01 quanttech-basic-class-present

varsha varde 1

Quantitative Methods

Essential Basics

varsha varde 22

Varsha Varde\

• M. Sc; Ph. D. in Statistics (O. R.)

• Taught Advanced Stats to PG Students

• Quantitative Faculty in NIBM

• Visiting Faculty at JBIMS

• Officer in Bank Of India

• General Manager At AFC

• Handled consultancy in Various Fields

varsha varde 3

QUANTITATIVE METHODS

• It is a broad term

• Two branches of relevance to us are statistics and operations research

• Each of these offers several tools and techniques to tackle real life problems in scientific manner

varsha varde 4

STATISTICS

• Word derived from Latin word status• It came into existence as collection of

certain data of states• It continued to expand its boundaries to

include planning and organising of data collection ,analysis of data and drawing meaningful conclusions from data

• Data are input, statistics is process and information is output

varsha varde 5

TOOLS IN STATISTICS

Broadly classified into • Descriptive statistics-describes principal

features of the collected data• Inferential statistics-says something about

future or for present but for larger group of data than actually collected

• Sampling- designing of sample survey, selection of representative sample

• Probability- quantifying uncertainties

varsha varde 6

History of OR• Origin: research in military operations• 1930’s: British scientists helped in solving

problems of military operations, such as:• Effective use of radar, Anti-submarine

warfare, civilian defence, deployment of convoy vessels

• Team: Experts from various disciplines• Inter disciplinary character of OR still

continues• World war II: Military operations research in

US.

varsha varde 7

History of OR

• Post world war-II: Military continued using OR analysts

• But, OR as a discipline not accepted in outside world• Reason: OR solves only military problems• Two Events helped spread to non –military

establishments• Development of Simplex method in1947• Development and usage of high speed computers• OR as a discipline came into existencein1950’s• OR: Systematic and scientific approach to problem

solving

varsha varde 8

Models in Operations Research

• Linear programming• Transportation • Assignment • Inventory• Queuing• Project scheduling• Simulation• Decision analysis

varsha varde 99

Statistical Problems

1. A market analyst wants to know the effectiveness of a new diet.

2. A pharmaceutical Co. wants to know if a new drug is superior to already existing drugs, or possible side effects.

3. How fuel efficient a certain car model is?

varsha varde 1010


4. Is there any relationship between your Grades and employment opportunities.

5. If you answer all questions on a (T,F) (or multiple choice) examination completely randomly, what are your chances of passing?

6. What is the effect of package designs on sales

varsha varde 1111


7. How to interpret polls. How many individuals you need to sample for your inferences to be acceptable? What is meant by the margin of error?

8. What is the effect of market strategy on market share?

9. How to pick the stocks to invest in?

varsha varde 1212

Course Coverage

• Essential Basics Management• Data Classification & Presentation Tools• Preliminary Analysis & Interpretation of Data• Correlation Model• Regression Model• Time Series Model• Forecasting• Uncertainty and Probability• Probability Distributions• Sampling and Sampling Distributions• Estimation and Testing of Hypothesis• Chi-Square and Analysis of Variance• Decision Theory• Linear Programming

varsha varde 1313

Suggested Reading

• Statistics for Management by Richard I Levin-Prentice Hall Of India –New DelhiDavid C. Howell (2003)

• Quantitative Techniques for Management Decisions by U K Srivastava & Others-New Age International-New Delhi

• Quantitative Methods for Business by David R Anderson &Others-Thomson Learning-New Delhi

• Business Statistics by David M Levine & Others-Pearson Education-Delhi-2004

varsha varde 14


Essential Basics

varsha varde 1515

Types of Numbers

• Nominal Numbers

• Ordinal Numbers

• Cardinal Numbers

varsha varde 1616

Nominal Numbers

• Purpose: Identification of an Object

• Example: House Number (10 Janpath)

Telephone Number

Smart Card PINumber

Number on Cricket T-Shirt

• No Quantitative Properties Except Equivalence: Two Different Nominal Numbers Indicate Two Different Objects

Silent Disaster• Nominal Nos. look like normal numerals

• Prime Foods CEO’s Tel No.: 23249843

• Prime Foods Ltd. Sales: Rs. 23249843

• No computer will stop you if you ask it to add nominal numbers (or multiply, divide)

• But, resultant figure makes no sense

• Still, this mistake is made occasionally.

varsha varde 1818

Ordinal Numbers

• Purpose: Represent Position or Ranking• Example: WTA Ranking of Sania Mirza

Salary Grade Floor NumberPerformance Rating

• No Quantitative Properties Except Order & Equivalence: Different Ordinal Numbers Indicate Different Objects in Some Kind of Relationship with Each Other

Silent Disaster• Ordinal Nos. look like normal numerals• Sania Mirza’s weight (kg) : 53 • Sania Mirza’s WTA Ranking : 53• You can safely add weights & divide them• No computer will stop you if you ask it to

add ordinal numbers (or multiply, divide)• But, the resultant figure makes no sense• Still, this blunder is committed frequently.

varsha varde 2020

Cardinal Numbers

• Purpose: Represent Quantity • Example: Sales Turnover in Million Rs.

Production in TonsNumber of Employees Earning Per Share

• Truly Quantitative• Follow All Mathematical Properties: Order,

Equivalence, +, -, x, /.

varsha varde 21

Interval and Ratio Scales• Interval Scale employs arbitrary zero point• Ratio Scale employs a true zero point• Only ratio scale permits statements

concerning ratios of numbers in the scale; e.g 4kgs to 2 kgs is 2kgs to 1 kg

• Scale of Temperature measured in Celsius is Interval Scale.

• Height as measured from a table top has interval scale

• Height as measured from floor has ratio scale

• Apart from difference in the nature of zero point ,interval and ratio scales have same properties and both employ cardinal numbers

varsha varde 2222

Example

Zone Code No. Sales(Rs. In Million)

Rank

Northern 01 483 3

Western 02 738 1

Eastern 03 265 4

Southern 04 567 2

Type

varsha varde 2323

Example

Zone Code No. Sales(Rs. In Million)

Rank

Northern 01 483 3

Western 02 738 1

Eastern 03 265 4

Southern 04 567 2

Type Nominal Cardinal Ordinal

7 38

Primary Scales of MeasurementScaleNominal Numbers

Assigned to Runners

Ordinal Rank Orderof Winners

Interval PerformanceRating on a

0 to 10 Scale

Ratio Time to Finish, in

Seconds

Thirdplace

Secondplace

Firstplace

Finish

Finish

8.2 9.1 9.6

15.2 14.1 13.4

Primary Scales of MeasurementNominal Scale

• The numbers serve only as labels or tags for identifying and classifying objects.

• When used for identification, there is a strict one-to-one correspondence between the numbers and the objects.

• The numbers do not reflect the amount of the characteristic possessed by the objects.

• The only permissible operation on the numbers in a nominal scale is counting.

• Only a limited number of statistics, all of which are based on frequency counts, are permissible, e.g., percentages, and mode.

Illustration of Primary Scales of Measurement

Nominal Ordinal RatioScale Scale Scale

Preference $ spent last No. Store Rankings 3 months

1. Lord & Taylor2. Macy’s3. Kmart4. Rich’s5. J.C. Penney 6. Neiman Marcus 7. Target 8. Saks Fifth Avenue 9. Sears 10.Wal-Mart

IntervalScale Preference Ratings

1-7

7 5 02 7 2008 4 03 6 1001 7 2505 5 359 4 06 5 1004 6 010 2 10

Primary Scales of MeasurementOrdinal Scale

• A ranking scale in which numbers are assigned to objects to indicate the relative extent to which the objects possess some characteristic.

• Can determine whether an object has more or less of a characteristic than some other object, but not how much more or less.

• Any series of numbers can be assigned that preserves the ordered relationships between the objects.

• In addition to the counting operation allowable for nominal scale data, ordinal scales permit the use of statistics based on centiles, e.g., percentile, quartile, median.

Primary Scales of MeasurementInterval Scale

• Numerically equal distances on the scale represent equal values in the characteristic being measured.

• It permits comparison of the differences between objects.

• The location of the zero point is not fixed. Both the zero point and the units of measurement are arbitrary.

• Any positive linear transformation of the form y = a + bx will preserve the properties of the scale.

• It is not meaningful to take ratios of scale values. • Statistical techniques that may be used include all of

those that can be applied to nominal and ordinal data, and in addition the arithmetic mean, standard deviation, and other statistics commonly used in marketing research.

Primary Scales of MeasurementRatio Scale

• Possesses all the properties of the nominal, ordinal, and interval scales.

• It has an absolute zero point. • It is meaningful to compute ratios of scale values. • Only proportionate transformations of the form y = bx,

where b is a positive constant, are allowed. • All statistical techniques can be applied to ratio data.

Primary Scales of Measurement

Scale Basic Characteristics

Common Examples

Marketing Examples

Nominal Numbers identify & classify objects

Social Security nos., numbering of football players

Brand nos., store types

Percentages, mode

Chi-square, binomial test

Ordinal Nos. indicate the relative positions of objects but not the magnitude of differences between them

Quality rankings, rankings of teams in a tournament

Preference rankings, market position, social class

Percentile, median

Rank-order correlation,

Ratio Zero point is fixed, ratios of scale values can be compared

Length, weight Age, sales, income, costs

Geometric mean, harmonic mean

Coefficient of variation

Permissible Statistics Descriptive Inferential

Interval Differences between objects

Temperature (Fahrenheit)

Attitudes, opinions, index

Range,Arithmetic Mean,SD

Correlation,t tests,ANOVA

varsha varde 3131

Basic Definitions

• Constant: A Characteristic that never changes its Value (Your Height after 20)

• Variable: A Characteristic that assumes different Values (Your Weight after 20)

• Discrete Variable: Cannot take a Value Between Any Two Values (Staff Strength)

• Continuous Variable: Can take a Value Between Any Two Values (P-E Ratio)

varsha varde 3232

Discrete Measurement DataOnly certain values are possible (there are gaps between the possible values).

Continuous Measurement Data

Theoretically, any value within an interval is possible with a fine enough

measuring device.

varsha varde 3333

Discrete data -- Gaps between possible values

0 1 2 3 4 5 6 7

Continuous data -- Theoretically,no gaps between possible values

0 1000

varsha varde 3434

Examples: Discrete Measurement Data

• Number of students late for class

• Number of crimes reported in a police station

• Number of times a particular word is used

• Number of defectives in a lot

Generally, discrete data are counts.

varsha varde 3535

Examples:Continuous Measurement Data

• Cholesterol level

• Height

• Age

• Time to complete a homework assignment

Generally, continuous data come from measurements.

varsha varde 3636

Who Cares?

The type(s) of data collected in a study

determine the type of statistical analysis used.

varsha varde 3737

For example ...

• Categorical data are commonly summarized using “percentages” (or “proportions”).– 31% of students have a passport– 2%, 33%, 39%, and 26% of the students in

class are, respectively engineers, science, commerce and arts graduates

varsha varde 3838

And for example …

• Measurement data are typically summarized using “averages” (or “mean– Average weight of male students of this batch

is 75 kg.– Average weight of female students of this

batch is 55 kg.– Average growth rate of sales of a company is

18%.

varsha varde 3939

Course Coverage

• Essential Basics for Business Executives• Data Classification & Presentation Tools• Preliminary Analysis & Interpretation of Data• Correlation Model• Regression Model• Time Series Model• Forecasting• Uncertainty and Probability• Sampling Techniques• Estimation and Testing of Hypothesis

varsha varde 40


Data Classification and Presentation Tools

varsha varde 4141

Data Classification

• First Step: Organize Data Systematically

• Arrange the Data According to a Common Characteristic Possessed by All Items

• Methods: Array

Frequency Array

Frequency Distribution

varsha varde 4242

Example: Number of Sales Orders Booked by 50 Sales Execs April 2006

09 34 11 07 43 05 14 19 04 06

04 10 16 07 03 06 24 08 01 09

11 11 02 09 08 12 04 15 30 08

00 03 06 10 02 17 00 09 05 21

02 08 07 28 05 03 06 09 00 00

varsha varde 4343

Array

0, 0, 0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 12, 14, 15, 16, 17, 19, 21, 24, 28, 30, 34, 43

Array: Arrangement of Data in Order of Magnitude

varsha varde 4444

Frequency Array

A Table Showing the Number of Times Each Value Occurs

varsha varde 4545

Frequency Array

Orders: SEs Orders: SEs Orders: SEs Orders: SEs

00: 04 11: 03 22: 00 33: 00

01: 01 12: 01 23: 00 34: 01

02: 03 13: 00 24: 01 35: 00

03: 03 14: 01 25: 00 36: 00

04: 03 15: 01 26: 00 37: 00

05: 03 16: 01 27: 00 38: 00

06: 04 17: 01 28: 01 39: 00

07: 03 18: 00 29: 00 40: 00

08: 04 19: 01 30: 01 41: 00

09: 05 20: 00 31: 00 42: 00

10: 02 21: 01 32: 00 43: 01

varsha varde 4646

Frequency Array

Xi: fi Xi: fi Xi: fi Xi: fi

00: 04 11: 03 22: 00 33: 00

01: 01 12: 01 23: 00 34: 01

02: 03 13: 00 24: 01 35: 00

03: 03 14: 01 25: 00 36: 00

04: 03 15: 01 26: 00 37: 00

05: 03 16: 01 27: 00 38: 00

06: 04 17: 01 28: 01 39: 00

07: 03 18: 00 29: 00 40: 00

08: 04 19: 01 30: 01 41: 00

09: 05 20: 00 31: 00 42: 00

10: 02 21: 01 32: 00 43: 01

varsha varde 4747


A Table Showing the Number of Times Each Cluster of Values Occurs

varsha varde 4848

Constructing Frequency Distribution

• Find Maximum & Minimum Values in Data.

• Make Sub-Intervals to Cover Entire Range

• They are Called the ‘Class Intervals’.

• Class Intervals Need Not Be of Equal Length. But, it is Useful if They Are.

• Note the Number of Observation that Belong to Each Class Interval.

• They are Called the ‘Frequencies’.

varsha varde 4949


Number of Orders Number of SEs

00 – 04 14

05 - 09 19

10 – 14 07

15 – 19 04

20 – 24 02

25 – 29 01

30 – 34 02

35 – 39 00

40 – 44 01

TOTAL 50

varsha varde 5050

In This Example

• What is the Variable? Sales Executives or Sales Orders?

• Is it Nominal, Ordinal or Cardinal?

• Is it Discrete or Continuous?

• What are the frequencies (sometimes called as frequency values or score)?

varsha varde 5151

Data Presentation

• Some People are Averse to Numbers

• They Can’t Grasp Tabulated Data

• Pictures Speak with Them; Figures Don’t.

• Pictures Tell Them What A Thousand Numbers Can’t.

• If your Boss Fits in This Category, You Must Learn the Art and Methods of Data Presentation.

varsha varde 5252

For Nominal & Ordinal Variables

Bar Chart:• Horizontal Diagram of Bars of Equal Width

But of Different Heights • Bars Stand on a Common Base Line • Horizontal Axis: Nominal/Ordinal Variables• Vertical Axis: Their Frequencies • Height of Bar is Prop. to Frequency Value• Bars are Separated by Equal Distance

varsha varde 5353

Plant wise Production

Tons

Per

Month

April

2006

0

5

10

15

20

25

30

35

PlantA

PlantB

PlantC

PlantD

varsha varde 5454


Component Bar Chart:

• Illustration of A Total Divided Into Parts

• Divide Simple Bars Into Component Parts

• Part Prop. to Component Freq. Value

Multiple Bar Chart:

• Direct Comparison Among Variables

• Draw Bars By the Side of Each Other

varsha varde 5555

Multiple Bar Chart

0

10

20

30

40

50

60

70

80

90

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

East

West

North

varsha varde 5656


Pie Chart:

• Divide A Circle Into Sectors (Pie)

• Area of Each Sector Proportionate to Component Frequency Value

• Also called ‘Pizza Chart’

varsha varde 5757

A Pie ChartSALES(Rs Crores)

A37%

B15%

C7%

D11%

E30%

A

B

C

D

E

varsha varde 5858

Example

• What are the Occasions to Explain the Facts Using a Pizza Chart?

varsha varde 5959

For Cardinal Variables

Histogram:

• A Graph of Columns, Each Having a Class Interval as Base and Frequency as Height

• Plot Class Intervals Along Horizontal Axis

• Erect A Rectangle On Each Class Interval

• Area of Rectangle Prop. to Freq. Value

• Rectangles Touch Each Other

varsha varde 6060

Histogram

varsha varde 6161

For Cardinal Variables

Frequency Polygon:

• Plot Mid Points of Class Intervals Along Horizontal Axis

• Concerned Frequencies on Vertical Axis

• Joins All These Points

Frequency Curve:

• Join All These Points by a Smooth Curve

varsha varde 6262

Frequency Polygon & Curve

varsha varde 6363

Normal Distribution

varsha varde 64

Visual Characteristics of Frequency Distributions

• Skewness

• Kurtosis

• Modality

varsha varde 6565

Skewness

• Symmetrical Distribution (Normal Distn.)

• Asymmetrical Distribution: Positively Skewed or Negatively Skewed

• Symmetrical Distributions are Easy to Handle Mathematically.

• But, Asymmetric Distributions Are More Commonly Found.

• That Is Why We Need Statistical Methods.

varsha varde 6666

Shapes of Frequency Distribution

• Draw Histogram on Paper.

• Fold Paper In Half the Long Way.

• If Distribution Is Symmetrical, the Left Side of Histogram Would Be Mirror Image of the Right Side.

• Life is Rarely Symmetrical.

• If Distribution Is Asymmetrical, Two Sides Will Not Be Mirror Images of Each Other.

varsha varde 6767

Positively Skewed Distribution

• Frequencies Cluster Toward the Lower End of The Scale (That Is, The Smaller Numbers).

• Increasingly Fewer Scores At the Upper End of The Scale (That Is, The Larger Numbers).

varsha varde 6868

Positively Skewed Distribution

varsha varde 6969

Negatively Skewed Distribution

• Negatively Skewed Distribution Is Exactly The Opposite.

• Most of The Scores Occur Toward The Upper End of The Scale (That Is, The Larger Numbers).

• Increasingly Fewer Scores Occur Toward The Lower End (That Is, The Smaller Numbers).

varsha varde 7070

Negatively Skewed Distribution

varsha varde 71

Kurtosis

• Relative Concentration of Scores in the Center, the Upper and Lower Ends and the Shoulders of a Distribution

• Platykurtic: Flatter Curve

• Leptokurtic: More Peaked

• Mesokurtic : Medium Peaked

varsha varde 72

Modality

• Unimodal: Only One Major "Peak" in the Distribution of Scores When Represented as a Histogram

• Bimodal: Two Major Peaks

• Multimodal: More Than Two Major Peaks

varsha varde 73

Bimodal Distribution

01 quanttech-basic-class-present

Technology

Transcript of 01 quanttech-basic-class-present