12 correlation analysis

Quantitative Methods

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Quantitative Methods

Models for Data Analysis & Interpretation: Correlation Analysis

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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?

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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.

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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

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Scatter Plot Horizontal Axis: Reasoning Scores

Vertical Axis: Creativity Scores

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Basic Patterns of Scatter Plot

Both Move Together Move In Opposite Way No Relationship

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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

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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

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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

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Logically Related Variables

• Technical: 1.2.3.

• Marketing: 1.2.3.

• Corporate:1.2.3.

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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

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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.

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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.

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Annual Production of 7 Plants

Plant 2011 (X) 2012 (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

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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

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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

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Between Yrs in Co & Salary• Formula: Ratio of (Mean of Products of


(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

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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

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Between Discount & Sales• Formula: Ratio of (Mean of Products of


(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.

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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

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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

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Correlation Coefficient For Ordinal Variables

• Actual Measurements on Both Variables Not Available

• Data Are In the Form of Ranks6 x Sum Square of Rank

Diff • Formula: 1 - ---------------------------------------

n x {(Square of n) -1}where n denotes Number of Observations

• Name: Rank Correlation Coefficient

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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

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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

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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.

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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.

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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

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Rank Correlation Coefficient Between M/c Downtime & Production

• Formula: 6 x 38 228

1 - ------------------- = 1 - ------- = 1 - 1.9 = -0.95 x (25 -1) 120

• Interpretation: Strong Negative Correlation between M/c Downtime & Prod

• Recall: Pearson’s Corr. Coeff. was -0.73

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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 ..

12 correlation analysis

Technology