Corelation Analysis

7/28/2019 Corelation Analysis

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Quantitative Methods



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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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.



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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 13.904 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


• 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 ComputingCorrelation Coefficient between TwoCardinal Variables is Different from the

One for Two Ordinal Variables• Statutory Warning: Using One Formula

for the Other is Seriously Injurious toCorporate Health.

• So, First Identify the Type of the Variables At Hand: Cardinal or Ordinal.



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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 2004 (X) 2005 (Y) XY

A 1 4 4

B 3 7 21

C 5 10 50D 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 Values – Product of the Two Means) toProduct 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 StronglyCorrelated 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




(881 / 5) – (5 x 34) 176.2 - 170= ----------------------- = ------------- = 0.48

2.3 x 5.6 12.9

• Interpretation: Sales Do Improve WithDiscounts, 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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Between M/c Downtime & Production




(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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For Ordinal Variables

• Actual Measurements on Both VariablesNot 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



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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 15 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 = 0.15 x (25 -1) 120

• Interpretation: Age Has Very Little To DoWith Performance



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Frequent Blunders

• People Treat All Variables As Cardinal.

• They Use Pearson’s Formula on OrdinalVariables 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 theOther Ordinal, Convert Cardinal ValuesInto Ordinal Ranks, and Then Compute

Rank Correlation Coefficient.• To Get a Quick Measure of the Extent of

Relationship Between Two CardinalVariables, Convert Both Sets of Data IntoOrdinal Ranks, and Compute RankCorrelation 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 41 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 = -0.9

5 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 ..

Corelation Analysis

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