Full Quantitative Techniques-I

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Quantitative Techniques-1

Prepared By: Kapil Verma 1


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Quantitative Technique

Quantitative Technique is a scientificapproach to managerial decision-making.

The successful use of Quantitative

Technique for management would help theorganization in solving complex problemson time, with greater accuracy and in themost economical way.



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BROAD CLASSIFICATION

Prepared By: Kapil Verma

QUANTITATIVETECHNIQUES

STATISTICALTECHNIQUES

OPERATION RESEARCH(OR PROGRAMMING)

TECHNIQUES

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STATISTICAL TECHNIQUES

Methods of collecting Data

Classification and tabulation of collecteddata

Probability theory and sampling analysis. Correlation and Regression Analysis

Index Numbers

Time Series Analysis Ratio Analysis



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OPERATION RESEARCH (OR

PROGRAMMING) TECHNIQUES Linear Programming Decision Theory

Theory of Games

Queuing Theory



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Statistics: What?

Conveys a variety of meaning

The term statistics refers to numerical factssuch as averages, medians, percents, and

index numbers that help us understand avariety of business and economic conditions.

Tables, Charts and Figures, commonly found in

newspaper, books, reports, classroom lectures.



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Statistics: What?

Definition of Statistics

E.g: LPU


C Collecting

O Organizing

D Displaying

I Interpreting

A Analyzing

Data

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What is Data?

Data is the plural ofDatum (Latin forgiven)

It is the generic term for numericalinformation that has been obtained on aset of objects/individuals etc.

The objects can be anything people,animals, etc.



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What is Data?

Variable: Some characteristic of the objects/individuals (e.g., height)

Can take on different values (e.g., 51 , 56 , 62). Data:

the values of a variable for a certain set of objects/individuals

(e.g., the height values of all the players on the basketball

team)



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Statistics: What?

Statistics Math (Dont Panic)

Statistics = Fundamental tool for all scientific inquiry

Way of making sense out of data



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Limitations of Statistics

Statistics does not deal with IndividualMeasurements.

Data are statistical when they relate to

measurement of masses, not statisticalwhen they relate to an individual.

Statistics deals only with Quantitative

Characteristics. Such characteristics as cannot be

expressed in numbers are incapable ofstatistical analysis: Honesty, Efficiency,Intelligence. Prepared By: Kapil Verma 19


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Statistical Results are true only on anAverage.

The Conclusions obtained statistically are

not universally true-they are true onlyunder certain conditions.

Statistics is only one of the Method of

studying a Problem. Do not provide the best solution under all

circumstances.



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Statistics can be Misused.

Based on incomplete information, one mayarrive fallacious conclusions. E.g:

Can be moulded in any manner so as toestablish wrong or write conclusions.

It requires experience and skill to drawsensible conclusions.


Wrong presentation may misled the

reader.

E.g: comparison of profits of two firms.

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APPLICATIONS OF STATISTICS IN BUSINESS &

MANAGEMENT

MANAGEMENT

i) Marketing:

Selection of product mix

Sales resources allocation

Analysis market research information

Sales forecasting

ii) Production

Production planning, control and analysis

Evaluation of machine performance Quality control requirements

Inventory control measures



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iii) Finance, Accounting and Investment:

Financial forecast

Budget preparation

Cash flow analysis

Capital budgeting

Dividend and Portfolio management Financial planning

iv) Personnel

Labour turnover rate Employment trends

Performance appraisal

Wage rates and incentive plans



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ECONOMICS

Measure of GNP.

Determination of business cycle

Comparison of market prices

Analysis of population

Formulation of appropriate economicpolicies

RESEARCH AND DEVELOPMENT

Development of new product lines Optimum use of resources

Evaluation of existing products



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What Is a Survey?

A survey is a series of questionsasked of a group of people in order togain information

Information gathered can be facts,attitudes, feelings, beliefs



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Survey

Marketing: A detailed study ofa market or geographical area togather data on attitudes, impressions,

opinions, satisfaction level, etc., bypolling a section of the population.

To examine as to condition, situation,or value :

To query (someone) in order to collectdata for the analysis of some aspectof a group or area



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Examples of SURVEY

A total of 250 city residentswere surveyedabout the project.

64 percent of the people surveyedsaid thatthe economy was doing well.



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Why Do a Survey?

The goal of any survey is to collectdata which can be analyzed, andused to aid decisions.

Prepared By: Kapil Verma28


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Planning the Survey

Doing a survey requires planningplanning what you want to ask; how toask it; how many people to survey and

how to reach them, either by mail, inperson, or by telephone. Will you haveteam members asking the questions orwill the respondents, those completing

the survey, fill it out themselves? Thisguide addresses these issues step-by-step, so that you can make choices andinitiate a survey as systematically as

possible. Prepared By: Kapil Verma 29


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Example 1: RSVP - Planning



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Example 1: RSVP - PlanningObjectives:

Whatinformation do you need todefine?

How many will attend the RetirementRecognition and how many of this groupwill eat lunch.

What decisions will this information impact?

How much space is needed for theRetirement Recognition and how manylunches are needed.

Who is the audience for the survey?

Everybody in Extension

Who is the audience for the report?

In what format is the report needed?

Excel spreadsheet with names andnumbers or head count

How will you deliver the survey to youraudience?

Email



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Planning a Survey

Deciding on a research question

Choosing the format of yourquestions/Choosing the format of your

interview--if you use an interview Editing your questions

Sequencing your questions

Refining your survey instrument Choosing a sampling strategy



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Choosing the Format of YourQuestions

Fixed alternative

Yes/No Reliable

Not powerful Scale

Open-ended

May not be properly answered

May be difficult to score



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Choosing the Format of YourInterview

Unstructured Interviewer bias is a serious problem

Data may not be hard to analyze

Semi-structured Follow-up questions allowed

Probably best for pilot studies

Structured Standardized, reducing interviewer bias



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Editing Questions: Mistakes toAvoid


1. Avoid leadingquestions

You were at Duffy's bar on the night of July

15, weren't you?

2. Avoid double-barreledquestions

Do you think professors should havemore contact with university staff and

university administrators?

3. Avoid long questions

4. Avoid negations

5. Avoid irrelevantquestions

6. Avoid poorlyworded responseoptions

7. Avoid big words

8. Avoid ambiguouswords & phrases

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Sequencing Questions

To boost response rate, put innocuousquestions first, personal questions last

To increase accuracy, keep similarquestions together

To boost response rate, putdemographic questions last



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Putting the Final Touches onYour Survey Instrument

Professional appearance

Proof reading

Practice coding responses--may leadto refining questionnaire so that it iseasier to code responses



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Choosing a SamplingStrategy

Random sampling

Proportionate stratified random sampling

Convenience sampling, etc



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Data Collection

Data collection is the act of assemblingand gathering the needed information inthe context of a specified research

investigation. Primary Data

Secondary Data



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Secondary Data

Data gathered by another source (e.g.research study, survey, interview)

Secondary data is gathered BEFORE primarydata. WHY?

Because you want to find out what is alreadyknown about a subject before you dive intoyour own investigation.

Because some of your questions can

possibly have been already answered byother investigators or authors. Why reinventthe wheel?



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Primary Data Data never gathered before

Advantage: find data you need to suit

your purpose Disadvantage: usually more costly

and time consuming than collectingsecondary data

Collected after secondary data iscollected



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Issues in Secondary DataCollection Identifying the sources of data, various

publications where the required data may befound available.

Examining the available data, if theses satisfythe needs of the proposed researchinvestigation.

Compiling and recognizing the available data

in the manner necessary for the investigationat hand.



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Characteristics of Secondary Data

Readymade and readily available

No original control

Limited in time and space. That is, theresearcher using them need not have beenpresent when and where they were

gathered.



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Advantages of Secondary Data

Secondary data, if available, can be securedquickly and cheaply.

Wider geographical area and longer referenceperiod may be covered without much cost.Thus the use of secondary data extends theresearcher's space and time reach.

The use of secondary data enables a

researcher to verify the findings based onprimary data.


i d li i i


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Disadvantages/limitations ofSecondary Data

The most important limitation is the availabledata may not meet, our specific researchneeds.

The available data may not be as accurate asdesired.

The secondary data are not up-to-date andbecome obsolete when they appear in print,

because of time lag in producing them.



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Sources of Secondary Data Internal Sources of secondary data

Accounting records

Sales force report

Miscellaneous report: Previous marketing research studies,special audit reports, reports purchased from outside.

Internal Experts External Sources of Secondary data

Computerized databases

Associations

Government Agencies Syndicated Services: Commercial organizations

Other Published sources: Books, Dissertations, News Paper

External Experts.



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Classification of Data

The process of arranging the data in groups orclasses according to their commoncharacteristics is technically known asclassification.

Or"Classification is the process of arranging data

into sequences and groups according to theircommon characteristics or separating them into

different but related parts. It is the first step in tabulation.

Classification is the grouping of related facts intoclasses.



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Classification of Data

Why Classification of Data isimportant??

Raw Data are so voluminous and huge.

After collection of data next step is toorganize, so as to present it in a

manner to highlight the importantcharacteristics of the data.



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Example: Classification of Data

The process of sorting letters in a postoffice, the letters are classified accordingto the cities and further arrangedaccording to streets.

Number of students registered for LPU in2011 may be classified on the basis ofany of the following criterion: Sex

Age Country to which they belong Region Course(B-Tech, BA, B-Com, MBA)



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Functions of Classification

It Condenses the data: Present thehuge raw data into condensed form, Highlightthe significant features contained in the data.

It facilitates comparisons.

It helps to study relationship.

It facilitates the statistical treatment of the

data: Makes the more useful, intelligible.



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Bases of Classification

There are four important bases ofclassification:

(1) Qualitative Base

(2) Quantitative Base

(3)Geographical Base

(4) Chronological or Temporal Base



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(1) Qualitative Base:When the data are classified

according to some quality or attributes

such as sex, religion, literacy,intelligence etc

(2) Quantitative Base:When the data are classified by

quantitative characteristics likeheights, weights, ages, income etc



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(3) Geographical Base:When the data are classified by

geographical regions or location, like

states, provinces, cities, countriesetc

(4) Chronological or TemporalBase:

When the data are classified orarranged by their time of occurrence,

such as years, months, weeks, daysPrepared By: Kapil Verma53


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Types of Classification:

(1) One -way Classification:If we classify observed data keeping in

view single characteristic, this type ofclassification is known as one-way classification.

For Example: The population of world may beclassified by religion as Muslim, Christians etc

(2) Two -way Classification:If we consider two characteristics at a

time in order to classify the observed data thenwe are doing two way classifications.For Example: The population of world may beclassified by Religion and Sex.



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Types of Classification:

(3) Multi -way Classification:We may consider more than

two characteristics at a time to classify

given data or observed data. In thisway we deal in multi-wayclassification.

For Example: The population of worldmay be classified by Religion, Sexand Literacy.



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Tabulation of data

It is cumbersome to study or interpretlarge data without grouping it, even ifit is arranged sequentially. For this,

the data are usually organized intogroups called classes and presentedin a table which gives the frequencyin each group. Such a frequency

table gives a better overall view of thedistribution of data and enables aperson to rapidly comprehendimportant characteristics of the data



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For example, a test of 50 marks isadministered on a class of 40 students andthe marks obtained by these students are as

listed below.


B i th h th k f 40 t d t li t d


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By going through the marks of 40 students listedin Table 12.5, you may be able to see that themarks vary from 16 to 48, but if you try to

comprehend the overall performance it is a difficultproposition.

Now consider the same set of marks, arranged in atabular form



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Types of Table

Single-column or single-row table Multiple-column and multiple-row

tables

Reference vs. summary tables



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Components of Table

Table number Title of table

Head Note

Stub and Stub-Heads: Main headings of rows Box Head and Stub-Heads: Data provided in

various columns

Body of the Table: r*c

Footnote:

Sources



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Contingency table

Cross-section presentation of observeddata in terms of any two attributes.

Last column provides row table, last row

gives column total.



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Frequency Distribution

A frequency distribution is anarrangement of the values that one or morevariables take in a sample.

The frequency is the number of values in aspecific class of data.

The researches organizes the raw data byusing frequency distribution.

A frequency distribution is the organizing ofraw data in table form, using classes andfrequencies.



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100 people rate a five-point Likert scale assessing theiragreement with a statement on a scale on which 1

denotes strong agreement and 5 strong disagreement, thefrequency distribution of their responses might look like



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Cumulative Frequency

The cumulative frequency is the sumof the frequencies accumulated up tothe upper boundary of a class in the

distribution. They are used to visually represent

how many values are below a certain

upper class boundary.


Example of Cumulative Frequency


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Example of Cumulative FrequencyDistribution


Methods of constructing


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Methods of constructingFrequency Distribution

Tally Method:

Entry Form Method


Class Tally Frequency

A ///// 5

B ///// // 7

O ///// //// 9

AB ///// 4

Total 25

67

Concerns in constructing a


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Concerns in constructing aFrequency Distribution

Number of Classes: 5-15 Width of class Interval(s): 5,10, 15 and should

remain constant

Establishing the Initial Class



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Histogram

The histogram is a graph that displays the data by usingvertical bars of various heights to represent the frequencies.

In statistics, a histogram is a graphical representationshowing a visual impression of the distribution of data. It is anestimate of the probability distribution of a continuousvariable and was first introduced by Karl Pearson. Ahistogram consists of tabular frequencies, shown asadjacent rectangles, erected over discrete intervals (bins),with an area equal to the frequency of the observations in the

interval.


C t hi t f th


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Create a histogram for thefollowing data:

5, 6, 4, 7, 5, 9, 11, 12, 4, 5, 6, 7, 9,19



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Pie Chart

A pie chart is a circle that is dividedinto sections according to thepercentage of frequencies in each

category of the distribution.



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Example



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Example Cont.

5%3%8%

51%

33% ConvertiblesStation wagons

Compacts

Coupes

Sedans



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Bar Chart

A bar chart is a broader concept thanhistogram.

A bar chart may be used to display

concepts other than frequency of anobservations. For example, a barchart may display the average examresults.

Histogram is a bar chart of frequencydistribution.


FREQUENCY BAR CHART (cont )


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FREQUENCY BAR CHART (cont.)

Prepared By: KapilVerma 75


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Cumulative Frequency Curve or Ogive

For plotting a cumulative frequencycurve or Ogive, first of all cumulativefrequencies against each of the

intervals are to be written. If we takethe frequency distribution of Table itwill be like



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f C


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Measures of Central Tendency

One of the most important objectives ofstatistical analysis is to get one singlevalue that describes the characteristics ofthe entire mass of the unwieldy data.Such a value is called the central value oran average or the expected value of thevariable.


Functions or objectives of an


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Functions or objectives of an

Average

It facilitates quick understanding ofcomplex data.

It facilitates comparison

To know about the universe from thesample

To get the single value that describes

the characteristic of the entire group.


T f A


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Types of Averages

Arithmetic Mean

Median

Mode

Geometric Mean

Harmonic Mean

Percentile Methods

Decile

Quartile


A ith ti M


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Arithmetic Mean


Arithmetic Mean


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Seventy efficiency apartments were randomlysampled in a small college town. The monthly rent

prices for these apartments are given below.

Example: Apartment Rents

445 615 430 590 435 600 460 600 440 615

440 440 440 525 425 445 575 445 450 450

465 450 525 450 450 460 435 460 465 480

450 470 490 472 475 475 500 480 570 465

600 485 580 470 490 500 549 500 500 480

570 515 450 445 525 535 475 550 480 510

510 575 490 435 600 435 445 435 430 440


Arithmetic Mean


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Arithmetic Mean

445 615 430 590 435 600 460 600 440 615

440 440 440 525 425 445 575 445 450 450

465 450 525 450 450 460 435 460 465 480

450 470 490 472 475 475 500 480 570 465

600 485 580 470 490 500 549 500 500 480

570 515 450 445 525 535 475 550 480 510

510 575 490 435 600 435 445 435 430 440

34,356490.80

70

ix

x

n



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Marks 20 30 40 50 60 70

No of

students

8 12 20 10 6 4


Example: From the following data of the marksobtained by 60 students of a class.

86

Arithmetic Mean-Discrete


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Series

Marks (x) No. of students(f) fx

20

30

40

50

60

70

8

12

20

10

6

4

160

360

800

500

360

280

N=60 2,460


fx

4160

2460 x

87


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There are three different basketball teams andeach has played five games. You have eachteam's score from each of its games.

Suppose you want to join one of the threebasketball teams. You want to join the one that isdoing the best so far. If you rank each team bytheir mean scores, which team would you join?


Class Interval Arithmetic


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Mean

Class Interval Arithmetic Mean :Arithmetic Mean = fX/fwhere

X = Midpointf = Frequency


Find the Arithmetic Mean of


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Find the Arithmetic Mean of



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Step 1: Find f.f = 17

Step 2:Then, Find the Midpoint for

the class interval. Midpoint(X) = (10+20)/2, (20+30)/2,

(30+40)/2 = 15, 25, 35

Step 3: Now, Find fX.

fX =((3*15)+(9*25)+(5*35)) = (45+225+175) = 445



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Step 4:Now, Substitute in the aboveformula given.

Arithmetic mean = fX/f =445/17 = 26.1765


Find the Mean for Grouped


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pData

Time (Days) Frequency10-14 4

15-19 8

20-24 5

25-29 2

30-34 1



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mean = fX/f

=380/20

=19 days


Problem


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A service station recorded the followingfrequency distribution for the number of

gallons of gasoline sold per car in a sample of689 cars.

Compute the mean

Answer: 10.74

Gasoline(Gallons) Frequency0-4 74

5-9 19210-14 280

15-19 105

20-24 23

25-29 6


Weighted arithmetic mean


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In calculating simple arithmetic mean, it isassumed that all the items in the seriescarry equal importance.

But in practice, there are many cases where

relative importance should be given todifferent items.




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Example:

A student obtained 40, 50, 60, 80,and 45 marks in the subjects of Math,Statistics, Physics, Chemistry and

Biology respectively. Assuming weights5, 2, 4, 3, and 1 respectively for theabove mentioned subjects. FindWeighted Arithmetic Mean per subject.



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Combined Mean


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Combined Mean

The arithmetic mean of several sets of data maybe combined into a single arithmetic mean for thecombined sets of data.

If a sample size of 22 items has a mean of 15and another sample size of 18 items has amean of 20. Find the mean of the combinedsample?


Correcting Incorrect Values


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Correcting Incorrect Values

The mean marks of 100 students werefound to be 40. Later on, it was discoveredthat a score of 53 was misread as 83. Findthe correct mean corresponding to the

correct score.

Mean of 100 observations is found to be 40.

If at the time of computation two items aretaken as 30 and 27 instead of 3 and 72, findthe correct mean.


Finding Missing Frequencies


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No. of

Accidents

0 1 2 3 4 5

Frequency 46 ? ? 25 10 5


For a distribution based on 200 observations partlyreproduces below, mean is 1.46 .Find the missingfrequencies

102



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Ans: f1=76, f2=38


Merits and Limitations of AM


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Merits and Limitations of AM

Merits Simplest and easiest to compute.

Arraying of data is not required.

Affected by the value of every item in the series

It is a calculated value, and not based onposition in the series.

Limitations May affected by extreme values

Distribution with open-end classes the mean cannot becomputed.


Geometric Mean


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Geometric Mean

In many business and economicsproblems, we deal with quantities thatchange over a period of time. In such

case the aim is to know an averagerate of change rather than simpleaverage value to represent theaverage growth or decline rate in a

data set over a period of time. Thuswe need to calculate another measureof central tendency, called geometricmean(G.M).


Geometric Mean


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Geometric Mean

E.g: we have two observations, say, 4and 16 then GM= ??


Difference between A.M & G.M


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The arithmetic mean is relevant any time several quantities addtogether to produce a total. The arithmetic mean answers the

question, "if all the quantities had the same value, what would thatvalue have to be in order to achieve the same total?

In the same way, the geometric mean is relevant any time severalquantities multiply together to produce a product. The geometric meananswers the question, "if all the quantities had the same value, what

would that value have to be in order to achieve the same product?" For example, suppose you have an investment which earns 10% the

first year, 50% the second year, and 30% the third year. What is itsaverage rate of return? It is not the arithmetic mean, because whatthese numbers mean is that on the first year your investment

was multiplied (not added to) by 1.10, on the second year it wasmultiplied by 1.60, and the third year it was multiplied by 1.20. Therelevant quantity is the geometric mean of these three numbers.


Merits and Limitations of


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Geometric Mean

Merits: Used to find the average per cent increase in sales,

production, population or other economic or businessseries.

Considered best average in construction of index

number

LimitationsDifficult to compute and interpret.

Can not be computed if one or more of the values are

zero.



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Year Growth Rate(%) Output at the end ofthe year

1998 5.0 105

1999 7.5 112.87

2000 2.5 115.69

2001 5.0 121.47

2002 10.0 133.61

The Simple arithmetic mean of the growth rate is, 6%.

GM=???


Example: Geometric Mean


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Example: Geometric Mean

A person has invested Rs 5000 in the stockmarket. At the end of the first year theamount has grown to Rs 6250; he has had a25% profit. If at the end of second year his

principal has grown to Rs 8750, the rate ofincrease is 40% for the year. What is theaverage rate of increase if his investmentduring the two years?

GM= sq of 1.25*1.40= 1.323 The average rate in the value of investment

is therefore 1.323-1=0.323, which ismultiplied by 100,gives the rate if increase as


Harmonic Mean


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Harmonic Mean

Harmonic mean (formerly sometimescalled the subcontrary mean) is one ofseveral kinds of average.

HM is particularly useful in averaging ratesand ratios, It is the most appropriateaverage where unit of observation(such asper day, per hour, per unit, per worker etc)

remains the same and the act beingperformed, such as covering distance, isconstant


Harmonic Mean Group Data


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Harmonic Mean Group Data

The harmonic mean Hof the positivereal numbers x1,x2, ..., xn is defined tobe

i

i

xf

nH

ix

nH

1

Ungroup Data Group Data


Relationship between AM, GM,


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HM

AM>GM>HMIf Computed for the same data



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Harmonic Mean =N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)where

X = Individual scoreN = Sample size (Number of

scores)



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Example: To find the Harmonic Mean of1,2,3,4,5.

Step 1: Calculate the total number ofvalues.

N = 5

Step 2: Now find Harmonic Mean using theabove formula.

N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)

= 5/(1/1+1/2+1/3+1/4+1/5)= 5/(1+0.5+0.33+0.25+0.2)= 5/2.28So, Harmonic Mean = 2.19


Harmonic Mean: cont


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Harmonic Mean: cont

Assume that a investor buys Rs 1200worth share of a company eachmonth. If he has bought shares at aprice of Rs 10, Rs 12, Rs 15, Rs, 20and Rs 24 per share during each ofthe first five month of the year. Findthe average share price.

Ans: 14.63


Weighted Harmonic Mean


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Weighted Harmonic Mean

Example: A computer operator who boughtthree different brands of computer printoutpaper A, B and C at the rate of Rs. 110, Rs.120, and Rs. 140 per packet of 100 pieces,

respectively. Having spent Rs. 320 on brand A,Rs. 480 on brand B and Rs. 280 on band C.He may be interested to know the averageprice of the three brands of paper.

Solution: w=330+480+280= 1090

w/x= 330/110 +480/120 +280/140 = 9

H M=1090/90= Rs 121 11(per packet)Prepared By: Kapil Verma 117

Example: H.M


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Example: H.M

A cyclist covers his first five km; at anaverage speed of 10 km/h. another 3 kmat 8km/h and the last two km. at 5 km findaverage speed

Ans: 7.85



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Median


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Median

Whenever a data set has extreme values, the medianis the preferred measure of central location.

A few extremely large incomes or property valuescan inflate the mean.

The median of a data set is the value in the middle

when the data items are arranged in ascending order.


Advantages and Disadvantagesf M di


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of Median

Advantages: Extreme values do not affect the median as strongly

as they do the mean.

Can be calculated with qualitative data

Disadvantages: We can perform any calculation only after making it

in order, may be a time consuming task


Median


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12 14 19 26 2718 27

For an odd number of observations:

in ascending order

26 18 27 12 14 27 19 7 observations

the median is the middle value.

Median = 19


Median


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12 14 19 26 2718 27

For an even number of observations:

in ascending order

26 18 27 12 14 27 30 8 observations

the median is the average of the middle two values.

Median = (19 + 26)/2 = 22.5

19

30


Median


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Averaging the 35th and 36th data values:

Median = (475 + 475)/2 = 475

Note: Data is in ascending order.

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615


Median for Grouped Data


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p



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Mode


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The mode of a data set is the value that occurs with

greatest frequency. The greatest frequency can occur at two or more

different values.

If the data have exactly two modes, the data are

bimodal. If the data have more than two modes, the data are

multimodal.

Use????


Mode.. Contd


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Advantages: Can be used with quantitative as well as with

qualitative data.

Not unduly affected by extreme values.

We can use mode even when one or moreclasses are open

Disadvantages: What if there is no modal value.

It is difficult to interpret if data set contain twoor more modal values.



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Quartile


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The values of observations in a data set,when arranged in an ordered sequence,can be divided into four equal parts, orquarters, using three quartiles namelyQ1, Q2, and Q3.

The first quartile Q1 divides a distributionin such a way that 25% of observations

have values less than Q1 and 75% havea value more than Q1.



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Deciles:


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The values of observations in a dataset when arranged in an orderedsequence can be divided into ten

equal parts, using nine deciles


Percentile


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The values of observations in a datawhen arranged in an orderedsequence into hundred equal partsusing ninety nine percentiles,



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The following distribution gives thepattern of overtime work per weekdone by 100 employees of acompany. Calculate median, first

quartile, and seventh decile.Overtime

Hours:10-15 15-20 20-25 25-30 30-35 35-40

No. of

employees:11 20 35 20 8 6



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Calculate the deciles of thedistribution for the following table


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distribution for the following table


Dispersion


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p

Variability among individual observationscomprising a set of data. Absolute Dispersion

Relative Dispersion

Measures of Dispersion Range

Mean Deviation

Standard Deviation

Variance Skewness


Range


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g

Value of highest observation-Value oflowest observation


Range = largest value - smallest value

Range = 615 - 425 = 190

Note: Data is in ascending order.

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

142

Range: Grouped Data


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1) By taking the difference between upperlimit of last class and lower limit of firstclass.

2)By taking the difference between the

midpoints of the first and last class


Interfractile Range


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Measure of spread between twofractiles in a frequency distribution.


INTERQUARTILE RANGE


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Difference between the third quartile and the first quartile.

Interquartile range = Q3- Ql

Semi interquartile range or Quartile deviation= (Q3 Ql)/2

When quartile deviation is small, it means that there is asmall deviation in the central 50 percent items. In contrast, ifthe quartile deviation is high, it shows that the central 50percent items have a large variation.


Do yourselfFind Interquartile Range & Quartiled i i


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deviation


Class Intervals Frequencies

161-162.9 3

163-164.9 7

165-166.9 14

167-168.9 12

169-170.9 10

171-172.9 4

146

Ans.


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Q1=165.3 Q3=169.3

QD=169


MEAN DEVIATION


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The mean deviation is also known asthe average deviation. As the nameimplies, it is the average of absoluteamounts by which the individual items

deviate from the mean. Since the positive deviations from the

mean are equal to the negative

deviations, while computing the meandeviation, we ignore positive andnegative signs.



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Where MD = mean deviation,

|d| = deviation of an item from themean ignoring positive and negativesigns,

n = the total number of observations.



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Standard Deviation


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The standard deviation is similar to the meandeviation in that here too the deviations aremeasured from the mean.

At the same time, the standard deviation is

preferred to the mean deviation or the quartiledeviation or the range because it has desirablemathematical properties.

Before defining the concept of the standarddeviation, we introduce another concept viz.variance


Variance


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The variance is a measure of variability that utilizes

all the data.

It is based on the difference between the value ofeach observation (x

i) and the mean (for a sample,

m for a population).x


Variance


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The variance is computed as follows:

The variance is the average of the squared

differences between each data value and the mean.

for asample

for apopulation

m22

( )xN

is xi x

n

2

2

1

( )


Standard Deviation


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The standard deviation of a data set is the positive

square root of the variance.

It is measured in the same units as the data, makingit more easily interpreted than the variance.


Standard Deviation


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The standard deviation is computed as follows:

for asample

for apopulation

s s 2 2



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PreparedBy:Kap

ilVerma157

Variance: Example


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Clippers' Player Height Deviation Squared deviation

1 Earl Boykins 65 -13.54 183.29

2 Keyon Dooling 75 -3.54 12.52

3 Jeff McInnis 76 -2.54 6.44

4 Quentin Richardson 78 -0.54 0.29

5 Corey Maggette 78 -0.54 0.29

6 Eric Piatkowski 78 -0.54 0.29

7 Elton Brand 80 1.46 2.14

8 Harold Jamison 81 2.46 6.06

9 Darius Miles 81 2.46 6.06

10 Obinna Ekezie 81 2.46 6.06

11 Sean Rooks 82 3.46 11.98

12 Lamar Odom 82 3.46 11.98

13 Michael Olowokandi 84 5.46 29.83Mean 78.54

Sum 277.23

N-1 12

Variance (s2) 23.10

s=(23.10)1/2=4.81inches

PreparedBy:Kap

ilVerma158

Example:


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The owner of the Ches Tahoerestaurant is interested in how muchmost of the people spend at the

restaurant. Heexamines 10 randomly selectedreceipts for parties of four and writesdown the following data.

44, 50, 38, 96, 42, 47, 40, 39,46, 50



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Now 2600.4/ 10 1 = 288.7

Hence the variance is 289 and the standard deviation is the square


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Hence the variance is 289 and the standard deviation is the squareroot of 289 = 17.

Since the standard deviation can be thought of measuring how farthe data values lie from the mean, we take the mean and moveone standard deviation in either direction. The mean for thisexample was about 49.2 and the standard deviation was 17. Wehave:

49.2 - 17 = 32.2

and

49.2 + 17 = 66.2

Wh t thi i th t t f th t b bl d

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