Quantitative Methods Varsha Varde Quantitative Methods Models for

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Quantitative Methods Varsha Varde

Quantitative Methods Varsha Varde

Quantitative Methods Models for Data Analysis & Interpretation: Correlation Analysis

Quantitative Methods Models for Data Analysis & Interpretation: Correlation Analysis

Quotable Quotes • There is a Great Correlation Between Music and Images. – Graham

Quotable Quotes • There is a Great Correlation Between Music and Images. – Graham Nash • There is Little Correlation Between the Conditions of People's Lives and How Happy They Are. – Dennis Prager • Even Pop Singer and Talk Show Host Talk About Correlation. • What Is It? Varsha Varde 3

Scatter Plot • Scatter Plot is a Visual Representation of the Relationship Between Two

Scatter Plot • Scatter Plot is a Visual Representation of the Relationship Between Two Variables. • Use the Horizontal Axis for Values of One Variable. • Use the Vertical Axis for Values of the Other Variable. • Plot the Actual Data. Varsha Varde 4

Reasoning & Creativity Scores of Twenty Job Applicants Apl No, 01 02 03 Rsn.

Reasoning & Creativity Scores of Twenty Job Applicants Apl No, 01 02 03 Rsn. Sc 15. 2 9. 9 7. 1 04 05 06 07 08 09 10 17. 9 5. 1 10. 0 7. 2 17. 1 15. 2 9. 2 Crv. Sc 11. 9 13. 1 8. 9 Apl No, 11 12 13 17. 4 14 6. 9 15 8. 8 16 14. 0 17 15. 8 18 9. 7 19 Varsha Varde 20 12. 1 Rsn. Sc 8. 1 15. 2 10. 9 Crv. Sc 6. 8 13. 0 13. 9 17. 2 8. 2 10. 8 12. 0 13. 1 17. 9 7. 1 19. 1 10. 1 15. 9 12. 1 16. 0 19. 2 11. 9 5

Scatter Plot Horizontal Axis: Reasoning Scores Vertical Axis: Creativity Scores Varsha Varde 6

Scatter Plot Horizontal Axis: Reasoning Scores Vertical Axis: Creativity Scores Varsha Varde 6

Basic Patterns of Scatter Plot Both Move Together Move In Opposite Way Varsha Varde

Basic Patterns of Scatter Plot Both Move Together Move In Opposite Way Varsha Varde No Relationship 7

Positive Correlation • Both Variables Increase Simultaneously or Decrease Simultaneously. • Examples: Ø Your

Positive Correlation • Both Variables Increase Simultaneously or Decrease Simultaneously. • Examples: Ø Your Income and Jeweler's Bills Ø Exercise and Appetite Ø Rainfall and Absenteeism Ø Discount and Sales Varsha Varde 8

Negative Correlation • As One Variables Increases the Other Variable Decreases. • Examples: Ø

Negative Correlation • As One Variables Increases the Other Variable Decreases. • Examples: Ø TV Viewing and Book Reading Ø Age and Sleep Ø Price and Demand Ø Machine Downtime and Production Varsha Varde 9

Correlation Coefficient • It Measures the Extent of Quantitative Relationship Between Two Variables •

Correlation Coefficient • It Measures the Extent of Quantitative Relationship Between Two Variables • Examples: Rainfall & Sales of Agro-Chemicals Gold Price & Real Estate Price Snowfall in Alps & Onion Price in Dadar • Compute Correlation Coefficient Only Between Logically Related Factors Varsha Varde 10

Logically Related Variables • Technical: • Marketing: • Corporate: 1. 2. 3. Varsha Varde

Logically Related Variables • Technical: • Marketing: • Corporate: 1. 2. 3. Varsha Varde 11

Features of Correlation Coefficient • • Value Ranges Between -1 and +1. Perfect Positive

Features of Correlation Coefficient • • Value Ranges Between -1 and +1. Perfect Positive Correlation = +1 Perfect Negative Correlation = -1 Positive Corr. Coeff. : Two Variables Go Up or Down Simultaneously • Negative Corr. Coeff. : Exactly Opposite • Zero Corr. Coeff. : No Relationship At All Varsha Varde 12

Computing Correlation • Caution: Method for Computing Correlation Coefficient between Two Cardinal Variables is

Computing Correlation • Caution: Method for Computing Correlation Coefficient between Two Cardinal Variables is Different from the One for Two Ordinal Variables • Statutory Warning: Using One Formula for the Other is Seriously Injurious to Corporate Health. • So, First Identify the Type of the Variables At Hand: Cardinal or Ordinal. Varsha Varde 13

Correlation Coefficient For Cardinal Variables • Data: Actual Measurements on Both Variables • Formula:

Correlation Coefficient For Cardinal Variables • Data: Actual Measurements on Both Variables • Formula: Ratio of {Mean of Products of Values – Product of the Two Means} to Product of the Two Standard Deviations Mean of Products of Values – Product of the Two Means = -------------------------------------Product of the Two Standard Deviations • Name: Pearson’s Correlation Coefficient • But, Your Statistician Calls It Pearson’s r. Varsha Varde 14

Annual Production of 7 Plants Plant A B C 2011 (X) 1 3 5

Annual Production of 7 Plants Plant A B C 2011 (X) 1 3 5 2012 (Y) 4 7 10 XY 4 21 50 D E F G Total Arith Mean Std Deviation 7 9 11 13 49 7 4 13 16 19 22 91 13 6 91 144 209 286 805 Varsha Varde 15

Pearson’s Correlation Coefficient of Plant Production • Formula: Ratio of (Mean of Products of

Pearson’s Correlation Coefficient of Plant Production • Formula: Ratio of (Mean of Products of Values – Product of the Two Means) to Product of the Two Std. Deviations (805 / 7) – (7 x 13) 115 - 91 = ------------ = 1 4 x 6 24 • Interpretation: Perfect Correlation 1 Varsha Varde 16

One More Example Empl. No. 1 2 3 4 5 Total Arith Mean Std.

One More Example Empl. No. 1 2 3 4 5 Total Arith Mean Std. Dev. Yrs in Co. Salary (‘ 000) Product 2 25 50 3 30 90 5 37 185 7 38 266 8 40 320 25 170 911 5 34 2. 3 5. 6 Varsha Varde 17

Pearson’s Correlation Coefficient Between Yrs in Co & Salary • Formula: Ratio of (Mean

Pearson’s Correlation Coefficient Between Yrs in Co & Salary • Formula: Ratio of (Mean of Products of Values – Product of the Two Means) to Product of the Two Std. Deviations (911 / 5) – (5 x 34) 182. 2 - 170 = ------------ = 0. 94 2. 3 x 5. 6 12. 9 • Interpretation: Salary and Years of Service in the Company are Strongly Correlated With Each Other Varsha Varde 18

One More for Practice Month Discount% Nov 2 Dec 5 Jan 3 Feb 7

One More for Practice Month Discount% Nov 2 Dec 5 Jan 3 Feb 7 March 8 Total 25 Arith Mean 5 Std. Dev. 2. 3 Varsha Varde Sales 25 38 37 30 40 170 34 5. 6 Product 50 190 111 210 320 881 19

Pearson’s Correlation Coefficient Between Discount & Sales • Formula: Ratio of (Mean of Products

Pearson’s Correlation Coefficient Between Discount & Sales • Formula: Ratio of (Mean of Products of Values – Product of the Two Means) to Product of the Two Std. Deviations (881 / 5) – (5 x 34) 176. 2 - 170 = ------------ = 0. 48 2. 3 x 5. 6 12. 9 • Interpretation: Sales Do Improve With Discounts, But Not Very Significantly. Varsha Varde 20

One More for Practice Month Nov Dec Jan Feb March Total Mean S. D.

One More for Practice Month Nov Dec Jan Feb March Total Mean S. D. M/c. Downtime 8 5 7 3 2 25 5 2. 3 Production 25 30 37 38 40 170 34 5. 6 Varsha Varde Product 200 150 259 114 80 803 21

Pearson’s Correlation Coefficient Between M/c Downtime & Production • Formula: Ratio of (Mean of

Pearson’s Correlation Coefficient Between M/c Downtime & Production • Formula: Ratio of (Mean of Products of Values – Product of the Two Means) to Product of the Two Std. Deviations (803 / 5) – (5 x 34) 160. 6 - 170 = ------------ = -0. 73 2. 3 x 5. 6 12. 9 • Interpretation: Significant Negative Correlation between M/c Downtime & Prod Varsha Varde 22

Correlation Coefficient For Ordinal Variables • Actual Measurements on Both Variables Not Available •

Correlation Coefficient For Ordinal Variables • Actual Measurements on Both Variables Not Available • Data Are In the Form of Ranks 6 x Sum Square of Rank Diff • Formula: 1 - -------------------n x {(Square of n) -1} where n denotes Number of Observations • Name: Rank Correlation Coefficient Varsha Varde 23

Rank Correlation Coefficient Between Age & Performance Age Rank Performance Difference Rank 1 4

Rank Correlation Coefficient Between Age & Performance Age Rank Performance Difference Rank 1 4 3 Square 9 2 2 0 0 3 1 2 4 4 5 1 1 5 3 2 4 Total 18 Varsha Varde 24

Rank Correlation Coefficient Between Age & Performance • Formula: 6 x 18 108 1

Rank Correlation Coefficient Between Age & Performance • Formula: 6 x 18 108 1 - ---------- = 1 - 0. 9 = 0. 1 5 x (25 -1) 120 • Interpretation: Age Has Very Little To Do With Performance Varsha Varde 25

Frequent Blunders • People Treat All Variables As Cardinal. • They Use Pearson’s Formula

Frequent Blunders • People Treat All Variables As Cardinal. • They Use Pearson’s Formula on Ordinal Variables and Create Havoc with Wrong Interpretations. • Even for Ranking Data on Cardinal Variables, They Use Pearson’s Formula and Draw Misleading Conclusions. • This is an International Disease. • DO NOT FALL PREY TO IT. Varsha Varde 26

Tips to Busy Executives • If One Set of Data is Cardinal and the

Tips to Busy Executives • If One Set of Data is Cardinal and the Other Ordinal, Convert Cardinal Values Into Ordinal Ranks, and Then Compute Rank Correlation Coefficient. • To Get a Quick Measure of the Extent of Relationship Between Two Cardinal Variables, Convert Both Sets of Data Into Ordinal Ranks, and Compute Rank Correlation Coefficient. Varsha Varde 27

Rank Correlation Coefficient Between M/c Downtime & Production M/c Down Rank 5 Prod Rank

Rank Correlation Coefficient Between M/c Downtime & Production M/c Down Rank 5 Prod Rank Difference Square 1 4 16 3 2 1 1 4 3 1 1 2 4 1 5 4 16 Total 38 Varsha Varde 28

Rank Correlation Coefficient Between M/c Downtime & Production • Formula: 6 x 38 228

Rank Correlation Coefficient Between M/c Downtime & Production • Formula: 6 x 38 228 1 - ---------- = 1 - 1. 9 = -0. 9 5 x (25 -1) 120 • Interpretation: Strong Negative Correlation between M/c Downtime & Prod • Recall: Pearson’s Corr. Coeff. was -0. 73 Varsha Varde 29

How Will You Proceed To Work Out Correlation In Following Pairs • • •

How Will You Proceed To Work Out Correlation In Following Pairs • • • Adult IQ and Annual Income Consumer Price Index and Sensex Dealer Seniority and Dealer Performance Gold Prices and Real Estate Prices Birth Rate in Germany and Voter Turnout in Kerala • WTA Ranking and Height. . Varsha Varde 30