  collected even by observing all
the models comprising the inhabitants. This whole  protection known as
census. Getting the specified worth for the inhabitants by means of census  is
not at all times possible and sensible for numerous causes. Other than time and
cash  concerns making the census operations prohibitive, observing
every particular person  unit of the inhabitants on the subject of any information
attribute could at instances contain  even harmful testing. In such
instances, clearly, the one recourse out there is to  make use of the partial or
incomplete info gathered by means of a pattern for the  goal. That is
exactly what inferential statistics does. Thus, acquiring a selected 
worth from the pattern info and utilizing it for drawing an inference about
the  total inhabitants underlies the subject material of inferential
statistics. Contemplate a 

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scenario wherein one is required to know
the typical physique weight of all the faculty  college students in a given
cosmopolitan metropolis throughout a sure yr. A fast and straightforward approach to  do that
is to document the load of solely 500 college students, from out of a complete power of, 
say, 10000, or an unknown whole power, take the typical, and use this
common primarily based  on incomplete weight information to symbolize the typical physique
weight of all the faculty  college students. In a distinct scenario, one could
must repeat this train for some future  yr and use the short
estimate of common physique weight for a comparability. This will likely  be wanted, for
instance, to resolve whether or not the load of the faculty college students has 
undergone a major change over time in contrast.  

Inferential statistics helps to guage the
dangers concerned in reaching inferences or  generalizations about an unknown
inhabitants on the premise of pattern info. for  instance, an inspection
of a pattern of 5 battery cells drawn from a given lot could  reveal that
all of the 5 cells are in completely good situation. This info could
be  used to conclude that your complete lot is sweet sufficient to purchase or
not.  

Since this inference relies on the
examination of a pattern of restricted variety of  cells, it’s equally possible
that each one the cells within the lot aren’t so as. Additionally it is potential  that
all of the objects which may be included within the pattern are unsatisfactory. This will likely
be  used to conclude that your complete lot is of unsatisfactory high quality,
whereas the very fact could  certainly be in any other case. It could, thus, be seen that
there’s at all times a threat of an inference  a few inhabitants being incorrect
when primarily based on the data of a restricted pattern.  The rescue in such
conditions lies in evaluating such dangers. For this, statistics offers 
the mandatory strategies. These centres on quantifying in probabilistic time period the
probabilities  of selections taken on the premise of pattern info being
incorrect. This requires an  understanding of the what, why, and the way of
likelihood and likelihood distributions  to equip ourselves with strategies
of drawing statistical inferences and estimating the 

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diploma of reliability of those inferences.  

1.5 SCOPE OF STATISTICS 

Other than the strategies comprising the scope
of descriptive and inferential branches  of statistics, statistics additionally
consists of strategies of coping with just a few different problems with  particular
nature. Since these strategies are basically descriptive in nature, they
have  been mentioned right here as a part of the descriptive statistics. These are
primarily involved  with the next:  

(i) It usually turns into crucial to look at how two paired information
units are associated.  For instance, we could have information on the gross sales of a product
and the expenditure  incurred on its commercial for a specified quantity
of years. On condition that gross sales  and commercial expenditure are associated to
one another, it’s helpful to look at  the character of relationship between
the 2 and quantify the diploma of that  relationship. As this requires
use of acceptable statistical strategies, these falls  underneath the purview of
what we name regression and correlation evaluation.  

(ii) Conditions happen very often once we require averaging
(or totalling) of information  on costs and/or portions expressed in several
models of measurement. For  instance, worth of fabric could also be quoted per meter
of size and that of wheat  per kilogram of weight. Since bizarre
strategies of totalling and averaging do  not apply to such worth/amount
information, particular methods wanted for the  goal are developed underneath index
numbers.  

(iii) Many a time, it turns into crucial to look at the previous
efficiency of an  exercise with a view to figuring out its future
behaviour. For instance, when  engaged within the manufacturing of a commodity,
month-to-month product gross sales are an  essential measure of evaluating efficiency.
This requires compilation and  evaluation of related gross sales information over time.
The extra advanced the exercise, the 

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extra assorted the info necessities. For revenue
maximising and future gross sales  planning, forecast of possible gross sales progress
fee is essential. This wants cautious  assortment and evaluation of previous gross sales
information. All such issues are taken care of  underneath time collection
evaluation.  

(iv) Acquiring the most certainly future estimates on any
facet(s) regarding a  enterprise or financial exercise has certainly been
participating the minds of all  involved. That is notably essential when
it pertains to product gross sales and  demand, which serve the mandatory foundation
of manufacturing scheduling and  planning. The regression, correlation, and
time collection analyses collectively assist  develop the fundamental methodology to do
the needful. Thus, the examine of strategies  and methods of acquiring the
possible estimates on enterprise/financial  variables contains the scope of
what we do underneath enterprise forecasting.  

Protecting in view the significance of inferential
statistics, the scope of statistics could  lastly be restated as consisting
of statistical strategies which facilitate decision– making underneath circumstances of
uncertainty. Whereas the time period statistical strategies is usually  used to cowl
the topic of statistics as a complete, specifically it refers to strategies
by  which statistical information are analysed, interpreted, and the inferences
drawn for determination making.  

Although generic in nature and versatile in
their functions, statistical strategies have  come to be extensively used,
particularly in all issues regarding enterprise and economics.  These are
additionally being more and more utilized in biology, medication, agriculture,
psychology,  and schooling. The scope of software of those strategies has
began opening and  increasing in a variety of social science disciplines
as effectively. Even a political scientist  finds them of accelerating relevance
for analyzing the political behaviour and it’s, of  course, no shock
to seek out even historians statistical information, for historical past is basically previous 

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information introduced in sure precise format.  

1.6 IMPORTANCE OF STATISTICS IN BUSINESS  

There are three main capabilities in any
enterprise enterprise wherein the statistical  strategies are helpful. These
are as follows:  

(i) The planning of operations: This will likely relate to both particular initiatives or
to  the recurring actions of a agency over a specified
interval.  

(ii) The establishing of requirements: This will likely relate to the scale of
employment,  quantity of gross sales, fixation of high quality norms for the
manufactured product,  norms for the day by day output, and so
forth.  

(iii) The operate of management: This includes comparability of precise
manufacturing  achieved towards the norm or goal set earlier. In case the
manufacturing has  fallen in need of the goal, it offers remedial measures so
that such a deficiency  doesn’t happen once more.  

A price noting level is that though these
three functions-planning of operations,  setting requirements, and
control-are separate, however in follow they’re very a lot 
interrelated.  

Completely different authors have highlighted the
significance of Statistics in enterprise. For  occasion, Croxton and Cowden
give quite a few makes use of of Statistics in enterprise reminiscent of  undertaking planning,
budgetary planning and management, stock planning and management,  high quality
management, advertising and marketing, manufacturing and personnel administration. Inside these
additionally  they’ve specified sure areas the place Statistics may be very related.
One other creator,  Irwing W. Burr, coping with the place of statistics in
an industrial organisation,  specifies a variety of areas the place statistics
is extraordinarily helpful. These are: buyer  desires and market analysis,
improvement design and specification, buying, 

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manufacturing, inspection, packaging and
transport, gross sales and complaints, stock and  upkeep, prices,
administration management, industrial engineering and analysis.  Statistical
issues arising in the midst of enterprise operations are multitudinous.
As  such, one could do not more than spotlight among the extra essential
ones to  emphasis the relevance of statistics to the enterprise world. In
the sphere of manufacturing,  for instance, statistics will be helpful in
numerous methods.  

Statistical high quality management strategies are used
to make sure the manufacturing of high quality items.  Figuring out and rejecting
faulty or substandard items obtain this. The sale targets  will be
fastened on the premise of sale forecasts, that are accomplished by utilizing various
strategies  of forecasting. Evaluation of gross sales affected towards the targets
set earlier would point out  the deficiency in achievement, which can be on
account of a number of causes: (i) targets  have been too excessive and unrealistic (ii)
salesmen’s efficiency has been poor (iii)  emergence of improve in competitors
(iv) poor high quality of firm’s product, and so  on. These elements will be
additional investigated.  

One other sphere in enterprise the place statistical
strategies can be utilized is personnel  administration. Right here, one is worried with
the fixation of wage charges, incentive norms  and efficiency appraisal of
particular person worker. The idea of productiveness is  very related right here.
On the premise of measurement of productiveness, the productiveness  bonus is
awarded to the employees. Comparisons of wages and productiveness are  undertaken
with a purpose to guarantee will increase in industrial productiveness.  

Statistical strategies may be used to
confirm the efficacy of a sure product,  say, medication. For instance,
a pharmaceutical firm has developed a brand new  medication within the therapy of
bronchial bronchial asthma. Earlier than launching it on business  foundation, it desires to
confirm the effectiveness of this medication. It undertakes an 
experimentation involving the formation of two comparable teams of
bronchial asthma 

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sufferers. One group is given this new
medication for a specified interval and the opposite  one is handled with the
traditional medicines. Information are maintained for the 2 teams for  the
specified interval. This document is then analysed to determine if there’s
any  vital distinction within the restoration of the 2 teams. If the
distinction is actually  vital statistically, the brand new medication is
commercially launched.  

1.7 LIMITATIONS OF STATISTICS  

Statistics has a variety of limitations,
pertinent amongst them are as follows:  (i) There are particular
phenomena or ideas the place statistics can’t be used. This  is as a result of
these phenomena or ideas aren’t amenable to measurement.  For instance,
magnificence, intelligence, braveness can’t be quantified. Statistics has  no place
in all such instances the place quantification just isn’t potential.  

(ii) Statistics reveal the typical behaviour, the traditional or
the overall development. An  software of the ‘common’ idea if utilized to
a person or a selected  scenario could result in a improper conclusion
and generally could also be disastrous.  For instance, one could also be misguided when
advised that the typical depth of a  river from one financial institution to the opposite is 4
ft, when there could also be some factors in  between the place its depth is way
greater than 4 ft. On this understanding, one  could enter these factors
having better depth, which can be hazardous.  

(iii) Since statistics are collected for a selected goal,
such information might not be  related or helpful in different conditions or instances. For
instance, secondary information  (i.e., information initially collected by another person)
might not be helpful for the opposite  particular person.  

(iv) Statistics aren’t 100 per cent exact as is Arithmetic
or Accountancy.  Those that use statistics ought to concentrate on this
limitation. 

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(v) In statistical surveys, sampling is mostly used because it
just isn’t bodily potential  to cowl all of the models or parts comprising
the universe. The outcomes could not  be acceptable so far as the universe
is worried. Furthermore, totally different surveys  primarily based on the identical dimension of
pattern however totally different pattern models could yield  totally different
outcomes.  

(vi) At instances, affiliation or relationship between two or extra
variables is studied  in statistics, however such a relationship doesn’t
point out trigger and impact’  relationship. It merely exhibits the similarity
or dissimilarity within the motion of  the 2 variables. In such instances, it
is the person who has to interpret the outcomes  fastidiously, stating the
kind of relationship obtained.  

(vii) A serious limitation of statistics is that it doesn’t
reveal all pertaining to a  sure phenomenon. There may be some background
info that statistics  doesn’t cowl. Equally, there are some
different features associated to the issue  readily available, that are additionally not coated.
The person of Statistics needs to be effectively  knowledgeable and will interpret
Statistics conserving in thoughts all different features  having relevance on the
given drawback.  

Other than the restrictions of statistics
talked about above, there are misuses of it. Many  individuals, knowingly or
unknowingly, use statistical information in improper method. Allow us to see  what the
major misuses of statistics are in order that the identical could possibly be averted when one 
has to make use of statistical information. The misuse of Statistics could take a number of varieties
a few of  that are defined beneath.  

(i) Sources of information not given: At instances, the supply of information just isn’t given. In
the  absence of the supply, the reader doesn’t understand how far the info are
dependable.  Additional, if he desires to discuss with the unique supply, he’s
unable to take action. 

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(ii) Faulty information: One other misuse is that generally one offers
faulty information.  This can be accomplished knowingly with a purpose to defend one’s
place or to show a  explicit level. This aside, the definition used
to indicate a sure  phenomenon could also be faulty. For instance, in case of
information regarding unem 

ployed individuals, the definition could embody
even those that are employed,  although partially. The query right here is how
far it’s justified to incorporate partially  employed individuals amongst
unemployed ones.  

(iii) Unrepresentative pattern: In statistics, a number of instances one has to
conduct a  survey, which necessitates to decide on a pattern from the given
inhabitants or  universe. The pattern could become unrepresentative of
the universe. One  could select a pattern simply on the premise of comfort.
He could acquire the  desired info from both his buddies or close by
respondents in his  neighbourhood despite the fact that such respondents don’t
represent a  consultant pattern.  

(iv) Insufficient pattern: Earlier, we’ve got seen {that a} pattern that
is  unrepresentative of the universe is a significant misuse of statistics. This
aside, at  instances one could conduct a survey primarily based on a particularly insufficient
pattern.  For instance, in a metropolis we could discover that there are 1, 00,000
households. When  we’ve got to conduct a family survey, we could take a
pattern of merely 100  households comprising solely 0.1 per cent of the
universe. A survey primarily based on  such a small pattern could not yield proper
info.  

(v) Unfair Comparisons: An essential misuse of statistics is making
unfair  comparisons from the info collected. As an illustration, one could
assemble an index  of manufacturing selecting the bottom yr the place the
manufacturing was a lot much less.  Then he could examine the following yr’s
manufacturing from this low base. 

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Such a comparability will undoubtedly give a
rosy image of the manufacturing  although in actuality it isn’t so. One other
supply of unfair comparisons could possibly be  when one makes absolute comparisons
as an alternative of relative ones. An absolute  comparability of two figures, say, of
manufacturing or export, could present a superb  improve, however in relative phrases it
could turnout to be very negligible. One other  instance of unfair comparability
is when the inhabitants in two cities is totally different,  however a comparability of
total loss of life charges and deaths by a selected illness is  tried. Such
a comparability is improper. Likewise, when information aren’t correctly  categorized or
when adjustments within the composition of inhabitants within the two years  aren’t
considered, comparisons of such information could be unfair as 
they might result in deceptive conclusions.  

(vi) Undesirable conclusions: One other misuse of statistics could also be on
account of  unwarranted conclusions. This can be because of making
false assumptions.  For instance, whereas making projections of inhabitants in
the subsequent 5 years,  one could assume a decrease fee of progress although the
previous two years point out  in any other case. Typically one might not be positive about
the adjustments in enterprise  atmosphere within the close to future. In such a case,
one could use an assumption that  could become improper. One other supply
of unwarranted conclusion could also be  the usage of improper common. Suppose in a
collection there are excessive values, one is  too excessive whereas the opposite is simply too
low, reminiscent of 800 and 50. Using an  arithmetic common in such a case
could give a improper thought. As an alternative, harmonic  imply could be correct in such a
case.  

(vii) Confusion of correlation and causation:
In statistics, a number of instances one has  to
look at the connection between two variables. A detailed relationship between
the  two variables could not set up a cause-and-effect-relationship in
the sense that one 

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variable is the trigger and the opposite is the
impact. It needs to be taken as one thing that  measures diploma of
affiliation somewhat than attempt to discover out causal relationship..  1.8 SUMMARY  

In a summarized method, ‘Statistics’ means
numerical info expressed in  quantitative phrases. As a matter of
reality, information don’t have any limits as to their reference,  protection, and scope. At
the macro degree, these are information on gross nationwide product and  shares of
agriculture, manufacturing, and providers in GDP (Gross Home Product). 
On the micro degree, particular person corporations, howsoever small or massive, produce
intensive  statistics on their operations. The annual experiences of corporations
comprise number of information  on gross sales, manufacturing, expenditure, inventories,
capital employed, and different actions.  These information are sometimes subject information,
collected by using scientific survey methods.  Except commonly
up to date, such information are the product of a one-time effort and have  restricted
use past the scenario which will have referred to as for his or her assortment. A
scholar  is aware of statistics extra intimately as a topic of examine like
economics, arithmetic,  chemistry, physics, and others. It’s a self-discipline,
which scientifically offers with information,  and is usually described because the
science of information. In coping with statistics as information,  statistics has
developed acceptable strategies of gathering, presenting, summarizing,  and
analysing information, and thus consists of a physique of those strategies.  

1.9 SELF-TEST QUESTIONS  

1. Outline Statistics. Clarify its sorts, and
significance to commerce, commerce and  enterprise.  

2. “Statistics is all-pervading”. Elucidate this
assertion.  

3. Write a observe on the scope and limitations of
Statistics.  

4. What are the foremost limitations of
Statistics? Clarify with appropriate examples.  5. Distinguish between
descriptive Statistics and inferential Statistics. 

18 

1.10 Relaxation Karlo Thoda 

 Khana
kha lo

19 

COURSE: BUSINESS STATISTICS  

COURSE CODE: MC-106 AUTHOR: SURINDER
KUNDU  LESSON: 02 VETTER: PROF. M. S. TURAN  

AN OVERVIEW OF CENTRAL TENDENCY 

OBJECTIVE: The current lesson imparts understanding of the
calculations and major  properties of measures of central tendency,
together with imply, mode,  median, quartiles, percentiles, and so forth.  

STRUCTURE:  

2.1 Introduction  

2.2 Arithmetic Imply  

2.3 Median  

2.4 Mode  

2.5 Relationships of the Imply, Median and
Mode  

2.6 The Greatest Measure of Central Tendency  

2.7 Geometric Imply  

2.8 Harmonic Imply  

2.9 Quadratic Imply  

2.10 Abstract  

2.11 Self-Take a look at Questions  

2.12 Shock

2.1 INTRODUCTION 

The outline of statistical information could also be
fairly elaborate or fairly temporary relying on  two elements: the character of
information and the aim for which the identical information have been  collected. Whereas
describing information statistically or verbally, one should make sure that the 
description is neither too temporary nor too prolonged. The measures of central
tendency  allow us to match two or extra distributions pertaining to the
identical time interval or  throughout the identical distribution over time. For instance,
the typical consumption of tea  in two totally different territories for a similar
interval or in a territory for 2 years, say, 2003  and 2004, will be
tried via a mean.  

20 

2.2 ARITHMETIC MEAN  

Including all of the observations and dividing the
sum by the variety of observations  outcomes the arithmetic imply. Suppose we
have the next observations:  10, 15,30, 7, 42, 79 and 83  

These are seven observations. Symbolically,
the arithmetic imply, additionally referred to as merely  imply is  

 x = ∑x/n, the place x is straightforward imply.  

10 +15 + 30 + 7 + 42 + 79 + 83 

 ⁼ 7 

⁼ ₇²⁶⁶ =
38  

It could be famous that the Greek letter μ is used to indicate the imply of the
inhabitants  and n to indicate the whole variety of observations in a
inhabitants. Thus the inhabitants  imply μ = ∑x/n. The system given above is the fundamental system that varieties
the  definition of arithmetic imply and is utilized in case of ungrouped information
the place weights are  not concerned.  

2.2.1 UNGROUPED DATA-WEIGHTED AVERAGE  

In case of ungrouped information the place weights are
concerned, our strategy for calculating  arithmetic imply will probably be totally different
from the one used earlier.  

Instance 2.1: Suppose a scholar has secured the next
marks in three checks:   Mid-term take a look at 30  

 Laboratory 25  

 Ultimate 20  

^{30 25 20} ₌ + +

 The easy arithmetic imply will probably be 25 

3 

21 

Nonetheless, this will probably be improper if the three
checks carry totally different weights on the premise of  their relative significance.
Assuming that the weights assigned to the three checks are:   Mid-term take a look at
2 factors  

 Laboratory 3 factors  

 Ultimate 5 factors  

Answer: On the premise of this info, we will now calculate a
weighted imply as  proven beneath:  

Desk 2.1: Calculation of a Weighted Imply  

Sort of Take a look at Relative Weight (w) Marks (x)
(wx)  Mid-term 2 30 60  Laboratory 3 25 75  Ultimate 5 20 100 
Complete ∑ w = 10 235  

^{+ + =} _∑^∑ = 

wx 

w x w x w x 1 1 2 2 3 3 

^x+ + 

w 

w w w 

1 2 3 

^{60 75 100 =} + + 

^{+ +} marks  

 =
23.5 

2 3 5 

It will likely be seen that weighted imply offers a
extra sensible image than the easy or  unweighted imply.  

Instance 2.2: An investor is keen on investing in fairness
shares. Throughout a interval of  falling costs within the inventory alternate, a inventory
is bought at Rs 120 per share on at some point, Rs  105 on the subsequent and Rs 90 on
the third day. The investor has bought 50 shares on  the primary day, 80
shares on the second day and 100 shares on the third’ day. What  common
worth per share did the investor pay? 

22 

Answer:  

Desk 2.2: Calculation of Weighted Common
Worth  

Day Worth per Share (Rs) (x) No of Shares
Bought (w) Quantity Paid (wx)  1 120 50 6000  2 105 80 8400  3
90 100 9000  Complete – 230 23,400  

+ + 

w x w x w x 

^{∑ =} + + 

Weighted common = _wwx 1 1 2 2 3 3 

w w w 1 2
3 

∑ 

^{+ +} marks  

^{6000 8400 9000 =} + + 

 =
101.7 

50 80 100 

Due to this fact, the investor paid a mean worth of Rs 101.7
per share.  

It will likely be seen that if merely costs of the
shares for the three days (whatever the  variety of shares bought)
have been considered, then the typical worth  would
be  

^{120 105 90} _{. =} ^{+ +} Rs 3 

105 

That is an unweighted or easy common and
because it ignores the-quantum of shares  bought, it fails to provide an accurate
image. A easy common, it could be famous, can also be  a weighted common
the place weight in every case is similar, that’s, only one. After we  use the
time period common alone, we at all times imply that it’s an unweighted or easy 
common.  

2.2.2 GROUPED DATA-ARITHMETIC MEAN  

For grouped information, arithmetic imply could also be
calculated by making use of any of the  following strategies:  

(i) Direct methodology, (ii) Brief-cut methodology , (iii)
Step-deviation methodology 

23 

Within the case of direct methodology, the system x
= ∑fm/n is used. Right here m is mid-point of  numerous
lessons, f is the frequency of every class and n is the whole
variety of  frequencies. The calculation of arithmetic imply by the direct
methodology is proven beneath.  Instance 2.3: The next desk offers the
marks of 58 college students in Statistics.  Calculate the typical marks of this
group.  

 Marks No. of College students  

0-10 4  

10-20 8  

20-30 11  

30-40 15  

40-50 12  

50-60 6  

60-70 2  

 Complete 58  

Answer:  

Desk 2.3: Calculation of Arithmetic Imply by Direct
Methodology  

_{Marks Mid-point m} No. of
College students  

 _f fm  

0-10 5 4 20  

10-20 15 8 120  

20-30 25 11 275  

30-40 35 15 525  

40-50 45 12 540  

50-60 55 6 330  

60-70 65 2 130  

 ∑fm =
1940  

The place,  

_{= = =} ∑₅₈ 

fm 

1940 

x 33.45
marks or 33 marks roughly.  

n 

It could be famous that the mid-point of every
class is taken as a superb approximation of  the true imply of the category.
That is primarily based on the belief that the values are  distributed pretty
evenly all through the interval. When massive numbers of frequency  happen,
this assumption is normally accepted. 

24 

Within the case of short-cut methodology, the idea
of arbitrary imply is adopted. The  system for calculation of the
arithmetic imply by the short-cut methodology is given  beneath:  

_{x A} ∑ _{= +} 

fd 

n 

The place A = arbitrary or assumed imply  

 f = frequency  

 d = deviation from the arbitrary or assumed
imply  

When the values are extraordinarily massive and/or in
fractions, the usage of the direct methodology  could be very cumbersome. In such
instances, the short-cut methodology is preferable. That is  as a result of the
calculation work within the short-cut methodology is significantly diminished 
notably for calculation of the product of values and their respective
frequencies.  Nonetheless, when calculations aren’t made manually however by a
machine calculator, it  might not be essential to resort to the short-cut
methodology, as the usage of the direct methodology  could not pose any
drawback.  

As will be seen from the system used within the
short-cut methodology, an arbitrary or assumed  imply is used. The second time period
within the system (∑fd ⎟ n) is the correction issue for the 
distinction between the precise imply and the assumed imply. If the assumed imply
turns  out to be equal to the precise imply, (∑fd ⎟ n) will probably be zero. Using the
short-cut  methodology relies on the precept that the whole of deviations
taken from an precise imply  is the same as zero. As such, the deviations taken
from another determine will depend upon  how the assumed imply is expounded to
the precise imply. Whereas one could select any worth  as assumed imply, it could
be correct to keep away from excessive values, that’s, too small or too  excessive to
simplify calculations. A price apparently near the arithmetic imply
ought to  be chosen. 

25 

For the figures given earlier pertaining to
marks obtained by 58 college students, we calculate  the typical marks by utilizing
the short-cut methodology.  

Instance 2.4:  

Desk 2.4: Calculation of Arithmetic Imply by Brief-cut
Methodology  

_Marks Mid-point  

_m f d
fd  

0-10 5 4 -30 -120  

10-20 15 8 -20 -160  

20-30 25 11 -10 -110  

30-40 35 15 0 0  

40-50 45 12 10 120  

50-60 55 6 20 120  

60-70 65 2 30 60  

 ∑fd =
-90  

It could be famous that we’ve got taken arbitrary
imply as 35 and deviations from  midpoints. In different phrases, the arbitrary
imply has been subtracted from every worth of  mid-point and the resultant
determine is proven in column d.  

^fd _{x A} ∑ _{= +} 

 

n 

^⎟_⎠^{⎞ ⎜}_⎝⎛ ^{− = +}₅₈⁹⁰ 35 

= 35 – 1.55 = 33.45 or 33 marks
roughly.  

Now we take up the calculation of arithmetic
imply for a similar set of information utilizing the  step-deviation methodology. That is
proven in Desk 2.5.  

Desk 2.5: Calculation of Arithmetic Imply by
Step-deviation Methodology  

Marks Mid-point f d d’= d/10 Fd’  

0-10 5 4 -30 -3 -12  

10-20 15 8 -20 -2 -16  

20-30 25 11 -10 -1 -11  

30-40 35 15 0 0 0  

40-50 45 12 10 1 12  

50-60 55 6 20 2 12  

60-70 65 2 30 3 6  

 ∑fd’
=-9 

26 

_{x = A+ ⋅} ∑ ^‘ 

fd 

 C 

n 

^{9 10} 35 =
33.45 or 33 marks roughly.  

 ^⎟_⎠^{⎞ ⎜}_⎝⎛ ^{− ⋅ =
+}58 

It will likely be seen that the reply in every of
the three instances is similar. The step deviation methodology is probably the most handy
on account of simplified calculations. It could  even be famous that if we
choose a distinct arbitrary imply and recalculate deviations  from that
determine, we’d get the identical reply.  

Now that we’ve got learnt how the arithmetic
imply will be calculated by utilizing totally different  strategies, we’re able
to deal with any drawback the place calculation of the  arithmetic imply is
concerned.  

Instance 2.6: The
imply of the next frequency distribution was discovered to be 1.46.  

No. of Accidents No. of Days (frequency)  

0 46  

1 ?  

2 ?  

3 25  

4 10  

5 5  

 Complete 200 days  

Calculate the lacking frequencies.  

Answer:  

Right here we’re given the whole variety of
frequencies and the arithmetic imply. Now we have  to find out the 2
frequencies which are lacking. Allow us to assume that the frequency  towards 1
accident is x and towards 2 accidents is y. If we will set up
two  simultaneous equations, then we will simply discover the values of X and Y.  

(0.46) + (1. x) + (2. y)
+ (3. 25) + (4.l0) + (5.5)

^{Imply =} 200 

27 

x + 2y +140 

^{1.46 =} 200 

x + 2y
+ 140 = (200) (1.46)  

x + 2y
= 152  

x + y=200- {46+25
+ 1O+5}  

x + y =
200 – 86  

x + y
= 114  

Now subtracting equation (ii) from equation (i), we
get  

x + 2y
= 152  

x + y
= 114  

 – – –  

 y = 38  

Substituting the worth of y = 38 in equation (ii)
above, x + 38 = 114  

Due to this fact, x = 114 – 38 = 76  

Therefore, the lacking frequencies are:  

In opposition to accident 1 : 76  

In opposition to accident 2 : 38  

2.2.3 CHARACTERISTICS OF THE ARITHMETIC MEAN  

Among the essential traits of the arithmetic
imply are:  

1. The sum of the deviations of the
particular person objects from the arithmetic imply is  at all times zero. This implies I: (x
– x ) = 0, the place x is the worth of an merchandise and x is 
the arithmetic imply. For the reason that sum of the deviations within the optimistic
course  is the same as the sum of the deviations within the destructive
course, the arithmetic  imply is thought to be a measure of central
tendency.  

2. The sum of the squared deviations of the
particular person objects from the arithmetic  imply is at all times minimal. In different
phrases, the sum of the squared deviations  taken from any worth aside from
the arithmetic imply will probably be increased. 

28 

3. Because the arithmetic imply relies on all of the
objects in a collection, a change within the  worth of any merchandise will result in a
change within the worth of the arithmetic imply.  4. Within the case of extremely
skewed distribution, the arithmetic imply could get  distorted on account of
just a few objects with excessive values. In such a case, it  could stop to be the
consultant attribute of the distribution.  

2.3 MEDIAN  

Median is outlined as the worth of the center
merchandise (or the imply of the values of the  two center objects) when the info
are organized in an ascending or descending order of  magnitude. Thus, in
an ungrouped frequency distribution if the n values are organized 
in ascending or descending order of magnitude, the median is the center worth
if n is  odd. When n is even, the median is the imply of the
two center values.  

 Suppose we’ve got the next collection:  

15, 19,21,7, 10,33,25,18 and 5  

Now we have to first organize it in both
ascending or descending order. These figures are  organized in an ascending
order as follows:  

5,7,10,15,18,19,21,25,33  

Now because the collection consists of strange variety of
objects, to seek out out the worth of the center  merchandise, we use the
system  

n +1 

^{The place} 2 

n + ¹ = 5, that’s, the scale  

The place ^{n is the variety of objects. In
this case, n is 9, as such} 2 

of the fifth merchandise is the median. This occurs to be
18.  

Suppose the collection consists of another objects
23. We could, subsequently, have to incorporate  23 within the above collection at an
acceptable place, that’s, between 21 and 25. Thus, the  collection is now 5,
7, 10, 15, 18, 19, and 21,23,25,33. Making use of the above system, the 

29 

median is the scale of 5.5^th merchandise.
Right here, we’ve got to take the typical of the values of fifth  and sixth merchandise. This
means a mean of 18 and 19, which provides the median as 18.5.  n + ¹ itself just isn’t the system for the median; it  

^{It could be famous that the system} 2 

merely signifies the place of the median,
specifically, the variety of objects we’ve got to  depend till we arrive on the merchandise
whose worth is the median. Within the case of the even  variety of objects within the
collection, we determine the 2 objects whose values must be  averaged to
acquire the median. Within the case of a grouped collection, the median is 
calculated by linear interpolation with the assistance of the next
system:  

^{l l} ₋ + 

M = l₁ ( ) ^{2 1} m c 

f 

The place M = the median  

 l₁ = the
decrease restrict of the category wherein the median lies  

1₂ = the higher restrict of the category wherein the median
lies  

 f = the frequency of the category wherein the
median lies  

 m = the center merchandise or (n +
1)/2th, the place n stands for whole variety of   objects  

 c = the cumulative frequency of the
class previous the one wherein the median lies  Instance
2.7:  

Month-to-month Wages (Rs) No. of Staff  

 800-1,000
18  

1,000-1,200 25  

1,200-1,400 30  

1,400-1,600 34  

1,600-1,800 26  

1,800-2,000 10  

Complete 143 

With the intention to calculate median on this case, we
must first present cumulative  frequency to the desk. Thus, the desk
with the cumulative frequency is written as: 

30 

_{Month-to-month Wages Frequency} Cumulative Frequency 

800 -1,000 18 18  

1,000 -1,200 25 43  

1,200 -1,400 30 73  

1,400 -1,600 34 107  

1,600 -1,800 26 133  

1.800 -2,000 10 143  

^{l l} ₋ + 

 M = l₁ ( ) ^{2 1} m
c 

f 

1 ⁺ ₌ ^{n +} =
72 

143
1 

 M = ₂ 

2 

It means median lies within the class-interval Rs 1,200 –
1,400.  

^{1400 1200} ₋ − 

Now, M = 1200 + ^{(72 43)} 30 

²⁰⁰ =1200 + 

 ⁽²⁹⁾ 30 

 = Rs 1393.3  

At this stage, allow us to introduce two different
ideas viz. quartile and decile. To  perceive these, we should always first
know that the median belongs to a common class of  statistical
descriptions referred to as fractiles. A fractile is a price beneath that lays a
given  fraction of a set of information. Within the case of the median, this fraction
is one-half (1/2).  Likewise, a quartile has a fraction one-fourth (1/4).
The three quartiles Q₁, Q₂ and Q₃ 

are such that 25 p.c of the info fall
beneath Q₁, 25 p.c fall between Q₁ and Q₂,  25 p.c fall between Q₂ and Q₃ and 25
p.c fall above Q₃ It will likely be seen that Q₂ is the median. We will use the above system
for the calculation of quartiles as effectively.  The one distinction will probably be in
the worth of m. Allow us to calculate each Q₁ and Q₃ in  respect of the desk given in
Instance 2.7.  

^{l l} ₋ −

 Q₁ = l₁ ( ) ^{2 1} m c 

f 

31 

n + ^{1 =} 4 

Right here, ^{m will probably be =} 4 

143 +¹ = 36  

_{1 1000 −} − _{Q = +} 

1200 1000 

 ^{(36 18)} 25 ²⁰⁰ =1000 + 

 ⁽¹⁸⁾ 25  = Rs. 1,144  

n + ^{1 =} 4 

Within the case of Q₃^{, m
will probably be 3 =} 4 _{1 1600 −} − _{Q = +} 

1800 1600 

^{(108 107)} 26 

²⁰⁰ =1600 + 

⁽¹⁾ 26 

 Rs. 1,607.7 approx  

3⋅¹⁴⁴ =
108  

In the identical method, we will calculate deciles
(the place the collection is split into 10  components) and percentiles (the place the
collection is split into 100 components). It could be famous that  not like arithmetic
imply, median just isn’t affected in any respect by excessive values, as it’s a 
positional common. As such, median is especially very helpful when a distribution 
occurs to be skewed. One other level that goes in favour of median is that it
will be  computed when a distribution has open-end lessons. But, one other
advantage of median is  that when a distribution accommodates qualitative information, it
is the one common that may be  used. No different common is appropriate in
case of such a distribution. Allow us to take a pair  of examples to
illustrate what has been mentioned in favour of median. 

32 

Instance 2.8:Calculate probably the most appropriate common for the
following information:  Measurement of the Merchandise Under 50 50-100 100-150 150-200
200 and above  Frequency 15 20 36 40 10  Answer: Since
the info have two open-end classes-one to start with (beneath 50) and
the  different on the finish (200 and above), median needs to be the best alternative
as a measure of central  tendency.  

Desk 2.6: Computation of Median  

Measurement of Merchandise Frequency Cumulative Frequency 

Under 50 15 15  

50-100 20 35  

100-150 36 71  

150-200 40 111  

200 and above 10 121  

n ^{+ 1} th merchandise  

^{Median is the scale of} 2 

¹²¹⁺¹= 61^st merchandise  

⁼ 2 

Now, 61^st merchandise lies within the 100-150 class  

^{l l} ₋ − 

 Median = 1₁ = l₁ ( ) ^{2 1} m c 

f 

^{150 100} ₋ − 

= 100 + ^{(61 35)} 36 

 = 100 + 36.11 = 136.11 approx.  

Instance 2.9: The next information give the financial savings financial institution
accounts balances of 9 pattern  households chosen in a survey. The
figures are in rupees.  

745 2,000 1,500 68,000 461 549 3750 1800 4795  

(a) Discover the imply and the median for these
information; (b) Do these information comprise an outlier? In that case,  exclude this worth and
recalculate the imply and median. Which of those abstract measures 

33 

has a better change when an outlier is
dropped?; (c) Which of those two abstract measures  is extra acceptable for
this collection?  

Answer:  

745 + 2,000 +1,500 + 68,000 + 461+ 549 + 3,750 +1,800 + 4,795 

^{Imply = Rs.} 9 

^{Rs 83,600} = Rs
9,289  

⁼ 9 

n + ¹ th merchandise  

^{Median = Measurement of} 2 

9 + ¹ = 5^th merchandise  

⁼ 2 

Arranging the info in an ascending order, we
discover that the median is Rs 1,800.  (b) An merchandise of Rs 68,000 is excessively
excessive. Such a determine known as an ‘outlier’. We  exclude this determine and
recalculate each the imply and the median.  

83,600 − 68,000 

 ^{Imply = Rs.} 8 

^15,600 = Rs.
1,950  

 ^{= Rs} 8 

n + ¹ th merchandise  

^{Median = Measurement of} 2 

^{8 1} ₌ ⁺ merchandise.  

= 4.5th 

2 

^{1,500 −1,800} = Rs. 1,650  

 ^{= Rs.} 2 

It will likely be seen that the imply exhibits a far
better change than the median when the  outlier is dropped from the
calculations.  

(c) So far as these information are involved, the
median will probably be a extra acceptable measure  than the imply.  

Additional, we will decide the median graphically as
follows: 

34 

Instance 2.10: Suppose
we’re given the next collection:  

Class interval 0-10
10-20 20-30 30-40 40-50 50-60 60-70 

Frequency 6 12
22 37 17 8 5  

We’re requested to attract each forms of ogive from
these information and to find out the  median.  

Answer:  

To start with, we rework the given information
into two cumulative frequency distributions,  one primarily based on ‘lower than’ and
one other on ‘greater than’ strategies.  

Desk A  

Frequency  

Lower than 10 6  

Lower than 20 18  

Lower than 30 40  

Lower than 40 77  

Lower than 50 94  

Lower than 60 102  

Lower than 70 107  

  

 Desk B  

 Frequency 
Greater than 0 107  

 Greater than 10 101  

Greater than 20 89  

Greater than 30 67  

Greater than 40 30  

Greater than 50 13  

 Greater than 60 5  

It could be famous that the purpose of  

intersection of the 2 ogives offers the  

worth of the median. From this level of  

intersection A, we draw a straight line to 

35 

meet the X-axis at M. Thus, from the purpose of
origin to the purpose at M offers the worth  of the median, which involves
34, roughly. If we calculate the median by  making use of the system,
then the reply involves 33.8, or 34, roughly. It could be  pointed
out that even a single ogive can be utilized to find out the median. As we
have  decided the median graphically, so additionally we will discover the values of
quartiles, deciles  or percentiles graphically. For instance, to find out
we’ve got to take dimension of {3(n +  1)} /4 = 81^st merchandise.
From this level on the Y-axis, we will draw a perpendicular to  meet the
‘lower than’ ogive from which one other straight line is to be drawn to satisfy
the  X-axis. This level will give us the worth of the higher quartile. In
the identical method,  different values of Q₁ and deciles and percentiles will be
decided.  

2.3.1 CHARACTERISTICS OF THE MEDIAN  

1. Not like the arithmetic imply, the median can
be computed from open-ended  distributions. It is because it’s positioned
within the median class-interval, which  wouldn’t be an open-ended
class.  

2. The median may also be decided
graphically whereas the arithmetic imply  can’t be ascertained on this
method.  

3. As it isn’t influenced by the acute
values, it’s most popular in case of a  distribution having excessive
values.  

4. In case of the qualitative information the place the
objects aren’t counted or measured however  are scored or ranked, it’s the
most acceptable measure of central tendency.  2.4 MODE  

The mode is one other measure of central
tendency. It’s the worth on the level round  which the objects are most
closely concentrated. For instance, contemplate the next  collection: 8,9,
11, 15, 16, 12, 15,3, 7, 15 

36 

There are ten observations within the collection
whereby the determine 15 happens most  variety of instances three. The mode is
subsequently 15. The collection given above is a discrete  collection; as such, the
variable can’t be in fraction. If the collection have been steady, we  might
say that the mode is roughly 15, with out additional computation.  

Within the case of grouped information, mode is
decided by the next system:  ^{− +}( ) ( ) _{1 0 1 2} 

^{f f ⋅} − + − 

1 0 

Mode= l₁ i 

f f f f 

The place, l₁ = the
decrease worth of the category wherein the mode lies  f_l = the frequency of the category wherein the mode
lies  

f_o = the frequency of the category previous the modal
class  

f₂ = the frequency of the category succeeding the modal
class  

i = the
class-interval of the modal class  

Whereas making use of the above system, we should always
make sure that the class-intervals are  uniform all through. If the
class-intervals aren’t uniform, then they need to be made  uniform on the
assumption that the frequencies are evenly distributed all through the 
class. Within the case of inequal class-intervals, the appliance of the above
system will  give deceptive outcomes.  

Instance 2.11: Allow us to
take the next frequency distribution:  

Class intervals (1) Frequency (2)  

30-40 4  

40-50 6  

50-60 8  

60-70 12  

70-80 9  

80-90 7  

90-100 4  

Now we have to calculate the mode in respect of this
collection.  

Answer: We will see from Column (2) of the desk that the utmost
frequency of  12 lies within the class-interval of 60-70. This implies that
the mode lies on this class interval. Making use of the system given earlier, we
get: 

37 

^{12 – 8}⋅ 

Mode = 60 + 10 

+ 

12 – 8 (12 – 8) (12 – 9) 

⁴⋅ 

= 60 + 10 

+ 

4 3 

 = 65.7 approx.  

In a number of instances, simply by inspection one can
determine the class-interval wherein the  mode lies. One ought to see which
the very best frequency is after which determine to which  class-interval this
frequency belongs. Having accomplished this, the system given for  calculating
the mode in a grouped frequency distribution will be utilized.  

At instances, it isn’t potential to determine by
inspection the category the place the mode lies. In  such instances, it turns into
crucial to make use of the tactic of grouping. This methodology consists  of two components:  

(i) Preparation of a grouping desk: A
grouping desk has six columns, the primary  column exhibiting the frequencies
as given in the issue. Column 2 exhibits  frequencies grouped in two’s,
ranging from the highest. Leaving the primary  frequency, column 3 exhibits
frequencies grouped in two’s. Column 4 exhibits the  frequencies of the primary
three objects, then second to fourth merchandise and so forth.  Column 5 leaves the
first frequency and teams the remaining objects in three’s.  Column 6
leaves the primary two frequencies after which teams the remaining in 
three’s. Now, the utmost whole in every column is marked and proven both 
in a circle or in a daring kind.  

(ii) Preparation of an evaluation desk:
After having ready a grouping desk, an  evaluation desk is ready. On
the left-hand aspect, present the primary column for  column numbers and on
the right-hand aspect the totally different potential values of  mode. The best
values marked within the grouping desk are proven right here by a  bar or by merely
getting into 1 within the related cell equivalent to the values 

38 

they symbolize. The final row of this desk
will present the variety of instances a  explicit worth has occurred within the
grouping desk. The best worth within the  evaluation desk will point out the
class-interval wherein the mode lies. The  process of making ready each
the grouping and evaluation tables to find the  modal class will probably be clear
by taking an instance.  

Instance 2.12: The
following desk offers some frequency information:  

Measurement of Merchandise Frequency  

10-20 10  

20-30 18  

30-40 25  

40-50 26  

50-60 17  

60-70 4  

Answer:  

 Grouping Desk  

Measurement of merchandise 1 2 3 4 5 6  

  

 10-20 10  

 28  

 20-30 18 53  

 43  

 30-40 25 69  

 51  

 40-50 26 68  

 43  

 50-60 17 47  

 21  

 60-70 4  

Evaluation desk  

  

Measurement of merchandise  

 Col. No. 10-20 20-30 30-40 40-50 50-60  

1 1  

2 1 1  

3 1 1 1 1  4 1 1 1  

5 1 1 1 

39 

6 1 1 1  

Complete 1 3 5 5 2    

This can be a bi-modal collection as is clear from
the evaluation desk, which exhibits that the  two lessons 30-40 and 40-50 have
occurred 5 instances every within the grouping. In such a  scenario, we could
have to find out mode not directly by making use of the next 
system:  

 Mode = 3 median – 2 imply  

Median = Measurement of (n + l)/2th merchandise,
that’s, 101/2 = 50.fifth merchandise. This lies within the class  30-40. Making use of the
system for the median, as given earlier, we get  

^{40 – 30} − 

= 30 + ^{(50.5 28)} 25 

 = 30 + 9 = 39  

Now, arithmetic imply is to be calculated. That is proven
within the following desk.  

Class- interval Frequency Mid- factors d d’ = d/10 fd’  

10-20 10 15 -20 -2 -20  

20-30 18 25 -10 -I -18  

30-40 25 35 0 0 0  

40-50 26 45 10 1 26  

50-60 17 55 20 2 34  

60-70 4 65 30 3 12  

Complete 100 34  

Deviation is taken from arbitrary imply = 35  

^fd_⋅ ∑ ^‘  

 Imply = A + i 

n 

³⁴⋅ 

 = 35 + 10 

100 

= 38.4  

Mode = 3 median – 2 imply  

= (3 x 39) – (2 x 38.4)  

= 117 -76.8 

40 

= 40.2  

This system, Mode = 3 Median-2 Imply, is an
empirical system solely. And it may well  give solely approximate outcomes. As
such, its frequent use needs to be averted. Nonetheless,  when mode is ailing
outlined or the collection is bimodal (as is the case within the current  instance)
it could be used.  

               2.5   RELATIONSHIPS OF THE MEAN, MEDIAN AND
MODE 

              Having
mentioned imply, median and mode, we now flip to the connection amongst 
these three measures of central tendency. We will focus on the connection
assuming  that there’s a unimodal frequency distribution.  

(i) When a distribution is symmetrical, the
imply, median and mode are the identical,  as is proven beneath within the following
determine.  

In case, a distribution is  skewed to the best, then  imply> median> mode.  

Typically, revenue distribution is skewed to the best the place a big
variety of households have comparatively  low revenue and a small variety of
households have extraordinarily excessive revenue. In  such a case, the imply is pulled
up by the acute excessive incomes and the  relation amongst these three
measures is as proven in Fig. Right here, we discover that  imply> median>
mode.  

(ii) When a distribution is skewed to  

 the left, then mode> median>  

 imply. It is because right here imply is  

 pulled down beneath the median  

 by extraordinarily low values. That is 

41 

 proven as within the determine.  

(iii) Given the imply and median of a unimodal
distribution, we will decide  whether or not it’s skewed to the  proper or left. When imply>  median, it’s skewed to the  proper; when median> imply, it  is skewed to the left. It could be famous that
the median is at all times within the center  between imply and mode.  

2.6 THE BEST MEASURE OF CENTRAL
TENDENCY  At
this stage, one could ask as to which of those three measures of central tendency
the  finest is. There isn’t any easy reply to this query. It’s as a result of
these three measures  are primarily based upon totally different ideas. The arithmetic
imply is the sum of the values  divided by the whole variety of observations
within the collection. The median is the worth of  the center statement that
divides the collection into two equal components. Mode is the worth  round which
the observations have a tendency to pay attention. As such, the usage of a selected 
measure will largely depend upon the aim of the examine and the character of the
information;  For instance, once we are fascinated about figuring out the customers
preferences for  totally different manufacturers of tv units or totally different sorts of
promoting, the selection ought to  go in favour of mode. Using imply and
median wouldn’t be correct. Nonetheless,  the median can generally be utilized in
the case of qualitative information when such information can  be organized in an
ascending or descending order. Allow us to take one other instance.  Suppose we
invite functions for a sure emptiness in our firm. A big quantity 
of candidates apply for that put up. We are actually to know as to which
age or  age group has the biggest focus of candidates. Right here,
clearly the mode will  be probably the most acceptable alternative. The arithmetic
imply might not be acceptable as it could 

42 

be influenced by some excessive values.
Nonetheless, the imply occurs to be probably the most  generally used measure of central
tendency as will probably be evident from the dialogue in  the following
chapters.  

2.7 GEOMETRIC MEAN  

Other than the three measures of central
tendency as mentioned above, there are two  different means which are used
generally in enterprise and economics. These are the  geometric imply and the
harmonic imply. The geometric imply is extra essential than  the harmonic
imply. We focus on beneath each these means. First, we take up the  geometric
imply. Geometric imply is outlined on the nth root of the product of n 
observations of a distribution.  

Symbolically, GM = …. ….. … _{1 2 n} n x x x If we’ve got solely two observations, say, 4
and  16 then GM = 4⋅16 = 64 = 8. Equally, if there are three
observations, then we  must calculate the dice root of the product of
these three observations; and so forth.  When the variety of objects is massive,
it turns into extraordinarily troublesome to multiply the  numbers and to calculate
the basis. To simplify calculations, logarithms are used.  

Instance 2.13: If we’ve got to seek out out the geometric imply of
2, 4 and eight, then we discover   ^{Log GM =} _n^x ∑ _i log 

Log2 + Log4 + Log8 

 ⁼ 3 

0.3010 + 0.6021+ 0.9031 

 ⁼ 3 

^1.8062 = 

 = 0.60206 

3 

 GM = Antilog 0.60206  

 = 4 

43 

When the info are given within the type of a
frequency distribution, then the geometric  imply will be obtained by the
system:  

+ + + 

f . x f . x … f . x _{l n n} 

log log log 

Log GM = _{f f fn} 

1 2 2 

∑ 

f x .log 

+ +
1 2 

………. 

= _{f f fn} 

₁ + ₂ + 

Then, GM = Antilog n  

………. 

The geometric imply is most fitted within the following
three instances:  

1. Averaging charges of change.  

2. The compound curiosity system.  

3. Discounting, capitalization.  

Instance 2.14: An individual has invested Rs 5,000 within the inventory
market. On the finish of the  first yr the quantity has grown to Rs 6,250; he
has had a 25 p.c revenue. If on the  finish of the second yr his
principal has grown to Rs 8,750, the speed of improve is 40  p.c for
the yr. What’s the common fee of improve of his funding throughout
the  two years?  

Answer:  

 GM = 1.25⋅1.40 = 1.75. =
1.323  

The typical fee of improve within the worth of
funding is subsequently 1.323 – 1 = 0.323,  which if multiplied by 100, offers
the speed of improve as 32.3 p.c.  

Instance 2.15: We will additionally derive a compound
curiosity system from the above set of  information. That is proven
beneath:  

Answer: Now, 1.25 x 1.40 = 1.75. This may be written as 1.75 = (1
+ 0.323)².  Let P₂ = 1.75, P₀ = 1, and r = 0.323, then the above equation will be
written as P₂ = (1  + r)² or P₂ = P₀ (1 + r)². 

44 

The place P2 is the worth of funding at
the tip of the second yr, P₀ is the preliminary  funding and r is the speed
of improve within the two years. This, in actual fact, is the acquainted  compound
curiosity system. This may be written in a generalised type as P_n = P₀(1 +  r)ⁿ. In our
case Po is Rs 5,000 and the speed of improve in funding is 32.3
p.c.  Allow us to apply this system to determine the worth of Pn, that is,
funding on the finish of  the second yr.  

 P_n = 5,000 (1 + 0.323)² 

 =
5,000 x 1.75  

 = Rs 8,750  

It could be famous that within the above instance, if
the arithmetic imply is used, the resultant  ^{25 + 40}p.c  

^{determine will probably be improper. On this case, the
common fee for the 2 years is} 2 ¹⁶⁵
x 5,000  

per yr, which involves 32.5. Making use of this fee, we get
P_n ⁼ 100 

 = Rs 8,250  

That is clearly improper, because the determine ought to have been
Rs 8,750.  

Instance 2.16: An economic system has grown at 5 p.c within the
first yr, 6 p.c within the  second yr, 4.5 p.c within the third yr,
3 p.c within the fourth yr and seven.5 p.c  within the fifth yr. What’s
the typical fee of progress of the economic system through the 5 
years?  

Answer:  

12 months Fee of Progress Worth on the finish of the
Log x   ( p.c) 12 months x (in Rs)  

1 5 105 2.02119  2 6 106 2.02531  3 4.5 104.5
2.01912  4 3 103 2.01284  5 7.5 107.5 2.03141   ∑ log X = 10.10987 

45 

_⎜^⎜_⎝^⎛∑_nlog
x 

GM = Antilog _⎟^⎟_⎠⎞ 

10.10987 

= Antilog ^⎟_⎠^{⎞ ⎜}_⎝^⎛5 

= Antilog 2.021974  

= 105.19  

Therefore, the typical fee of progress through the
five-year interval is 105.19 – 100 = 5.19  p.c every year. In case of a
easy arithmetic common, the corresponding fee of  progress would have
been 5.2 p.c every year.  

2.7.1 DISCOUNTING  

The compound curiosity system given above was  

P 

 P_n=P₀(1+r)ⁿ This may be written as P₀ = _n 

n 

_{(1+ )}  

r 

This can be expressed as follows:  

If the longer term revenue is P_n rupees
and the current fee of curiosity is 100 r p.c, then  the
current worth of P n rupees will probably be P₀ rupees. For instance, if we’ve got a
machine  that has a lifetime of 20 years and is predicted to yield a web revenue
of Rs 50,000 per yr,  and on the finish of 20 years it will likely be out of date and
can’t be used, then the machine’s  current worth is  

50,000 

50,000 

50,000 

50,000 

+ r⁺³ (1
) 

₊+² (1 ) ⁿ (1 r) 

+ r^{+……………..
20} (1 ) + r 

This strategy of ascertaining the current
worth of future revenue by utilizing the curiosity  fee is named
discounting. 

In conclusion, it could be mentioned that when there
are excessive values in a collection, geometric  imply needs to be used as it’s
a lot much less affected by such values. The arithmetic imply  in such instances will
give deceptive outcomes. 

46 

Earlier than we shut our dialogue on the
geometric imply, we should always concentrate on its  benefits and
limitations.  

2.7.2 ADVANTAGES OF G. M.  

1. Geometric imply relies on every
statement within the information set.  2. It’s rigidly outlined.  

3. It’s extra appropriate whereas averaging ratios
and percentages as additionally in calculating  progress charges.  

4. As in comparison with the arithmetic imply, it
offers extra weight to small values and  much less weight to massive values. As a
results of this attribute of the geometric  imply, it’s typically much less
than the arithmetic imply. At instances it could be equal to  the arithmetic
imply.  

5. It’s able to algebraic manipulation.
If the geometric imply has two or extra  collection is understood together with their
respective frequencies. Then a mixed  geometric imply will be calculated
by utilizing the logarithms.  

2.7.3 LIMITATIONS OF G.M.  

1. As in comparison with the arithmetic imply,
geometric imply is troublesome to  perceive.  

2. Each computation of the geometric imply and
its interpretation are somewhat  troublesome.  

3. When there’s a destructive merchandise in a collection
or a number of observations have  zero worth, then the geometric imply
can’t be calculated.  

In view of the restrictions talked about above,
the geometric imply just isn’t steadily  used.  

2.8 HARMONIC MEAN 

47 

The harmonic imply is outlined because the
reciprocal of the arithmetic imply of the  reciprocals of particular person
observations. Symbolically,  

_ciprocal ⁿ ₌ ∑ 

^HM=_nx 

1/ 

Re 

1/ x₁ 1/ x₂ 1/ x₃ … 1/ x_n 

+ + + + 

The calculation of harmonic imply turns into very
tedious when a distribution has a  massive variety of observations. Within the
case of grouped information, the harmonic imply is  calculated by utilizing the
following system:  

_{− ⎟}^⎟_⎠⎞ 

n 

 ^{HM = Reciprocal of} ∑

_⎜^⎜_⎝^⎛⋅ f 

1 

 or    

ⁱx 

ⁱ i 

1 

n 

_⎜^⎜_⎝^⎛⋅ 

_{− ⎟}^⎟_⎠⎞ 

n 

∑

f 

1 

ⁱx 

ⁱ i 

1 

The place n is the whole variety of
observations.  

Right here, every reciprocal of the unique determine
is weighted by the corresponding  frequency (f).  

The primary benefit of the harmonic
imply is that it’s primarily based on all observations in a  distribution and is
amenable to additional algebraic therapy. After we want to provide  better
weight to smaller observations and fewer weight to the bigger observations, then 
the usage of harmonic imply will probably be extra appropriate. As towards these benefits,
there  are sure limitations of the harmonic imply. First, it’s
obscure as effectively  as troublesome to compute. Second, it
can’t be calculated if any of the observations is  zero or destructive.
Third, it’s only a abstract determine, which might not be an precise 
statement within the distribution.  

It’s price noting that the harmonic imply is
at all times decrease than the geometric imply,  which is decrease than the arithmetic
imply. It is because the harmonic imply assigns 

48 

lesser significance to increased values. For the reason that
harmonic imply relies on reciprocals,  it turns into clear that as
reciprocals of upper values are decrease than these of decrease  values, it’s
a decrease common than the arithmetic imply in addition to the geometric imply.  Instance
2.17: Suppose we’ve got three observations 4, 8 and 16. We’re required
to  ^{calculate the harmonic imply. Reciprocals of 4,8 and 16 are:} ₄^1
,₈^{1 ,}₁₆¹
respectively  

n 

 Since HM = _{1/ x
1/ x 1/ x 1} + ₂ + ₃ 

3 

 = _{1/ 4
1/ 8 1/ 16} 

+ + 

3 

 = _{0.25 0.125 0.0625} 

+ + 

 = 6.857 approx.  

Instance 2.18: Contemplate
the next collection:  

Class-interval 2-4 4-6 6-8 8-10 
Frequency 20 40 30 10  

Answer:  

Allow us to arrange the desk as follows:  

Class-interval Mid-value Frequency Reciprocal of MV f x
1/x  

2-4 3 20 0.3333 6.6660  

4-6 5 40 0.2000 8.0000  

6-8 7 30 0.1429 4.2870  

8-10 9 10 0.1111 1.1111  

 Complete 20.0641  

_⎜^⎜_⎝^⎛⋅ 

_{− ⎟}^⎟_⎠⎞ 

n 

∑ i f 

1 

= _nx 

1 

ⁱ i 

¹⁰⁰ =
4.984 approx. 

⁼ 20.0641 

49 

Instance 2.19: In a small firm, two typists are employed.
Typist A sorts one  web page in ten minutes whereas typist B takes twenty
minutes for a similar. (i) Each are  requested to kind 10 pages. What’s the
common time taken for typing one web page? (ii) Each  are requested to kind for
one hour. What’s the common time taken by them for typing one 
web page?  

Answer: Right here Q-(i) is on arithmetic imply whereas Q-(ii) is on
harmonic imply.  (10 10) (20 20)(min ) 

⋅ + ⋅ 

 ^{(i) M =} 10 2( ) 

utes 

⋅ 

pages 

 = quarter-hour  ⋅ 

60 (min ) 

utes 

^{HM =} 60 /10
60 / 20( ) 

+ 

pages 

^{120 = =} ₊and 20 seconds.  

40 

= 13min utes 

120 60 20 

3 

Instance 2.20: It takes ship A ten days to cross the Pacific
Ocean; ship B takes 15  days and ship C takes 20 days. (i) What’s the
common variety of days taken by a ship  to cross the Pacific Ocean? (ii)
What’s the common variety of days taken by a cargo  to cross the Pacific
Ocean when the ships are employed for 60 days?  

Answer: Right here once more Q-(i) pertains to easy arithmetic imply whereas
Q-(ii) is  involved with the harmonic imply.  

10 +15 + ²⁰ = 15
days  

 ^{(i) M =} 3 

⋅ days 

60 3( ) _ 

^{(ii) HM =} 60 /10
60 /15 60 / 20 

+ + 

=  

180 

360 240 180 

+ +

60 

50 

= 13.8 days approx.  

2.9 QUADRATIC MEAN  

Now we have seen earlier that the geometric imply
is the antilogarithm of the arithmetic  imply of the logarithms, and the
harmonic imply is the reciprocal of the arithmetic  imply of the
reciprocals. Likewise, the quadratic imply (Q) is the sq. root of the 
arithmetic imply of the squares. Symbolically,  

2 

2 2 

₁ + + ……
+ 

x x _n 

2 

^{Q =} n 

As an alternative of utilizing authentic values, the
quadratic imply can be utilized whereas averaging  deviations when the usual
deviation is to be calculated. This will probably be used within the  subsequent chapter on
dispersion.  

2.9.1 Relative Place of Completely different Means  

The relative place of various means will at all times
be:  

Q> x >G>H supplied that each one the person observations
in a collection are optimistic and  all of them aren’t the identical.  

2.9.2 Composite Common or Common of Means  

Typically, we could must calculate an
common of a number of averages. In such instances,  we should always use the identical methodology
of averaging that was employed in calculating the  authentic averages.
Thus, we should always calculate the arithmetic imply of a number of values of  x, the
geometric imply of a number of values of GM, and the harmonic imply of a number of 
values of HM. It will likely be improper if we use another common in averaging of
means.  

2.10 SUMMARY  

It’s crucial goal of
statistical evaluation is to get one single worth that  describes the traits
of your complete mass of cumbersome information. Such a price is  discovering out, which
is named central worth to serve our goal. 

51 

2.11 SELF-TEST QUESTIONS  

1. What are the desiderata (necessities) of
a superb common? Evaluate the imply,  the median and the mode within the gentle of
these desiderata? Why averages are  referred to as measures of central
tendency?  

2. “Each common has its personal peculiar
traits. It’s troublesome to say which  common is one of the best.”
Clarify with examples.  

3. What do you perceive .by ‘Central
Tendency’? Below what circumstances is the  median extra appropriate than different
measures of central tendency?  

4. The typical month-to-month wage paid to all
staff in an organization was Rs 8,000.  The typical month-to-month salaries paid to
female and male staff of the  firm have been Rs 10,600 and Rs 7,500
respectively. Discover out the chances  of men and women employed by
the corporate.  

5. Calculate the arithmetic imply from the next
information:  

Class 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89  Frequency
2 4 9 11 12 6 4 2  6. Calculate the imply, median and mode from the
following information:   Peak in Inches Variety of Individuals  

62-63 2  

63-64 6  

64-65 14  

65-66 16  

66-67 8  

67-68 3  

68-69 1  

Complete 50  

7. Quite a lot of explicit articles have been
categorized in line with their weights.  After drying for 2 weeks, the
identical articles have once more been weighed and  equally categorized. It’s
identified that the median weight within the first weighing 

52 

was 20.83 gm whereas within the second weighing it
was 17.35 gm. Some  frequencies a and b within the first
weighing and x and y within the second are lacking.  It’s identified
that a = 1/3x and b = 1/2 y. Discover out the values of
the lacking  frequencies.  

Class Frequencies  

First Weighing Second Weighing  

0- 5 a z  

5-10 b y  

 10-15 11 40  

15-20 52 50  

20-25 75 30  

25-30 22 28  

8 Cities A, Band C are equidistant from every
different. A motorist travels from A to  B at 30 km/h; from B to C at 40 km/h
and from C to A at 50 km/h. Decide  his common pace for your complete
journey.  

9 Calculate the harmonic imply from the next
information:  

Class-Interval 2-4 4-6 6-8 8-10  Frequency 20 40
30 10  

10 A car when climbing up a gradient,
consumes petrol @ 8 km per litre.  Whereas coming down it runs 12 km per
litre. Discover its common consumption for  from side to side journey between two
locations located on the two ends of 25 Ian lengthy  gradient. 

53 

2.12 Relaxation Karlo Thoda

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COURSE: BUSINESS STATISTICS  

 

DISPERSION AND SKEWNESS 

OBJECTIVE: The
goal of the current lesson is to impart the data of  measures of
dispersion and skewness and to allow the scholars to  distinguish between
common, dispersion, skewness, moments and  kurtosis.  

STRUCTURE:  

3.1 Introduction  

3.2 That means and Definition of Dispersion  

3.3 Significance and Properties of Measuring
Variation  

3.4 Measures of Dispersion  

3.5 Vary  

3.6 Interquartile Vary or Quartile Deviation  

3.7 Imply Deviation  

3.8 Commonplace Deviation  

3.9 Lorenz Curve  

3.10 Skewness: That means and Definitions  

3.11 Checks of Skewness  

3.12 Measures of Skewness  

3.13 Moments  

3.14 Kurtosis  

3.15 Abstract  

3.16 Self-Take a look at Questions  

3.17 shock

3.1 INTRODUCTION  

Within the earlier chapter, we’ve got defined
the measures of central tendency. It could  be famous that these measures do
not point out the extent of dispersion or variability in a  distribution.
The dispersion or variability offers us another step in growing our 
understanding of the sample of the info. Additional, a excessive diploma of uniformity
(i.e. low  diploma of dispersion) is a fascinating high quality. If in a enterprise
there’s a excessive diploma of  variability within the uncooked materials, then it might
not discover mass manufacturing economical.  

55 

Suppose an investor is searching for an appropriate
fairness share for funding. Whereas  analyzing the motion of share
costs, he ought to keep away from these shares which are extremely  fluctuating-having
generally very excessive costs and at different instances going very low.  Such
excessive fluctuations imply that there’s a excessive threat within the funding in
shares.  The investor ought to, subsequently, favor these shares the place threat is
not so excessive.  

3.2 MEANING AND DEFINITIONS OF
DISPERSION  The
numerous measures of central worth give us one single determine that represents
the  total information. However the common alone can not adequately describe a set
of observations,  until all of the observations are the identical. It’s
crucial to explain the variability or  dispersion of the observations.
In two or extra distributions the central worth could also be  the identical however nonetheless
there will be large disparities within the formation of distribution.  Measures
of dispersion assist us in learning this essential attribute of a  distribution.  

Some essential definitions of dispersion are given
beneath:  

1. “Dispersion is the measure of the variation of
the objects.” -A.L. Bowley  2. “The diploma to which numerical information
are inclined to unfold about a mean worth is  referred to as the variation of dispersion
of the info.” -Spiegel  

3. Dispersion or unfold is the diploma of the scatter or
variation of the variable  a few central worth.” -Brooks &
Dick  4. “The measurement of the scatterness of the mass of figures
in a collection about an  

common known as measure of variation or
dispersion.” -Simpson & Kajka  It’s clear from above that
dispersion (often known as scatter, unfold or variation)  measures the
extent to which the objects differ from some central worth. Since measures  of
dispersion give a mean of the variations of assorted objects from an
common,  they’re additionally referred to as averages of the second order. A median is
extra significant  when it’s examined within the gentle of dispersion. For
instance, if the typical wage of the 

56 

staff of manufacturing unit A is Rs. 3885 and that of
manufacturing unit B Rs. 3900, we can not  essentially conclude that the employees of
manufacturing unit B are higher off as a result of in manufacturing unit B  there could also be a lot better
dispersion within the distribution of wages. The examine of  dispersion is of
nice significance in follow as might effectively be appreciated from the 
following instance:  

Collection A Collection B Collection C  

100 100 1  

100 105 489  

100 102 2  

100 103 3  

100 90 5  

Complete 500 500 500  

x ^{100
100 100}  

Since arithmetic imply is similar in all
three collection, one is prone to conclude that  these collection are alike
in  nature. However an in depth  examination shall reveal  that distributions differ  extensively from each other.  In collection A, (In Field-3.1)  every merchandise is  completely represented by the  arithmetic imply or in different  phrases not one of the objects of  collection A deviates from the 

57 

arithmetic imply and therefore there is no such thing as a
dispersion. In collection B, just one merchandise is  completely represented by the
arithmetic imply and the opposite objects differ however the variation  may be very small
as in comparison with collection C. In collection C. not a single merchandise is represented by 
the arithmetic imply and the objects differ extensively from each other. In collection
C,  dispersion is far better in comparison with collection B. Equally, we could
have two teams  of labourers with the identical imply wage and but their
distributions could differ extensively.  The imply wage might not be so essential
a attribute because the variation of the objects  from the imply. To the
scholar of social affairs the imply revenue just isn’t so vitally  essential as
to understand how this revenue is distributed. Are a big quantity receiving the 
imply revenue or are there just a few with huge incomes and thousands and thousands with incomes
far  beneath the imply? The three figures given in Field 3.1 symbolize
frequency distributions  with among the traits. The 2 curves
in diagram (a) symbolize two  distractions with the identical imply X ,
however with totally different dispersions. The 2 curves in  (b) symbolize two
distributions with the identical dispersion however with unequal means X _l and X
₂, (c) represents two distributions with unequal
dispersion. The measures of  central tendency are, subsequently inadequate.
They have to be supported and supplemented  with different measures.  

Within the current chapter, we will be
particularly involved with the measures of  variability or unfold or
dispersion. A measure of variation or dispersion is one which  measures the
extent to which there are variations between particular person statement and 
some central or common worth. In measuring variation we will be fascinated about
the  quantity of the variation or its diploma however not within the course. For
instance, a measure  of 6 inches beneath the imply has simply as a lot dispersion
as a measure of six inches  above the imply. 

58 

Actually that means of dispersion is
‘scatteredness’. Common or the measures of central  tendency offers us an
thought of the focus of the observations in regards to the central  a part of
the distribution. If we all know the typical alone, we can not type a whole
thought  in regards to the distribution. However with the assistance of dispersion, we’ve got an
thought about  homogeneity or heterogeneity of the distribution.  

3.3 SIGNIFICANCE AND PROPERTIES OF
MEASURING  VARIATION  

 Measures
of variation are wanted for 4 fundamental functions:  

1. Measures of variation level out as to how
far a mean is consultant of  the mass. When dispersion is small,
the typical is a typical worth within the sense  that it intently represents
the person worth and it’s dependable within the sense that  it’s a good
estimate of the typical within the corresponding universe. On the opposite  hand,
when dispersion is massive, the typical just isn’t so typical, and until the 
pattern may be very massive, the typical could also be fairly unreliable.  

2. One other goal of measuring dispersion is
to find out nature and reason behind  variation with a purpose to management the
variation itself. In issues of well being variations  in physique temperature,
pulse beat and blood strain are the fundamental guides to  prognosis.
Prescribed therapy is designed to manage their variation. In 
industrial manufacturing environment friendly operation requires management of high quality
variation  the causes of that are sought by means of inspection is fundamental to
the management of  causes of variation. In social sciences a particular drawback
requiring the  measurement of variability is the measurement of
“inequality” of the  distribution of revenue or wealth
and so forth.  

3. Measures of dispersion allow a comparability
to be product of two or extra collection  with regard to their variability. The
examine of variation may additionally be regarded 

59 

upon as a way of figuring out uniformity of
consistency. A excessive diploma of  variation would imply little uniformity or
consistency whereas a low diploma of  variation would imply nice uniformity
or consistency.  

4. Many highly effective analytical instruments in
statistics reminiscent of correlation evaluation. the  testing of speculation,
evaluation of variance, the statistical high quality management,  regression
evaluation relies on measures of variation of 1 variety or one other.  A very good
measure of dispersion ought to possess the next properties  

1. It needs to be easy to grasp.  

2. It needs to be simple to compute.  

3. It needs to be rigidly outlined.  

4. It needs to be primarily based on every merchandise of the
distribution.  

5. It needs to be amenable to additional algebraic
therapy.  

6. It ought to have sampling
stability.  

7. Excessive
objects mustn’t unduly have an effect on it. 

3.4 MEAURES OF DISPERSION  

There are 5 measures of dispersion: Vary,
Inter-quartile vary or Quartile  Deviation, Imply deviation, Commonplace
Deviation, and Lorenz curve. Amongst them, the  first 4 are mathematical
strategies and the final one is the graphical methodology. These  are mentioned in
the following paragraphs with appropriate examples.  

3.5 RANGE  

The best measure of dispersion is the
vary, which is the distinction between the  most worth and the minimal
worth of information.  

Instance 3.1: Discover
the vary for the next three units of information:  

Set 1: 05 15 15 05 15 05 15 15 15 15 
Set 2: 8 7 15 11 12 5 13 11 15 9 

60 

Set 3: 5 5 5 5 5 5 5 5 5 5  Answer: In every of those three units, the very best quantity is 15 and
the bottom quantity  is 5. For the reason that vary is the distinction between the
most worth and the minimal  worth of the info, it’s 10 in every case.
However the vary fails to provide any thought in regards to the  dispersal or unfold of the
collection between the very best and the bottom worth. This  turns into evident
from the above information.  

In a frequency distribution, vary is
calculated by taking the distinction between the  higher restrict of the
highest class and the decrease restrict of the bottom class.  Instance 3.2: Discover
the vary for the next frequency distribution:  

Measurement of Merchandise Frequency  

20- 40 7  

40- 60 11  

60- 80 30  

80-100 17  

100-120 5  

Complete 70  

Answer: Right here, the higher restrict of the very best class is 120 and the
decrease restrict of the  lowest class is 20. Therefore, the vary is 120 – 20 =
100. Observe that the vary just isn’t  influenced by the frequencies.
Symbolically, the vary is calculated b the system L –  S, the place L is the
largest worth and S is the smallest worth in a distribution. The 
coefficient of vary is calculated by the system: (L-S)/ (L+S). That is the
relative  measure. The coefficient of the vary in respect of the sooner
instance having three  units of information is: 0.5.The coefficient of vary is
extra acceptable for functions of  comparability as will probably be evident from the
following instance:  

Instance 3.3: Calculate the coefficient of vary individually
for the 2 units of information  given beneath:  

Set 1 8 10 20 9 15 10 13 28  Set 2 30 35
42 50 32 49 39 33 

61 

Answer: It may be seen that the vary in each the units of information is
the identical:  Set 1 28 – 8 = 20  

Set 2 50 – 30 = 20  

Coefficient of vary in Set 1 is:  

28 – 8 =
0.55  

28+8  

Coefficient of vary in set 2 is:  

50 – 30  50
+30  

= 0.25 

3.5.1 LIMITATIONS OF RANGE 

 There
are some limitations of vary, that are as follows:  

1. It’s primarily based solely on two objects and doesn’t
cowl all of the objects in a distribution.  2. It’s topic to large
fluctuations from pattern to pattern primarily based on the identical 
inhabitants.  

3. It fails to provide any thought in regards to the
sample of distribution. This was evident from  the info given in Examples
1 and three.  

4. Lastly, within the case of open-ended
distributions, it isn’t potential to compute  the vary.  

Regardless of these limitations of the vary, it’s
primarily utilized in conditions the place one desires  to shortly have some thought of
the variability or’ a set of information. When the pattern dimension is  very small,
the vary is taken into account fairly sufficient measure of the variability. Thus,
it  is extensively utilized in high quality management the place a steady test on the
variability of uncooked  supplies or completed merchandise is required. The vary is
additionally an appropriate measure in  climate forecast. The meteorological
division makes use of the vary by giving the  most and the minimal temperatures.
This info is kind of helpful to the  frequent man, as he can know the
extent of potential variation within the temperature on a  explicit
day.  

62 

3.6 INTERQUARTILE RANGE OR QUARTILE
DEVIATION  The
interquartile vary or the quartile deviation is a greater measure of variation
in a  distribution than the vary. Right here, avoiding the 25 p.c of the
distribution at each  the ends makes use of the center 50 p.c of the
distribution. In different phrases, the  interquartile vary denotes the distinction
between the third quartile and the primary  quartile.  

Symbolically, interquartile vary = Q₃– Q_l 

Many instances the interquartile vary is diminished
within the type of semi-interquartile vary  or quartile deviation as proven
beneath:  

Semi interquartile vary or Quartile deviation = (Q₃ – Q_l)/2  

When quartile deviation is small, it means
that there’s a small deviation within the central  50 p.c objects. In
distinction, if the quartile deviation is excessive, it exhibits that the central  50
p.c objects have a big variation. It could be famous that in a symmetrical 
distribution, the 2 quartiles, that’s, Q3 and QI are equidistant from the
median.  Symbolically,  

 M-Q_I = Q₃-M  

Nonetheless, that is seldom the case as most of
the enterprise and financial information are  asymmetrical. However, one can assume that
roughly 50 p.c of the observations  are contained within the
interquartile vary. It could be famous that interquartile vary or the 
quartile deviation is an absolute measure of dispersion. It may be became
a  relative measure of dispersion as follows:  

 Coefficient of QD
= 

Q₃ –Q₁ Q₃ +Q₁ 

The computation of a quartile deviation is
quite simple, involving the computation of  higher and decrease quartiles. As
the computation of the 2 quartiles has already been  defined within the
previous chapter, it isn’t tried right here.  

63 

 3.6.1 MERITS OF QUARTILE DEVIATION  

 The
following deserves are entertained by quartile deviation:  

1. As in comparison with vary, it’s thought of a
superior measure of dispersion.  2. Within the case of open-ended
distribution, it’s fairly appropriate.  

3. Since it isn’t influenced by the acute
values in a distribution, it’s  notably appropriate in extremely skewed or
erratic distributions.  

3.6.2 LIMITATIONS OF QUARTILE DEVIATION 

1. Just like the vary, it fails to cowl all of the objects in a
distribution.  

2. It isn’t amenable to mathematical
manipulation.  

3. It varies extensively from pattern to pattern
primarily based on the identical inhabitants.  4. Since it’s a positional common, it’s
not thought of as a measure of dispersion.  It merely exhibits a distance on
scale and never a scatter round a mean.  In view of the above-mentioned
limitations, the interquartile vary or the quartile  deviation has a
restricted sensible utility.  

3.7 MEAN DEVIATION  

The imply deviation is often known as the
common deviation. Because the identify implies, it’s  the typical of absolute
quantities by which the person objects deviate from the imply.  For the reason that
optimistic deviations from the imply are equal to the destructive deviations, 
whereas computing the imply deviation, we ignore optimistic and destructive indicators. 
Symbolically,  

∑^{| x |} The place MD = imply deviation, |x| = deviation of an
merchandise  MD = _n 

from the imply ignoring optimistic and destructive
indicators, n = the whole variety of  observations. 

64 

 Instance 3.4:  

Measurement of Merchandise Frequency  

2-4 20  

4-6 40  

6-8 30  

8-10 10  

Answer: 

^{Measurement of Merchandise Mid-points (m) Frequency (f) fm} d from x ^{f |d|}  

2-4 3 20 60 -2.6 52  

4-6 5 40 200 -0.6 24  

6-8 7 30 210 1.4 42  

8-10 9 10 90 3.4 34  

Complete 100 560 152
 

⁵⁶⁰ _{= =} ∑_nfm 

x = 5.6 

100 

^{| | 152} _{= =} ∑_nf
d 

 MD ( x
) = 1.52 

100 

3.7.1 MERITS OF MEAN DEVIATION 

1. A serious benefit of imply deviation is
that it’s easy to grasp and  simple to calculate.  

2. It takes into consideration every
merchandise within the distribution. Consequently,  a change within the worth of any merchandise
may have its impact on the magnitude of imply  deviation.  

3. The values of maximum objects have much less
impact on the worth of the imply  deviation.  

4. As deviations are taken from a central
worth, it’s potential to have significant  comparisons of the formation of
totally different distributions.  

3.7.2 LIMITATIONS OF MEAN DEVIATION 

1. It isn’t able to additional algebraic
therapy. 

65 

2. At instances it could fail to provide correct
outcomes. The imply deviation offers finest  outcomes when deviations are taken
from the median as an alternative of from the imply.  However in a collection, which has large
variations within the objects, median just isn’t a  passable measure.  

3. Strictly on mathematical concerns,
the tactic is improper because it ignores the  algebraic indicators when the deviations
are taken from the imply.  

In view of those limitations, it’s seldom
utilized in enterprise research. A greater measure  referred to as the usual
deviation is extra steadily used.  

3.8 STANDARD DEVIATION  

The usual deviation is much like the imply
deviation in that right here too the deviations  are measured from the imply. At
the identical time, the usual deviation is most popular to  the imply deviation
or the quartile deviation or the vary as a result of it has fascinating 
mathematical properties.  

Earlier than defining the idea of the usual
deviation, we introduce one other idea  viz. variance.  

Instance 3.5:  

X X-μ (X-μ)² 

20 20-18=12 4  

15 15-18= -3 9  

19 19-18 = 1 1  

24 24-18 = 6 36  

16 16-18 = -2 4  

14 14-18 = -4 16  

108 Complete 70  

Answer:  

^{Imply =} ₆¹⁰⁸ =
18 

66 

The second column exhibits the deviations from
the imply. The third or the final column  exhibits the squared deviations, the
sum of which is 70. The arithmetic imply of the  squared deviations
is:  

∑ ^{x − 2} 

 ( ) 

N 

^μ = 70/6=11.67 approx.  

This imply of the squared deviations is understood
because the variance. It could be famous that  this variance is described by
totally different phrases which are used interchangeably: the  variance of the
distribution X; the variance of X; the variance of the distribution; and  simply
merely, the variance.  

∑ ^{x − 2} 

_{Symbolically, Var (X) =} ( ) 

μ 

N 

^x ∑ _i ⁻ =² 

_{Additionally it is written as} ( ) 

σ 

₂ μ N 

The place σ² (referred to as sigma squared) is used to indicate the
variance.  

Though the variance is a measure of
dispersion, the unit of its measurement is  (factors). If a distribution
pertains to revenue of households then the variance is (Rs)² and 
not rupees. Equally, if one other distribution pertains to marks of scholars,
then the  unit of variance is (marks)². To
overcome this inadequacy, the sq. root of variance  is taken, which
yields a greater measure of dispersion referred to as the usual deviation. 
Taking our earlier instance of particular person observations, we take the sq. root
of the  variance  

 SD or σ = Variance = 11
= 3.42 factors .67 

^x ∑ _i − ² μ 

Symbolically,