This document provides information about quantitative methods and statistics. It discusses different types of data including discrete and continuous variables. Discrete data can only take certain values while continuous data can theoretically take any value. The document also covers different scales of measurement for data like nominal, ordinal, interval and ratio scales. Finally, it discusses tools for data classification including arrays, frequency arrays and frequency distributions which arrange data according to a common characteristic.

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Quantitative Methods
Essential Basics

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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

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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

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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

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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

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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.

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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

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Models in Operations Research
• Linear programming
• Transportation
• Assignment
• Inventory
• Queuing
• Project scheduling
• Simulation
• Decision analysis

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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?

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Statistical Problems
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

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Statistical Problems
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?

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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

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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

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Quantitative Methods
Essential Basics

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Types of Numbers
• Nominal Numbers
• Ordinal Numbers
• Cardinal Numbers

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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

17. 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.

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Ordinal Numbers
• Purpose: Represent Position or Ranking
• Example: WTA Ranking of Sania Mirza
Salary Grade
Floor Number
Performance Rating
• No Quantitative Properties Except Order &
Equivalence: Different Ordinal Numbers
Indicate Different Objects in Some Kind of
Relationship with Each Other

19. 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.

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Cardinal Numbers
• Purpose: Represent Quantity
• Example: Sales Turnover in Million Rs.
Production in Tons
Number of Employees
Earning Per Share
• Truly Quantitative
• Follow All Mathematical Properties: Order,
Equivalence, +, -, x, /.

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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

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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

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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

24. 7 38
Primary Scales of Measurement
Scale
Nominal Numbers
Assigned
to Runners
Ordinal Rank Order
of Winners
Interval Performance
Rating on a
0 to 10 Scale
Ratio Time to
Finish, in
Third
place
Second
place
First
place
Finish
Finish
8.2 9.1 9.6
15.2 14.1 13.4

25. Primary Scales of Measurement
Nominal 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.

26. Illustration of Primary Scales of
Measurement
Nominal Ordinal Ratio
Scale Scale Scale
Preference $ spent last
No. Store Rankings
3 months
1. Lord & Taylor
2. Macy’s
3. Kmart
4. Rich’s
5. J.C. Penney
6. Neiman Marcus
7. Target
8. Saks Fifth Avenue
9. Sears
10.Wal-Mart
Interval
Scale
Preference
Ratings
1-7
7 5 0
2 7 200
8 4 0
3 6 100
1 7 250
5 5 35
9 4 0
6 5 100
4 6 0
10 2 10

27. Primary Scales of Measurement
Ordinal 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.

28. Primary Scales of Measurement
Interval 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.

29. Primary Scales of Measurement
Ratio 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.

30. 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,Arithmeti
c Mean,SD
Correlation,t
tests,ANOVA

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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)

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Discrete Measurement Data
Only 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.

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Discrete data -- Gaps between possible values
0 1 2 3 4 5 6 7
Continuous data -- Theoretically,
no gaps between possible values
0 1000

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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.

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Examples:
Continuous Measurement Data
• Cholesterol level
• Height
• Age
• Time to complete a homework assignment
Generally, continuous data come from
measurements.

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Who Cares?
The type(s) of data
collected in a study
determine the type of
statistical analysis used.

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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

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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%.

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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

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Quantitative Methods
Data Classification and
Presentation Tools

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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

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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

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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

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Frequency Array
A Table Showing the Number of Times
Each Value Occurs

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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

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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

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

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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’.

49. varsha varde 4949
Frequency Distribution
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

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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)?

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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.

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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

53. varsha varde 5353
Plant wise Production
Tons
Per
Month
April
2006
0
5
10
15
20
25
30
35
Plant
A
Plant
B
Plant
C
Plant
D

54. varsha varde 5454
For Nominal & Ordinal Variables
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

55. 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

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For Nominal & Ordinal Variables
Pie Chart:
• Divide A Circle Into Sectors (Pie)
• Area of Each Sector Proportionate to
Component Frequency Value
• Also called ‘Pizza Chart’

57. varsha varde 5757
A Pie Chart
SALES(Rs Crores)
A
37%
B
15%
C
7%
D
11%
E
30%
A
B
C
D
E

58. varsha varde 5858
Example
• What are the Occasions to Explain the
Facts Using a Pizza Chart?

59. 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

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Histogram

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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

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Frequency Polygon & Curve

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Normal Distribution

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Visual Characteristics of
Frequency Distributions
• Skewness
• Kurtosis
• Modality

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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.

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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.

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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).

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Positively Skewed Distribution

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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).

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Negatively Skewed Distribution

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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

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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

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Bimodal Distribution