Quantitative Methods: Quantitative Methods
Quantitative Methods: Quantitative Methods Models for Data Analysis & Interpretation: Correlation Analysis
Quotable Quotes: piyoosh 3 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?
Correlation: piyoosh 4 Correlation Dictionary Says: Correlation is a Close Connection Between Two Things In Which One Thing Changes as the Other Does. Note the Phrase: Close Connection Remember: Correlation Does Not Necessarily Mean Causation. Importance : Use Information About One To Estimate Values of the Other.
Scatter Plot: piyoosh 5 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.
Reasoning & Creativity Scores of Twenty Job Applicants: piyoosh 6 Reasoning & Creativity Scores of Twenty Job Applicants Apl No, RsnSc CrvSc Apl No, RsnSc CrvSc 01 15.2 11.9 11 8.1 6.8 02 9.9 13.1 12 15.2 13.0 03 7.1 8.9 13 10.9 13.9 04 17.9 17.4 14 17.2 19.1 05 5.1 6.9 15 8.2 10.1 06 10.0 8.8 16 10.8 15.9 07 7.2 14.0 17 12.0 12.1 08 17.1 15.8 18 13.1 16.0 09 15.2 9.7 19 17.9 19.2 10 9.2 12.1 20 7.1 11.9
Scatter Plot Horizontal Axis: Reasoning Scores Vertical Axis: Creativity Scores: piyoosh 7 Scatter Plot Horizontal Axis: Reasoning Scores Vertical Axis: Creativity Scores
Basic Patterns of Scatter Plot: piyoosh 8 Basic Patterns of Scatter Plot Both Move Together Move In Opposite Way No Relationship
Positive Correlation: piyoosh 9 Positive Correlation Both Variables Increase Simultaneously or Decrease Simultaneously. Examples: Your Income and Jeweler's Bills Exercise and Appetite Rainfall and Absenteeism Discount and Sales
Negative Correlation: piyoosh 10 Negative Correlation As One Variables Increases the Other Variable Decreases. Examples: TV Viewing and Book Reading Age and Sleep Price and Demand Machine Downtime and Production
Correlation Coefficient: piyoosh 11 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
Logically Related Variables: piyoosh 12 Logically Related Variables Technical: 1. 2. 3. Marketing: 1. 2. 3. Corporate: 1. 2. 3.
Features of Correlation Coefficient: piyoosh 13 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
Computing Correlation: piyoosh 14 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.
Correlation Coefficient For Cardinal Variables: piyoosh 15 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.
Annual Production of 7 Plants: piyoosh 16 Annual Production of 7 Plants Plant 2004 (X) 2005 (Y) XY A 1 4 4 B 3 7 21 C 5 10 50 D 7 13 91 E 9 16 144 F 11 19 209 G 13 22 286 Total 49 91 805 Arith Mean 7 13 Std Deviation 4 6
Pearson’s Correlation Coefficient of Plant Production: piyoosh 17 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
One More Example: piyoosh 18 One More Example Empl. No. Yrs in Co. Salary (‘000) Product 1 2 25 50 2 3 30 90 3 5 37 185 4 7 38 266 5 8 40 320 Total 25 170 911 Arith Mean 5 34 Std. Dev. 2.3 5.6
Pearson’s Correlation Coefficient Between Yrs in Co & Salary: piyoosh 19 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
One More for Practice: piyoosh 20 One More for Practice Month Discount% Sales Product Nov 2 25 50 Dec 5 38 190 Jan 3 37 111 Feb 7 30 210 March 8 40 320 Total 25 170 881 Arith Mean 5 34 Std. Dev. 2.3 5.6
Pearson’s Correlation Coefficient Between Discount & Sales: piyoosh 21 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.
One More for Practice: piyoosh 22 One More for Practice Month M/cDowntime Production Product Nov 8 25 200 Dec 5 30 150 Jan 7 37 259 Feb 3 38 114 March 2 40 80 Total 25 170 803 Mean 5 34 S. D. 2.3 5.6
Pearson’s Correlation Coefficient Between M/c Downtime & Production: piyoosh 23 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
Correlation Coefficient For Ordinal Variables: piyoosh 24 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
Rank Correlation Coefficient Between Age & Performance: piyoosh 25 Rank Correlation Coefficient Between Age & Performance Age Rank Performance Rank Difference Square 1 4 3 9 2 2 0 0 3 1 2 4 4 5 1 1 5 3 2 4 Total 18
Rank Correlation Coefficient Between Age & Performance: piyoosh 26 Rank Correlation Coefficient Between Age & Performance Formula: 6 x 18 108 1 - ------------------- = 1 - ------- = 1 - 0.9 = 0.1 5 x (25 -1) 120 Interpretation: Age Has Very Little To Do With Performance
Frequent Blunders: piyoosh 27 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.
Tips to Busy Executives: piyoosh 28 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.
Rank Correlation Coefficient Between M/c Downtime & Production: piyoosh 29 Rank Correlation Coefficient Between M/c Downtime & Production M/c Down Rank Prod Rank Difference Square 5 1 4 16 3 2 1 1 4 3 1 1 2 4 2 4 1 5 4 16 Total 38
Rank Correlation Coefficient Between M/c Downtime & Production: piyoosh 30 Rank Correlation Coefficient Between M/c Downtime & Production Formula: 6 x 38 228 1 - ------------------- = 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
How Will You Proceed To Work Out Correlation In Following Pairs: piyoosh 31 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 ..