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 ...
varsha varde 3
QUANTITATIVE METHODS
• It is a broad term
• Two branches of relevance to us are
statistics and operations r...
varsha varde 4
STATISTICS
• Word derived from Latin word status
• It came into existence as collection of
certain data of ...
varsha varde 5
TOOLS IN STATISTICS
Broadly classified into
• Descriptive statistics-describes principal
features of the co...
varsha varde 6
History of OR
• Origin: research in military operations
• 1930’s: British scientists helped in solving
prob...
varsha varde 7
History of OR
• Post world war-II: Military continued using OR
analysts
• But, OR as a discipline not accep...
varsha varde 8
Models in Operations Research
• Linear programming
• Transportation
• Assignment
• Inventory
• Queuing
• Pr...
varsha varde 99
Statistical Problems
1. A market analyst wants to know the
effectiveness of a new diet.
2. A pharmaceutica...
varsha varde 1010
Statistical Problems
4. Is there any relationship between your
Grades and employment opportunities.
5. I...
varsha varde 1111
Statistical Problems
7. How to interpret polls. How many
individuals you need to sample for your
inferen...
varsha varde 1212
Course Coverage
• Essential Basics Management
• Data Classification & Presentation Tools
• Preliminary A...
varsha varde 1313
Suggested Reading
• Statistics for Management by Richard I Levin-
Prentice Hall Of India –New DelhiDavid...
varsha varde 14
Quantitative Methods
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 Nu...
Silent Disaster
• Nominal Nos. look like normal numerals
• Prime Foods CEO’s Tel No.: 23249843
• Prime Foods Ltd. Sales: R...
varsha varde 1818
Ordinal Numbers
• Purpose: Represent Position or Ranking
• Example: WTA Ranking of Sania Mirza
Salary Gr...
Silent Disaster
• Ordinal Nos. look like normal numerals
• Sania Mirza’s weight (kg) : 53
• Sania Mirza’s WTA Ranking : 53...
varsha varde 2020
Cardinal Numbers
• Purpose: Represent Quantity
• Example: Sales Turnover in Million Rs.
Production in To...
varsha varde 21
Interval and Ratio Scales
• Interval Scale employs arbitrary zero point
• Ratio Scale employs a true zero ...
varsha varde 2222
Example
Zone Code No. Sales
(Rs. In Million)
Rank
Northern 01 483 3
Western 02 738 1
Eastern 03 265 4
So...
varsha varde 2323
Example
Zone Code No. Sales
(Rs. In Million)
Rank
Northern 01 483 3
Western 02 738 1
Eastern 03 265 4
So...
7 38
Primary Scales of Measurement
Scale
Nominal Numbers
Assigned
to Runners
Ordinal Rank Order
of Winners
Interval Perfor...
Primary Scales of Measurement
Nominal Scale
• The numbers serve only as labels or tags for identifying
and classifying obj...
Illustration of Primary Scales of
Measurement
Nominal Ordinal Ratio
Scale Scale Scale
Preference $ spent last
No. Store Ra...
Primary Scales of Measurement
Ordinal Scale
• A ranking scale in which numbers are assigned to
objects to indicate the rel...
Primary Scales of Measurement
Interval Scale
• Numerically equal distances on the scale represent
equal values in the char...
Primary Scales of Measurement
Ratio Scale
• Possesses all the properties of the nominal, ordinal, and
interval scales.
• I...
Primary Scales of Measurement
Scale Basic
Characteristics
Common
Examples
Marketing
Examples
Nominal Numbers identify
& cl...
varsha varde 3131
Basic Definitions
• Constant: A Characteristic that never
changes its Value (Your Height after 20)
• Var...
varsha varde 3232
Discrete Measurement Data
Only certain values are possible (there
are gaps between the possible values)....
varsha varde 3333
Discrete data -- Gaps between possible values
0 1 2 3 4 5 6 7
Continuous data -- Theoretically,
no gaps ...
varsha varde 3434
Examples:
Discrete Measurement Data
• Number of students late for class
• Number of crimes reported in a...
varsha varde 3535
Examples:
Continuous Measurement Data
• Cholesterol level
• Height
• Age
• Time to complete a homework a...
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%...
varsha varde 3838
And for example …
• Measurement data are typically
summarized using “averages” (or “mean
– Average weigh...
varsha varde 3939
Course Coverage
• Essential Basics for Business Executives
• Data Classification & Presentation Tools
• ...
varsha varde 40
Quantitative Methods
Data Classification and
Presentation Tools
varsha varde 4141
Data Classification
• First Step: Organize Data Systematically
• Arrange the Data According to a Common
...
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 ...
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,...
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: 0...
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: ...
varsha varde 4747
Frequency Distribution
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 Cov...
varsha varde 4949
Frequency Distribution
Number of Orders Number of SEs
00 – 04 14
05 - 09 19
10 – 14 07
15 – 19 04
20 – 2...
varsha varde 5050
In This Example
• What is the Variable? Sales Executives or
Sales Orders?
• Is it Nominal, Ordinal or Ca...
varsha varde 5151
Data Presentation
• Some People are Averse to Numbers
• They Can’t Grasp Tabulated Data
• Pictures Speak...
varsha varde 5252
For Nominal & Ordinal Variables
Bar Chart:
• Horizontal Diagram of Bars of Equal Width
But of Different ...
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
varsha varde 5454
For Nominal & Ordinal Variables
Component Bar Chart:
• Illustration of A Total Divided Into Parts
• Divi...
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
For Nominal & Ordinal Variables
Pie Chart:
• Divide A Circle Into Sectors (Pie)
• Area of Each Sector Pr...
varsha varde 5757
A Pie Chart
SALES(Rs Crores)
A
37%
B
15%
C
7%
D
11%
E
30%
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 Frequen...
varsha varde 6060
Histogram
varsha varde 6161
For Cardinal Variables
Frequency Polygon:
• Plot Mid Points of Class Intervals Along
Horizontal Axis
• C...
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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 Ne...
varsha varde 6666
Shapes of Frequency Distribution
• Draw Histogram on Paper.
• Fold Paper In Half the Long Way.
• If Dist...
varsha varde 6767
Positively Skewed Distribution
• Frequencies Cluster Toward the Lower
End of The Scale (That Is, The Sma...
varsha varde 6868
Positively Skewed Distribution
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Negatively Skewed Distribution
• Negatively Skewed Distribution Is Exactly
The Opposite.
• Most of The S...
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...
varsha varde 72
Modality
• Unimodal: Only One Major "Peak" in the
Distribution of Scores When Represented
as a Histogram
•...
varsha varde 73
Bimodal Distribution
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01 quanttech-basic-class-present

  1. 1. varsha varde 1 Quantitative Methods Essential Basics
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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
  6. 6. 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.
  7. 7. 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
  8. 8. varsha varde 8 Models in Operations Research • Linear programming • Transportation • Assignment • Inventory • Queuing • Project scheduling • Simulation • Decision analysis
  9. 9. 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?
  10. 10. varsha varde 1010 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
  11. 11. varsha varde 1111 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?
  12. 12. 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
  13. 13. 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
  14. 14. varsha varde 14 Quantitative Methods Essential Basics
  15. 15. varsha varde 1515 Types of Numbers • Nominal Numbers • Ordinal Numbers • Cardinal Numbers
  16. 16. 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
  17. 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.
  18. 18. varsha varde 1818 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. 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.
  20. 20. varsha varde 2020 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, /.
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 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. 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. 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. 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. 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. 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. 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
  31. 31. 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)
  32. 32. varsha varde 3232 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.
  33. 33. 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
  34. 34. 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.
  35. 35. varsha varde 3535 Examples: Continuous Measurement Data • Cholesterol level • Height • Age • Time to complete a homework assignment Generally, continuous data come from measurements.
  36. 36. varsha varde 3636 Who Cares? The type(s) of data collected in a study determine the type of statistical analysis used.
  37. 37. 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
  38. 38. 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%.
  39. 39. 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
  40. 40. varsha varde 40 Quantitative Methods Data Classification and Presentation Tools
  41. 41. 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
  42. 42. 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
  43. 43. 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
  44. 44. varsha varde 4444 Frequency Array A Table Showing the Number of Times Each Value Occurs
  45. 45. 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
  46. 46. 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
  47. 47. varsha varde 4747 Frequency Distribution A Table Showing the Number of Times Each Cluster of Values Occurs
  48. 48. 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’.
  49. 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
  50. 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)?
  51. 51. 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.
  52. 52. 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
  53. 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. 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. 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
  56. 56. varsha varde 5656 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. 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. 58. varsha varde 5858 Example • What are the Occasions to Explain the Facts Using a Pizza Chart?
  59. 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
  60. 60. varsha varde 6060 Histogram
  61. 61. 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
  62. 62. varsha varde 6262 Frequency Polygon & Curve
  63. 63. varsha varde 6363 Normal Distribution
  64. 64. varsha varde 64 Visual Characteristics of Frequency Distributions • Skewness • Kurtosis • Modality
  65. 65. 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.
  66. 66. 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.
  67. 67. 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).
  68. 68. varsha varde 6868 Positively Skewed Distribution
  69. 69. 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).
  70. 70. varsha varde 7070 Negatively Skewed Distribution
  71. 71. 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
  72. 72. 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
  73. 73. varsha varde 73 Bimodal Distribution

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